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+String Field Theory – A Modern Introduction
+Harold Erbin1*
+*Center for Theoretical Physics
+Massachusetts Institute of Technology, Cambridge, MA 02139, Usa
+*Cea, List, Gif-sur-Yvette, F-91191, France
+7th January 2023
+1erbin@mit.edu
+arXiv:2301.01686v1  [hep-th]  4 Jan 2023
+
+Abstract
+This book provides an introduction to string field theory (SFT). String theory is usually
+formulated in the worldsheet formalism, which describes a single string (first-quantization).
+While this approach is intuitive and could be pushed far due to the exceptional properties of
+two-dimensional theories, it becomes cumbersome for some questions or even fails at a more
+fundamental level. These motivations have led to the development of SFT, a description of
+string theory using the field theory formalism (second-quantization). As a field theory, SFT
+provides a rigorous and constructive formulation of string theory.
+The main objective is to construct the closed bosonic SFT and to explain how to assess
+the consistency of string theory with it. The accent is put on providing the reader with
+the foundations, conceptual understanding and intuition of what SFT is. After reading this
+book, they should be able to study the applications from the literature.
+The book is organized in two parts. The first part reviews the topics of the worldsheet
+theory that are necessary to build SFT (worldsheet path integral, CFT and BRST quant-
+ization). The second part starts by introducing general concepts of SFT from the BRST
+quantization. Then, it introduces off-shell string amplitudes before providing a Feynman
+diagrams interpretation from which the building blocks of SFT are extracted. After con-
+structing the closed SFT, it is used to outline the proofs of several important consistency
+properties, such as background independence, unitarity and crossing symmetry. Finally, the
+generalization to the superstring is also discussed.
+This book grew up from lecture notes for a course given at the Ludwig-Maximilians-
+Universität LMU (winter semesters 2017–2018 and 2018–2019). The current document is
+the draft of the manuscript published by Springer.
+
+Preface
+This book grew up from lectures delivered within the Elite Master Program “Theoretical
+and Mathematical Physics” from the Ludwig-Maximilians-Universität during the winter
+semesters 2017–2018 and 2018–2019.
+The main focus of this book is the closed bosonic string field theory (SFT). While there
+are many resources available for the open bosonic SFT, a single review [71] has been written
+since the final construction of the bosonic closed SFT by Zwiebach [262]. For this reason,
+it makes sense to provide a modern and extensive study. Moreover, the usual approach to
+open SFT focuses on the cubic theory, which is so special that it is difficult to generalize the
+techniques to other SFTs. Finally, closed strings are arguably more fundamental than open
+strings because they are always present since they describe gravity, which further motivates
+my choice. However, the reader should not take this focus as denying the major achievements
+and the beauty of the open SFT; reading this book should provide most of the tools needed
+to feel comfortable also with this theory.
+While part of the original goal of SFT is to provide a non-perturbative definition of string
+theory and to address important questions such as classifying consistent string backgrounds
+or understanding dualities, no progress on this front has been achieved so far. Hence, there
+is still much to understand and the recent surge of developments provide a new chance
+to deepen our understanding of closed SFT. For example, several consistency properties of
+string theory have been proven rigorously using SFT. Moreover, the recent construction of
+the open-closed superstring field theory [165] together with earlier works [42, 218, 262, 264]
+show that all types of string theories can be recast as a SFT. This is why, I believe, it is a
+good time to provide a complete book on SFT.
+The goal of this book is to offer a self-contained description of SFT and all the tools
+necessary to build it. The emphasis is on describing the concepts behind SFT and to make
+the reader build intuitions on what it means.
+For this reason, there are relatively few
+applications.
+The reader is assumed to have some knowledge of QFT, and a basic knowledge of CFT
+and string theory (classical string, Nambu–Goto action, light-cone and old-covariant quant-
+izations).
+Organization
+The text is organized on three levels: the main content (augmented with examples), compu-
+tations, and remarks. The latter two levels can be omitted in a first lecture. The examples,
+computations and remarks are clearly separated from the text (respectively, by a half-box
+on the left and bottom, by a vertical line on the left, and by italics) to help the navigation.
+Many computations have been set aside from the main text to avoid breaking the flow
+and to provide the reader with the opportunity to check by themself first. In some occasions,
+computations are postponed well below the corresponding formula to gather similar compu-
+tations or to avoid breaking an argument. While the derivations contain more details than
+2
+
+usual textbooks and may look pedantic to the expert, I think it is useful for students and
+newcomers to have complete references where to check each step. This is even more the case
+when there are many different conventions in the literature. The remarks are not directly
+relevant to the core of the text but they make connections with other parts or topics. The
+goal is to broaden the perspectives of the main text.
+General references can be found at the end of each chapter to avoid overloading the
+text. In-text references are reserved for specific points or explicit quotations (of a formula, a
+discussion, a proof, etc.). I did not try to be exhaustive in the citations and I have certainly
+missed important references: this should be imputed to my lack of familiarity with them
+and not to their value.
+This text is a preprint of the textbook [64] and is reproduced with permission of Springer.
+My plan is to frequently update the draft of this book with new content. The last version
+can be accessed on my professional webpage, currently located at:
+http://www.lpthe.jussieu.fr/~erbin/
+Acknowledgements
+I have started to learn string field theory at Hri by attending lectures from Ashoke Sen.
+Since then, I have benefited from collaboration and many insightful discussions with him.
+Following his lectures have been much helpful in building an intuition that cannot be found
+in papers or reviews on the topic. Through this book, I hope being able to make some of
+these insights more accessible.
+I am particularly grateful to Ivo Sachs who proposed me to teach this course and to
+Michael Haack for continuous support and help with the organization, and to both of them
+for many interesting discussions during the two years I have spent at Lmu. Moreover, I have
+been very lucky to be assigned an excellent tutor for this course, Christoph Chiaffrino. After
+providing him with the topic and few references, Christoph has prepared all the tutorials
+and the corrections autonomously. His help brought a lot to the course.
+I am particularly obliged to all the students who have taken this course at Lmu for
+many interesting discussions and comments: Enrico Andriolo, Hrólfur Ásmundsson, Daniel
+Bockisch, Fabrizio Cordonnier, Julian Freigang, Wilfried Kaase, Andriana Makridou, Pouria
+Mazloumi, Daniel Panea, Martin Rojo.
+I am also grateful to all the string theory community for many exchanges. For discussions
+related to the topics of this book, I would like to thank more particularly: Costas Bachas,
+Adel Bilal, Subhroneel Chakrabarti, Atish Dabholkar, Benoit Douçot, Ted Erler, Dileep
+Jatkar, Carlo Maccaferri, Juan Maldacena, Yuji Okawa, Sylvain Ribault, Raoul Santachiara,
+Martin Schnabl, Dimitri Skliros, Jakub Vošmera. I have received a lot of feedback during
+the different stages of writing this book, and I am obliged to all the colleagues who sent me
+feedback.
+I am thankful to my colleagues at Lmu for providing a warm and stimulating envir-
+onment, with special thanks to Livia Ferro for many discussions around coffee. Moreover,
+the encouragements and advice from Oleg Andreev and Erik Plauschinn have been strong
+incentives for publishing this book.
+The editorial process at Springer has been very smooth.
+I would also like to thank
+Christian Caron and Lisa Scalone for their help and efficiency during the publishing process.
+I am also indebted to Stefan Theisen for having supported the publication at Springer and
+for numerous comments and corrections on the draft.
+Most of this book was written at the Ludwig–Maximilians–Universität (Lmu, Munich,
+Germany) where I was supported by a Carl Friedrich von Siemens Research Fellowship of the
+Alexander von Humboldt Foundation. The final stage has been completed at the University
+3
+
+of Turin (Italy). My research is currently funded by the European Union’s Horizon 2020
+research and innovation program under the Marie Skłodowska-Curie grant agreement No
+891169.
+Finally, writing this book would have been more difficult without the continuous and
+loving support from Corinne.
+November 2020
+Harold Erbin
+4
+
+Contents
+Preface
+2
+1
+Introduction
+11
+1.1
+Strings, a distinguished theory
+. . . . . . . . . . . . . . . . . . . . . . . . . .
+11
+1.2
+String theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+14
+1.2.1
+Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+14
+1.2.2
+Classification of superstring theories . . . . . . . . . . . . . . . . . . .
+18
+1.2.3
+Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+20
+1.3
+String field theory
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+23
+1.3.1
+From the worldsheet to field theory . . . . . . . . . . . . . . . . . . . .
+23
+1.3.2
+String field action
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+25
+1.3.3
+Expression with spacetime fields
+. . . . . . . . . . . . . . . . . . . . .
+25
+1.3.4
+Applications
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+26
+1.4
+Suggested readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+27
+I
+Worldsheet theory
+28
+2
+Worldsheet path integral: vacuum amplitudes
+29
+2.1
+Worldsheet action and symmetries . . . . . . . . . . . . . . . . . . . . . . . .
+29
+2.2
+Path integral
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+34
+2.3
+Faddeev–Popov gauge fixing . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+36
+2.3.1
+Metrics on Riemann surfaces
+. . . . . . . . . . . . . . . . . . . . . . .
+37
+2.3.2
+Reparametrizations and analysis of P1 . . . . . . . . . . . . . . . . . .
+42
+2.3.3
+Weyl transformations and quantum anomalies . . . . . . . . . . . . . .
+47
+2.3.4
+Ambiguities, ultralocality and cosmological constant . . . . . . . . . .
+48
+2.3.5
+Gauge-fixed path integral . . . . . . . . . . . . . . . . . . . . . . . . .
+49
+2.4
+Ghost action
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+51
+2.4.1
+Actions and equations of motion . . . . . . . . . . . . . . . . . . . . .
+51
+2.4.2
+Weyl ghost
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+52
+2.4.3
+Zero-modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+54
+2.5
+Normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+55
+2.6
+Summary
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+56
+2.7
+Suggested readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+56
+3
+Worldsheet path integral: scattering amplitudes
+58
+3.1
+Scattering amplitudes on moduli space . . . . . . . . . . . . . . . . . . . . . .
+58
+3.1.1
+Vertex operators and path integral . . . . . . . . . . . . . . . . . . . .
+58
+3.1.2
+Gauge fixing: general case . . . . . . . . . . . . . . . . . . . . . . . . .
+61
+3.1.3
+Gauge fixing: 2-point amplitude
+. . . . . . . . . . . . . . . . . . . . .
+64
+5
+
+3.2
+BRST quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+67
+3.2.1
+BRST symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+68
+3.2.2
+BRST cohomology and physical states . . . . . . . . . . . . . . . . . .
+69
+3.3
+Summary
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+71
+3.4
+Suggested readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+71
+4
+Worldsheet path integral: complex coordinates
+72
+4.1
+Geometry of complex manifolds . . . . . . . . . . . . . . . . . . . . . . . . . .
+72
+4.2
+Complex representation of path integral . . . . . . . . . . . . . . . . . . . . .
+75
+4.3
+Summary
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+77
+4.4
+Suggested readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+77
+5
+Conformal symmetry in D dimensions
+78
+5.1
+CFT on a general manifold
+. . . . . . . . . . . . . . . . . . . . . . . . . . . .
+78
+5.2
+CFT on Minkowski space
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+79
+5.3
+Suggested readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+80
+6
+Conformal field theory on the plane
+81
+6.1
+The Riemann sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+81
+6.1.1
+Map to the complex plane . . . . . . . . . . . . . . . . . . . . . . . . .
+81
+6.1.2
+Relation to the cylinder – string theory
+. . . . . . . . . . . . . . . . .
+83
+6.2
+Classical CFTs
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+84
+6.2.1
+Witt conformal algebra
+. . . . . . . . . . . . . . . . . . . . . . . . . .
+85
+6.2.2
+PSL(2, C) conformal group
+. . . . . . . . . . . . . . . . . . . . . . . .
+86
+6.2.3
+Definition of a CFT
+. . . . . . . . . . . . . . . . . . . . . . . . . . . .
+88
+6.3
+Quantum CFTs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+89
+6.3.1
+Virasoro algebra
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+90
+6.3.2
+Correlation functions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+90
+6.4
+Operator formalism and radial quantization . . . . . . . . . . . . . . . . . . .
+92
+6.4.1
+Radial ordering and commutators
+. . . . . . . . . . . . . . . . . . . .
+92
+6.4.2
+Operator product expansions . . . . . . . . . . . . . . . . . . . . . . .
+94
+6.4.3
+Hermitian and BPZ conjugation
+. . . . . . . . . . . . . . . . . . . . .
+96
+6.4.4
+Mode expansion
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+97
+6.4.5
+Hilbert space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
+6.4.6
+CFT on the cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
+6.5
+Suggested readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
+7
+CFT systems
+107
+7.1
+Free scalar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
+7.1.1
+Covariant action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
+7.1.2
+Action on the complex plane
+. . . . . . . . . . . . . . . . . . . . . . . 109
+7.1.3
+OPE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
+7.1.4
+Mode expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
+7.1.5
+Commutators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
+7.1.6
+Hilbert space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
+7.1.7
+Euclidean and BPZ conjugates . . . . . . . . . . . . . . . . . . . . . . 118
+7.2
+First-order bc ghost system
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
+7.2.1
+Covariant action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
+7.2.2
+Action on the complex plane
+. . . . . . . . . . . . . . . . . . . . . . . 119
+7.2.3
+OPE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
+7.2.4
+Mode expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
+7.2.5
+Commutators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
+6
+
+7.2.6
+Hilbert space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
+7.2.7
+Euclidean and BPZ conjugates . . . . . . . . . . . . . . . . . . . . . . 132
+7.2.8
+Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
+7.3
+Suggested readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
+8
+BRST quantization
+134
+8.1
+BRST for reparametrization invariance . . . . . . . . . . . . . . . . . . . . . . 134
+8.2
+BRST in the CFT formalism
+. . . . . . . . . . . . . . . . . . . . . . . . . . . 135
+8.2.1
+OPE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
+8.2.2
+Mode expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
+8.2.3
+Commutators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
+8.3
+BRST cohomology: two flat directions . . . . . . . . . . . . . . . . . . . . . . 138
+8.3.1
+Conditions on the states . . . . . . . . . . . . . . . . . . . . . . . . . . 139
+8.3.2
+Relative cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
+8.3.3
+Absolute cohomology, states and no-ghost theorem . . . . . . . . . . . 146
+8.3.4
+Cohomology for holomorphic and anti-holomorphic sectors . . . . . . . 147
+8.4
+Summary
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
+8.5
+Suggested readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
+II
+String field theory
+149
+9
+String field
+150
+9.1
+Field functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
+9.2
+Field expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
+9.3
+Summary
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
+9.4
+Suggested readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
+10 Free BRST string field theory
+154
+10.1 Classical action for the open string . . . . . . . . . . . . . . . . . . . . . . . . 154
+10.1.1 Warm-up: point-particle . . . . . . . . . . . . . . . . . . . . . . . . . . 154
+10.1.2 Open string action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
+10.1.3 Gauge invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
+10.1.4 Siegel gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
+10.2 Open string field expansion, parity and ghost number
+. . . . . . . . . . . . . 160
+10.3 Path integral quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
+10.3.1 Tentative Faddeev–Popov gauge fixing . . . . . . . . . . . . . . . . . . 162
+10.3.2 Tower of ghosts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
+10.4 Spacetime action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
+10.5 Closed string
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
+10.6 Summary
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
+10.7 Suggested readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
+11 Introduction to off-shell string theory
+171
+11.1 Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
+11.1.1 3-point function
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
+11.1.2 4-point function
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
+11.2 Off-shell states
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
+11.3 Off-shell amplitudes
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
+11.3.1 Amplitudes from the marked moduli space
+. . . . . . . . . . . . . . . 178
+11.3.2 Local coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
+11.4 Suggested readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
+7
+
+12 Geometry of moduli spaces and Riemann surfaces
+182
+12.1 Parametrization of Pg,n
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
+12.2 Tangent space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
+12.3 Plumbing fixture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
+12.3.1 Separating case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
+12.3.2 Non-separating case
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
+12.3.3 Decomposition of moduli spaces and degeneration limit
+. . . . . . . . 191
+12.3.4 Stubs
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
+12.4 Summary
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
+12.5 Suggested readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
+13 Off-shell amplitudes
+197
+13.1 Cotangent spaces and amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . 197
+13.1.1 Construction of forms . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
+13.1.2 Amplitudes and surface states . . . . . . . . . . . . . . . . . . . . . . . 199
+13.2 Properties of forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
+13.2.1 Vanishing of forms with trivial vectors . . . . . . . . . . . . . . . . . . 202
+13.2.2 BRST identity
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
+13.3 Properties of amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
+13.3.1 Restriction to ˆPg,n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
+13.3.2 Consequences of the BRST identity
+. . . . . . . . . . . . . . . . . . . 206
+13.4 Suggested readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
+14 Amplitude factorization and Feynman diagrams
+208
+14.1 Amplitude factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
+14.1.1 Separating case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
+14.1.2 Non-separating case
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
+14.2 Feynman diagrams and Feynman rules . . . . . . . . . . . . . . . . . . . . . . 213
+14.2.1 Feynman graphs
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
+14.2.2 Propagator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
+14.2.3 Fundamental vertices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
+14.2.4 Stubs
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
+14.2.5 1PI vertices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
+14.3 Properties of fundamental vertices
+. . . . . . . . . . . . . . . . . . . . . . . . 223
+14.3.1 String product
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
+14.3.2 Feynman graph interpretation . . . . . . . . . . . . . . . . . . . . . . . 224
+14.4 Suggested readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
+15 Closed string field theory
+226
+15.1 Closed string field expansion
+. . . . . . . . . . . . . . . . . . . . . . . . . . . 226
+15.2 Gauge fixed theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
+15.2.1 Kinetic term and propagator
+. . . . . . . . . . . . . . . . . . . . . . . 227
+15.2.2 Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
+15.2.3 Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
+15.3 Classical gauge invariant theory . . . . . . . . . . . . . . . . . . . . . . . . . . 232
+15.4 BV theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
+15.5 1PI theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
+15.6 Suggested readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
+8
+
+16 Background independence
+238
+16.1 The concept of background independence
+. . . . . . . . . . . . . . . . . . . . 238
+16.2 Problem setup
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
+16.3 Deformation of the CFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
+16.4 Expansion of the action
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
+16.5 Relating the equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . 242
+16.6 Idea of the proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
+16.7 Suggested readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
+17 Superstring
+245
+17.1 Worldsheet superstring theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
+17.1.1 Heterotic worldsheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
+17.1.2 Hilbert spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
+17.2 Off-shell superstring amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . 249
+17.2.1 Amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
+17.2.2 Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
+17.2.3 Spurious poles
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
+17.3 Superstring field theory
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
+17.3.1 String field and propagator . . . . . . . . . . . . . . . . . . . . . . . . 256
+17.3.2 Constraint approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
+17.3.3 Auxiliary field approach . . . . . . . . . . . . . . . . . . . . . . . . . . 257
+17.3.4 Large Hilbert space
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
+17.4 Suggested readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
+18 Momentum-space SFT
+260
+18.1 General form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260
+18.2 Generalized Wick rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
+18.3 Suggested readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
+A Conventions
+266
+A.1 Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
+A.2 Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268
+A.3 QFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268
+A.4 Curved space and gravity
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
+A.5 List of symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270
+B Summary of important formulas
+273
+B.1
+Complex analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
+B.2
+QFT, curved spaces and gravity . . . . . . . . . . . . . . . . . . . . . . . . . . 273
+B.2.1
+Two dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
+B.3
+Conformal field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
+B.3.1
+Complex plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
+B.3.2
+General properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
+B.3.3
+Hermitian and BPZ conjugations . . . . . . . . . . . . . . . . . . . . . 276
+B.3.4
+Scalar field
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276
+B.3.5
+Reparametrization ghosts . . . . . . . . . . . . . . . . . . . . . . . . . 277
+B.4
+Bosonic string . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280
+9
+
+C Quantum field theory
+281
+C.1 Path integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
+C.1.1
+Integration measure
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
+C.1.2
+Field redefinitions
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
+C.1.3
+Zero-modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
+C.2 BRST quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
+C.3 BV formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
+C.3.1
+Properties of gauge algebra . . . . . . . . . . . . . . . . . . . . . . . . 289
+C.3.2
+Classical BV
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
+C.3.3
+Quantum BV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
+C.4 Suggested readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294
+Bibliography
+295
+Index
+310
+10
+
+Chapter 1
+Introduction
+Abstract
+In this chapter, we introduce the main motivations for studying string theory,
+and why it is important to design a string field theory. After describing the central features
+of string theory, we describe the most important concepts of the worldsheet formulation.
+Then, we explain the reasons leading to string field theory (SFT) and outline the ideas
+which will be discussed in the rest of the book.
+1.1
+Strings, a distinguished theory
+The first and simplest reason for considering theories of fundamental p-branes (fundamental
+objects extended in p spatial dimensions) can be summarized by the following question:
+“Why would Nature just make use of point-particles?” There is no a priori reason forbidding
+the existence of fundamental extended objects and, according to Gell-Mann’s totalitarian
+principle, “Everything not forbidden is compulsory.” If a consistent theory cannot be built
+(after a reasonable amount of effort) or if it contradicts current theories (in their domains
+of validity) and experiments, then one can support the claim that only point-particles exist.
+On the other side, if such a theory can be built, it is of primary interest to understand it
+deeper and to see if it can solve the current problems in high-energy theoretical physics.
+The simplest case after the point particle is the string, so it makes sense to start with
+it. It happens that a consistent theory of strings can be constructed, and that the latter
+(in its supersymmetric version) contains all the necessary ingredients for a fully consistent
+high-energy model:1
+• quantum gravity (quantization of general relativity plus higher-derivative corrections);
+• grand unification (of matter, interactions and gravity);
+• no divergences, UV finiteness (finite and renormalizable theory);
+• fixed number of dimensions (26 = 25 + 1 for the bosonic string, 10 = 9 + 1 for the
+supersymmetric version);
+• existence of all possible branes;
+• no dimensionless parameters and one dimensionful parameter (the string length ℓs).
+It can be expected that a theory of fundamental strings (1-branes) occupies a distinguished
+place among fundamental p-branes for the following reasons.
+1There are also indications that a theory of membranes (2-branes) in 10+1 dimensions, called M-theory,
+should exist. No direct and satisfactory description of the latter has been found and we will thus focus on
+string theory in this book.
+11
+
+Figure 1.1: Locality of a particle interaction: two different observers always agree on the
+interaction point and which parts of the worldline are 1- and 2-particle states.
+Interaction non-locality
+In a QFT of point particles, UV divergences arise because
+interactions (defined as the place where the number and/or nature of the objects change)
+are arbitrarily localized at a spacetime point. In Feynman graphs, such divergences can be
+seen when the momentum of a loop becomes infinite (two vertices collide): this happens
+when trying to concentrate an infinite amount of energy at a single point. However, these
+divergences are expected to be reduced or absent in a field theory of extended objects:
+whereas the interaction between particles is perfectly local in spacetime and agreed upon by
+all observers (Figure 1.1), the spatial extension of branes makes the interactions non-local.
+This means that two different observers will neither agree on the place of the interactions
+(Figure 1.2), nor on the part of the diagram which describes one or two branes.
+The string lies at the boundary between too much local and too much non-local: in any
+given frame, the interaction is local in space, but not in spacetime. The reason is that a
+string is one-dimensional and splits or joins along a point. For p > 1, the brane needs to
+break/join along an extended spatial section, which looks non-local.
+Another consequence of the non-locality is a drastic reduction of the possible interactions.
+If an interaction is Lorentz invariant, Lorentz covariant objects can be attached at the vertex
+(such as momentum or gamma matrices): this gives Lorentz invariants after contracting with
+indices carried by the field. But, this is impossible if the interaction itself is non-local (and
+thus not invariant): inserting a covariant object would break Lorentz invariance.
+Brane degrees of freedom
+The higher the number of spatial dimensions of a p-brane, the
+more possibilities it has to fluctuate. As a consequence, it is expected that new divergences
+appear as p increases due to the proliferations of the brane degrees of freedom. From the
+worldvolume perspective, this is understood from the fact that the worldvolume theory
+describes a field theory in (p + 1) dimensions, and UV divergences become worse as the
+number of dimensions increase.
+The limiting case happens for the string (p = 1) since
+two-dimensional field theories are well-behaved in this respect (for example, any monomial
+interaction for a scalar field is power-counting renormalizable). This can be explained by the
+low-dimensionality of the momentum integration and by the enhancement of symmetries in
+two dimensions. Hence, strings should display nice properties and are thus of special interest.
+Worldvolume theory
+The point-particle (0-brane) and the string (1-brane) are also re-
+markable in another aspect: it is possible to construct a simple worldvolume field theory
+(and the associated functional integral) in terms of a worldvolume metric. All components
+of the latter are fixed by gauge symmetries (diffeomorphisms for the particle, diffeomorph-
+isms and Weyl invariance for the string). This ensures the reparametrization invariance of
+12
+
+(a) Observers at rest and boosted.
+(b) Observers close to the speed
+of light moving in opposite dir-
+ections.
+The interactions are
+widely separated in each case.
+Figure 1.2: Non-locality of string interaction: two different observers see the interaction
+happening at different places (denoted by the filled and empty circles) and they don’t agree
+on which parts of the worldsheet are 1- and 2-string states (the litigation is denoted by the
+grey zone).
+the worldvolume without having to use a complicated action. Oppositely, the worldvolume
+metric cannot be completely gauge fixed for p > 1.
+Summary
+As a conclusion, strings achieve an optimal balance between spacetime and
+worldsheet divergences, as well as having a simple description with reparametrization in-
+variance.
+Since the construction of a field theory is difficult, it is natural to start with a worldsheet
+theory and to study it in the first-quantization formalism, which will provide a guideline for
+writing the field theory. In particular, this allows to access the physical states in a simple
+way and to find other general properties of the theory. When it comes to the interactions
+and scattering amplitudes, this approach may be hopeless in general since the topology of
+the worldvolume needs to be specified by hand (describing the interaction process). In this
+respect, the case of the string is again exceptional: because Riemann surfaces have been
+classified and are well-understood, the arbitrariness is minimal. Combined with the tools of
+conformal field theory, many computations can be performed. Moreover, since the modes
+of vibrations of the strings provide all the necessary ingredients to describe the Standard
+model, it is sufficient to consider only one string field (for one type of strings), instead
+of the plethora found in point-particle field theory (one field for each particle). Similarly,
+non-perturbative information (such as branes and dualities) could be found only due to the
+specific properties of strings.
+Coming back to the question which opened this section, higher-dimensional branes of all
+the allowed dimensions naturally appear in string theory as bound states. Hence, even if
+the worldvolume formulation of branes with p > 1 looks pathological2, string theory hints
+towards another definition of these objects.
+2Entering in the details would take us too far away from the main topic of this book.
+Some of the
+problems found when dealing with (p > 2)-branes are: how to define a Wick rotation for 3-manifolds, the
+presence of Lorentz anomalies in target spacetime, problems with the spectrum, lack of renormalizability,
+impossibility to gauge-fix the worldvolume metric [5–11, 41, 44, 45, 62, 127, 152, 156–158, 181, 193].
+13
+
+1.2
+String theory
+1.2.1
+Properties
+The goal of this section is to give a general idea of string theory by introducing some concepts
+and terminology. The reader not familiar with the points described in this section is advised
+to follow in parallel some standard worldsheet string theory textbooks.
+Worldsheet CFT
+A string is characterized by its worldsheet field theory (Chapter 2).3
+The worldsheet is
+parametrized by coordinates σa = (τ, σ). The simplest description is obtained by endowing
+the worldsheet with a metric gab(σa) (a = 0, 1) and by adding a set of D scalar fields
+Xµ(σa) living on the worldsheet (µ = 0, . . . , D − 1). The latter represents the position
+of the string in the D-dimensional spacetime. From the classical equations of motion, the
+metric gab is proportional to the metric induced on the worldsheet from its embedding in
+spacetime.
+More generally, one ensures that the worldsheet metric is non-dynamical by
+imposing that the action is invariant under (worldsheet) diffeomorphisms and under Weyl
+transformations (local rescalings of the metric). The consistency of these conditions at the
+quantum level imposes that D = 26, and this number is called the critical dimension. Gauge
+fixing the symmetries, and thus the metric, leads to the conformal invariance of the resulting
+worldsheet field theory: a conformal field theory (CFT) is a field theory (possibly on a curved
+background) in which only angles and not distances can be measured (Chapters 5 to 7).
+This simplifies greatly the analysis since the two-dimensional conformal algebra (called the
+Virasoro algebra) is infinite-dimensional.
+CFTs more general than D free scalar fields can be considered: fields taking non-compact
+values are interpreted as non-compact dimensions while compact or Grassmann-odd fields
+are interpreted as compact dimensions or internal structure, like the spin.
+While the light-cone quantization allows to find quickly the states of the theory, the
+simplest covariant method is the BRST quantization (Chapter 8). It introduces ghosts (and
+superghosts) associated to the gauge fixing of diffeomorphisms (and local supersymmetry).
+These (super)ghosts form a CFT which is universal (independent of the matter CFT).
+The trajectory of the string is denoted by xc(τ, σ). It begins and ends respectively at the
+geometric shapes parametrized by xc(τi, σ) = xi(σ) and by xc(τf, σ) = xf(σ). Note that the
+coordinate system on the worldsheet itself is arbitrary. The spatial section of a string can be
+topologically closed (circle) or open (line) (Figure 1.3), leading to cylindrical or rectangular
+worldsheets as illustrated in Figures 1.4 and 1.5. To each topology is associated different
+boundary conditions and types of strings:
+• closed: periodic and anti-periodic boundary conditions;
+• open: Dirichlet and Neumann boundary conditions.
+While a closed string theory is consistent by itself, an open string theory is not and requires
+closed strings.
+Spectrum
+In order to gain some intuition for the states described by a closed string, one can write the
+Fourier expansion of the fields Xµ (in the gauge gab = ηab and after imposing the equations
+3We focus mainly on the bosonic string theory, leaving aside the superstring, except when differences
+are important.
+14
+
+(a) Open string
+(b) Closed string
+Figure 1.3: Open and closed strings.
+−−−−−−−−→
+Figure 1.4: Trajectory xµ
+c (τ, σ) of a closed string in spacetime (worldsheet). It begins and
+ends at the circles parametrized by xi(σ) and xf(σ).
+The worldsheet is topologically a
+cylinder and is parametrized by (τ, σ) ∈ [τi, τf] × [0, 2π).
+−−−−−−−−→
+Figure 1.5: Trajectory xµ
+c (τ, σ) of an open string in spacetime (worldsheet).
+It begins
+and ends at the lines parametrized by xi(σ) and xf(σ). The worldsheet is topologically a
+rectangle and is parametrized by (τ, σ) ∈ [τi, τf] × [0, ℓ].
+15
+
+of motion)
+Xµ(τ, σ) ∼ xµ + pµτ +
+i
+√
+2
+�
+n∈Z∗
+1
+n
+�
+αµ
+ne−in(τ−σ) + ¯αµ
+ne−in(τ+σ)�
+,
+(1.1)
+where xµ is the centre-of-mass position of the string and pµ its momentum.4
+Canonical
+quantization leads to the usual commutator:
+[xµ, pν] = iηµν .
+(1.2)
+With respect to a point-particle for which only the first two terms are present, there are an
+infinite number of oscillators αµ
+n and ¯αµ
+n which satisfy canonical commutation relations for
+creation n < 0 and annihilation operators n > 0
+[αµ
+m, αν
+n] = m ηµνδm+n,0 .
+(1.3)
+The non-zero modes are the Fourier modes of the excitations of the embedded string. The
+case of the open string is simply obtained by setting ¯αn = αn and p → 2p. The Hamiltonian
+for the closed and open strings read respectively
+Hclosed = −m2
+2 + N + ¯N − 2 ,
+(1.4a)
+Hopen = −m2 + N − 1
+(1.4b)
+where m2 = −pµpµ is the mass of the state (in Planck units), N and ¯N (level operators)
+count the numbers Nn and ¯Nn of oscillators αn and ¯αn weighted by their mode index n:
+N =
+�
+n∈N
+nNn ,
+Nn = 1
+n α−n · αn ,
+¯N =
+�
+n∈N
+n ¯Nn ,
+¯Nn = 1
+n ¯α−n · ¯αn .
+(1.5)
+With these elements, the Hilbert space of the string theory can be constructed. Invariance
+under reparametrization leads to the on-shell condition, which says that the Hamiltonian
+vanishes:
+H |ψ⟩ = 0
+(1.6)
+for any physical state |ψ⟩. Another constraint for the closed string is the level-matching
+condition
+(N − ¯N) |ψ⟩ = 0 .
+(1.7)
+It can be understood as fixing an origin on the string.
+The ground state |k⟩ with momentum k is defined to be the eigenstate of the momentum
+operator which does not contain any oscillator excitation:
+pµ |k⟩ = kµ |k⟩ ,
+∀n > 0 :
+αµ
+n |k⟩ = 0 .
+(1.8)
+A general state can be built by applying successively creation operators
+|ψ⟩ =
+�
+n>0
+D−1
+�
+µ=0
+(αµ
+−n)Nn,µ |k⟩ ,
+(1.9)
+4In the introduction, we set α′ = 1.
+16
+
+where Nn,µ ∈ N counts excitation level of the oscillator αµ
+−n. In the rest of this section, we
+describe the first two levels of states.
+The ground state is a tachyon (faster-than-light particle) because the Hamiltonian con-
+straint shows that it has a negative mass (in the units where α′ = 1):
+closed :
+m2 = −4 ,
+open :
+m2 = −1 .
+(1.10)
+The first excited state of the open string is found by applying α−1 on the vacuum |k⟩:
+αµ
+−1 |k⟩ .
+(1.11)
+This state is massless:
+m2 = 0
+(1.12)
+and since it transforms as a Lorentz vector (spin 1), it is identified with a U(1) gauge boson.
+Writing a superposition of such states
+|A⟩ =
+�
+dDk Aµ(k) αµ
+−1 |k⟩ ,
+(1.13)
+the coefficient Aµ(k) of the Fourier expansion is interpreted as the spacetime field for the
+gauge boson. Reparametrization invariance is equivalent to the equation of motion
+k2Aµ = 0 .
+(1.14)
+One can prove that the field obeys the Lorentz gauge condition
+kµAµ = 0 ,
+(1.15)
+which results from gauge fixing the U(1) gauge invariance
+Aµ −→ Aµ + kµλ .
+(1.16)
+It can also be checked that the low-energy action reproduces the Maxwell action.
+The first level of the closed string is obtained by applying both α−1 and ¯α−1 (this is the
+only way to match N = ¯N at this level)
+αµ
+−1¯αν
+−1 |k⟩
+(1.17)
+and the corresponding states are massless
+m2 = 0 .
+(1.18)
+These states can be decomposed into irreducible representations of the Lorentz group
+�
+αµ
+−1¯αν
+−1 + αν
+−1¯αµ
+−1 − 1
+D ηµνα−1 · ¯α−1
+�
+|p⟩ ,
+�
+αµ
+−1¯αν
+−1 − αν
+−1¯αµ
+−1
+�
+|p⟩ ,
+1
+D ηµναµ
+−1¯αν
+−1 |p⟩
+(1.19)
+which are respectively associated to the spacetime fields Gµν (metric, spin 2), Bµν (Kalb–
+Ramond 2-form) and Φ (dilaton, spin 0). The appearance of a massless spin 2 particle (with
+low-energy action being the Einstein–Hilbert action) is a key result and originally raised
+interest for string theory.
+17
+
+Remark 1.1 (Reparametrization constraints) Reparametrization invariance leads to
+other constraints than H = 0. They imply in particular that the massless fields have the
+correct gauge invariance and hence the correct degrees of freedom.
+Note that, after taking into account these constraints, the remaining modes correspond
+to excitations of the string in the directions transverse to it.
+Hence, each vibrational mode of the string corresponds to a spacetime field for a point-
+particle (and linear superpositions of modes can describe several fields). This is how string
+theory achieves unification since a single type of string (of each topology) is sufficient for
+describing all the possible types of fields encountered in the standard model and in gravity.
+They correspond to the lowest excitation modes, the higher massive modes being too heavy
+to be observed at low energy.
+Bosonic string theory includes tachyons and is thus unstable. While the instability of
+the open string tachyon is well understood and indicates that open strings are unstable and
+condense to closed strings, the status of the closed string tachyon is more worrisome (literally
+interpreted, it indicates a decay of spacetime itself). In order to solve this problem, one can
+introduce supersymmetry: in this case, the spectrum does not include the tachyon because
+it cannot be paired with a supersymmetric partner.
+Moreover, as its name indicates, the bosonic string possesses only bosons in its spectrum
+(perturbatively), which is an important obstacle to reproduce the standard model.
+By
+introducing spacetime fermions, supersymmetry also solves this problem. The last direct
+advantage of the superstring is that it reduces the number of dimensions from 26 to 10,
+which makes the compactification easier.
+1.2.2
+Classification of superstring theories
+In this section, we describe the different superstring theories (Chapter 17).
+In order to
+proceed, we need to introduce some new elements.
+The worldsheet field theory of the closed string is made of two sectors, called the left- and
+right-moving sectors (the αn and ¯αn modes). While they are treated symmetrically in the
+simplest models, they are in fact independent (up to the zero-mode) and the corresponding
+CFT can be chosen to be distinct.
+The second ingredient already evoked earlier is supersymmetry. This symmetry associ-
+ates a fermion to each boson (and conversely) through the action of a supercharge Q
+|boson⟩ = Q |fermion⟩ .
+(1.20)
+More generally, one can consider N supercharges which build up a family of several bosonic
+and fermionic partners. Since each supercharge increases the spin by 1/2 (in D = 4), there
+is an upper limit for the number of supersymmetries – for interacting theories with a finite
+number of fields5 – in order to keep the spin of a family in the range where consistent actions
+exist:
+• Nmax = 4 without gravity (−1 ≤ spin ≤ 1);
+• Nmax = 8 with gravity (−2 ≤ spin ≤ 2).
+This counting serves as a basis to determine the maximal number of supersymmetries in
+other dimensions (by relating them through dimensional reductions).
+Let’s turn our attention to the case of the two-dimensional worldsheet theory.
+The
+number of supersymmetries of the closed left- and right-moving sectors can be chosen in-
+dependently, and the number of charges is written as (NL, NR) (the index is omitted when
+5These conditions exclude the cases of free theories and higher-spin theories.
+18
+
+statements are made at the level of the CFT). The critical dimension (absence of quantum
+anomaly for the Weyl invariance) depends on the number of supersymmetry
+D(N = 0) = 26,
+D(N = 1) = 10.
+(1.21)
+Type II superstrings have (NL, NR) = (1, 1) and come in two flavours called IIA and IIB
+according to the chiraly of the spacetime gravitini chiralities. A theory is called heterotic if
+NL > NR; we will mostly be interested in the case NL = 1 and NR = 0.6 In such theories,
+there cannot be open strings since both sectors must be equal in the latter. Since the critical
+dimensions of the two sectors do not match, one needs to get rid of the additional dimensions
+of the right-moving sector; this leads to the next topic – gauge groups.
+Gauge groups associated with spacetime gauge bosons appear in two different places. In
+heterotic models, the compactification of the unbalanced dimensions of the left sector leads
+to the appearance of a gauge symmetry. The possibilities are scarce due to consistency
+conditions which ensure a correct gluing with the right-sector. Another possibility is to
+add degrees of freedom – known as Chan–Paton indices – at the ends of open strings:
+one end transforms in the fundamental representation of a group G, while the other end
+transforms in the anti-fundamental. The modes of the open string then reside in the adjoint
+representation, and the massless spin-1 particles become the gauge bosons of the non-Abelian
+gauge symmetry.
+Finally, one can consider oriented or unoriented strings. An oriented string possesses an
+internal direction, i.e. there is a distinction between going from the left to the right (for an
+open string) or circling in clockwise or anti-clockwise direction (for a closed string). Such
+an orientation can be attributed globally to the spacetime history of all strings (interacting
+or not). The unoriented string is obtained by quotienting the theory by the Z2 worldsheet
+parity symmetry which exchanges the left- and right-moving sectors. Applying this to the
+type IIB gives the type I theory.
+The tachyon-free superstring theories together with the bosonic string are summarized
+in Table 1.1.
+worldsheet
+susy
+D
+spacetime
+susy
+gauge group
+open string
+oriented
+tachyon
+bosonic
+(0, 0)
+26
+0
+any*
+yes
+yes / no
+yes
+type I
+(1, 1)
+10
+(1, 0)
+SO(32)
+yes
+no
+no
+type IIA
+(1, 1)
+10
+(1, 1)
+U(1)
+(yes)†
+yes
+no
+type IIB
+(1, 1)
+10
+(2, 0)
+none
+(yes)†
+yes
+no
+heterotic SO(32)
+(1, 0)
+10
+(1, 0)
+SO(32)
+no
+yes
+no
+heterotic E8
+(1, 0)
+10
+(1, 0)
+E8 × E8
+no
+yes
+no
+heterotic SO(16)
+(1, 0)
+10
+(0, 0)
+SO(16) × SO(16)
+no
+yes
+no
+* UV divergences beyond the tachyon (interpreted as closed string dilaton tadpoles) cancel only for the unoriented
+open plus closed strings with gauge group SO(213) = SO(8192).
+† The parenthesis indicates that type II theories don’t have open strings in the vacuum: they require a D-brane
+background. This is expected since there is no gauge multiplet in d = 10 (1, 1) or (2, 0) supergravities (the
+D-brane breaks half of the supersymmetry).
+Table 1.1: List of the consistent tachyon-free (super)string theories. The bosonic theory is
+added for comparison. There are additional heterotic theories without spacetime supersym-
+metry, but they contain a tachyon and are thus omitted.
+6The case NL < NR is identical up to exchange of the left- and right-moving sectors.
+19
+
+(a) Closed strings
+(b) Open strings
+Figure 1.6: Graphs corresponding to 1-loop 4-point scattering after a conformal mapping.
+1.2.3
+Interactions
+Worldsheet and Riemann surfaces
+After having described the spectrum and the general characteristics of string theory comes
+the question of interactions. The worldsheets obtained in this way are Riemann surfaces,
+i.e. 1-dimensional complex manifolds. They are classified by the numbers of handles (or
+holes) g (called the genus) and external tubes n. In the presence of open strings, surfaces
+have boundaries: in addition to the handles and tubes, they are classified by the numbers
+of disks b and of strips m.7 A particularly important number associated to each surface is
+the Euler characteristics
+χ = 2 − 2g − b ,
+(1.22)
+which is a topological invariant. It is remarkable that there is a single topology at every
+loop order when one considers only closed strings, and just a few more in the presence of
+open strings. The analysis is greatly simplified in contrast to QFT, for which the number
+of Feynman graphs increases very rapidly with the number of loops and external particles.
+Due to the topological equivalence between surfaces, a conformal map can be used in
+order to work with simpler surfaces. In particular, the external tubes and strips are collapsed
+to points called punctures (or marked points) on the corresponding surfaces or boundaries.
+A general amplitude then looks like a sphere from which holes and disks have been removed
+and to which marked points have been pierced (Figure 1.7).
+Amplitudes
+In order to compute an amplitude for the scattering of n strings (Chapters 3 and 4), one
+must sum over all the inequivalent worldsheets through a path integral weighted by the CFT
+action chosen to describe the theory.8 At fixed n, the sum runs over the genus g, such that
+each term is described by a Riemann surface Σg,n of genus g with n punctures.
+The interactions between strings follow from the graph topologies: since the latter are not
+encoded into the action, the dependence in the coupling constant must be added by hand.
+For closed strings, there is a unique cubic vertex with coupling gs. A direct inspection shows
+that the correct factor is gn−2+2g
+s
+:
+7We ignore unoriented strings in this discussion.
+They would lead to an additional object called a
+cross-cap, which is a place where the surface looses its orientation.
+8For simplicity we focus on closed string amplitudes in this section.
+20
+
+Figure 1.7: General Riemann surfaces with boundaries and punctures.
+• for n = 3 there is one factor gs, and every additional external string leads to the
+addition of one vertex with factor gs, since this process can be obtained from the n−1
+process by splitting one of the external string in two by inserting a vertex;
+• each loop comes with two vertices, so g-loops provide a factor g2g
+s .
+Remark 1.2 (Status of gs as a parameter) It was stated earlier that string theory has
+no dimensionless parameter, but gs looks to be one. In reality it is determined by the expect-
+ation value of the dilaton gs = e⟨Φ⟩. Hence the coupling constant is not a parameter defining
+the theory but is rather determined by the dynamics of the theory.
+Finally, the external states must be specified: this amounts to prescribe boundary condi-
+tions for the path integral or to insert the corresponding wave functions. Under the conformal
+mapping which brings the external legs to punctures located at zi, the states are mapped to
+local operators Vi(ki, zi) inserted at the points zi. The latter are built from the CFT fields
+and are called vertex operators: they are characterized by a momentum kµ which comes
+from the Fourier transformation of the Xµ fields representing the non-compact dimensions.
+These operators are inserted inside the path integral with integrals over the positions zi in
+order to describe all possible conformal mappings.
+Ultimately, the amplitude (amputated Green function) is computed as
+An(k1, . . . , kn) =
+�
+g≥0
+gn−2+2g
+s
+Ag,n
+(1.23)
+where
+Ag,n =
+�
+n
+�
+i=1
+d2zi
+�
+dgabdΨ e−Scft[gab,Ψ]
+n
+�
+i=1
+Vi(ki, zi)
+(1.24)
+is the g-loop n-point amplitude (for simplicity we omit the dependence on the states beyond
+the momentum). Ψ denotes collectively the CFT fields and gab is the metric on the surface.
+The integration over the metrics and over the puncture locations contain a huge redund-
+ancy due to the invariance under reparametrizations, which means that one integrates over
+many equivalent surfaces. To avoid this, Faddeev–Popov ghosts must be introduced and the
+integral is restricted to only finitely many (real) parameters tλ. They form the moduli space
+Mg,n of the Riemann surfaces Σg,n whose dimension is
+dimR Mg,n = 6g − 6 + 2n.
+(1.25)
+21
+
+The computation of the amplitude Ag,n can be summarized as:
+Ag,n =
+�
+Mg,n
+6g−6+2n
+�
+λ=1
+dtλ F(t).
+(1.26)
+The function F(t) is a correlation function in the worldsheet CFT defined on the Riemann
+surface Σg,n
+F(t) =
+� n
+�
+i=1
+Vi × ghosts × super-ghosts
+�
+Σg,n
+.
+(1.27)
+Note that the (super)ghost part is independent of the choice of the matter CFT.
+Divergences and Feynman graphs
+Formally the moduli parameters are equivalent to Schwinger (proper-time) parameters si in
+usual QFT: these are introduced in order to rewrite propagators as
+1
+k2 + m2 =
+� ∞
+0
+ds e−s(k2+m2),
+(1.28)
+such that the integration over the momentum k becomes a Gaussian times a polynomial.
+This form of the propagator is useful to display the three types of divergences which can be
+encountered:
+1. IR: regions si → ∞ (for k2 + m2 ≤ 0). These divergences are artificial for k2 + m2 < 0
+and means that the parametrization is not appropriate. Divergences for k2 + m2 = 0
+are genuine and translates the fact that quantum effects shift the vacuum and the
+masses.
+Taking these effects into account necessitates a field theory framework in
+which renormalization can be used.
+2. UV: regions si → 0 (after integrating over k). Such divergences are absent in string
+theories because these regions are excluded from the moduli space Mg,n (see Figure 1.8
+for the example of the torus).9
+3. Spurious: regions with finite si where the amplitude diverges. This happens typically
+only in the presence of super-ghosts and it translates a breakdown of the gauge fixing
+condition.10 Since these spurious singularities of the amplitudes are not physical, one
+needs to ensure that they can be removed, which is indeed possible to achieve.
+Hence, only IR divergences present a real challenge to string theory. Dealing with these
+divergences requires renormalizing the amplitudes, but this is not possible in the standard
+formulation of worldsheet string theory since the states are on-shell.11
+9There is a caveat to this statement: UV divergences reappear in string field theory in Lorentzian
+signature due to the way the theory is formulated.
+The solution requires a generalization of the Wick
+rotation.
+Moreover, this does not hold for open strings whose moduli spaces contains those regions: in this case,
+the divergences are reinterpreted in terms of closed strings propagating.
+10Such spurious singularities are also found in supergravity.
+11The on-shell condition is a consequence of the BRST and conformal invariance. While the first will be
+given up, the second will be maintained to facilitate the computations.
+22
+
+Figure 1.8: Moduli space of the torus: Re τ ∈ [−1/2, 1/2], Im τ > 0 and |τ| > 1.
+1.3
+String field theory
+1.3.1
+From the worldsheet to field theory
+The first step is to solve the IR divergences problem is to go off-shell (Chapters 11 and 13).
+This is made possible by introducing local coordinates around the punctures of the Riemann
+surface (Chapter 12).
+The IR divergences originate from Riemann surfaces close to degeneration, that is, sur-
+faces with long tubes. The latter can be of separating and non-separating types, depending
+on whether the Riemann surface splits in two pieces if the tube is cut (Figure 1.9). By
+exploring the form of the amplitudes in this limit (Chapter 14), the expression naturally
+separates into several pieces, to be interpreted as two amplitudes (of lower n and g) con-
+nected by a propagator. The latter can be reinterpreted as a standard (k2 + m2)−1 term,
+hence solving the divergence problem for k2 + m2 < 0. Taking this decomposition seriously
+leads to identify each contribution with a Feynman graph.
+Decomposing the amplitude recursively, the next step consists in finding the elementary
+graphs, i.e. the interaction vertices from which all other graphs (and amplitudes) can be
+built. These graphs are the building blocks of the field theory (Chapter 15), with the kinetic
+term given by the inverse of the propagator. Having Feynman diagrams and a field theory
+allows to use all the standard tools from QFT.
+However, this field theory is gauge fixed because on-shell amplitudes are gauge invariant
+and include only physical states.
+For this reason, one needs to find how to re-establish
+the gauge invariance. Due to the complicated structure of string theory, the full-fledged
+Batalin–Vilkovisky (BV) formalism must be used (Chapter 15): it basically amounts to
+introduce ghosts before the gauge fixing. The final stage is to obtain the 1PI effective action
+from which the physics is more easily extracted. But, it is useful to study first the free
+theory (Chapters 9 and 10) to gain some insights. The book ends with a discussion of the
+momentum-space representation and of background independence (Chapters 16 and 18).
+The procedure we will follow is a kind of reverse-engineering: we know what is the final
+23
+
+(a) Separating.
+(b) Non-separating.
+Figure 1.9: Degeneration of Riemann surfaces.
+result and we want to study backwards how it is obtained:
+on-shell amplitude → off-shell amplitude → Feynman graphs
+→ gauge fixed field theory → BV field theory
+In standard QFT, one follows the opposite process.
+Remark 1.3 There are some prescriptions (using for example analytic continuation, the
+optical theorem, some tricks. . . ) to address the problems mentioned above, but there is no
+general and universally valid procedure. A field theory is much more satisfactory because it
+provides a unique and complete framework.
+We can now summarise the disadvantages of the worldsheet approach over the spacetime
+field one:
+• no natural description of (relativistic) multi-particle states;
+• on-shell states:
+– lack of renormalization,
+– presence of infrared divergences,
+– scattering amplitudes only for protected states;
+• interactions added by hand;
+• hard to check consistency (unitarity, causality. . . );
+• absence of non-perturbative processes.
+Some of these problems can be addressed with various prescriptions, but it is desirable
+to dispose of a unified and systematic procedure, which is to be found in the field theory
+description.
+24
+
+1.3.2
+String field action
+A string field theory (SFT) for open and closed strings is based on two fields Φ[X(σ)] (open
+string field) and Ψ[X(σ)] (closed string field) governed by some action S[Φ, Ψ]. This action
+is built from a diagonal kinetic term
+S0 = 1
+2 KΨ(Ψ, Ψ) + 1
+2 KΦ(Φ, Φ)
+(1.29)
+and from an interaction polynomial in the fields
+Sint =
+�
+m,n
+Vm,n(Φm, Ψn)
+(1.30)
+where Vm,n is an appropriate product mapping m closed and n open string states to a number
+(the power is with respect to the tensor product). In particular, it contains the coupling
+constant. Contrary to the worldsheet approach where the cubic interaction looks sufficient,
+higher-order elementary interactions with m, n ∈ N are typically needed. A second specific
+feature is that the products also admit a loop (or genus g) expansion: a fundamental n-
+point interaction is introduced at every loop order g. These terms are interpreted as (finite)
+counter-terms needed to restore the gauge invariance of the measure. These two facts come
+from the decomposition of the moduli spaces in pieces (Section 1.2.3).
+Writing an action for a field Ψ[X(σ)] for which reparametrization invariance holds is
+highly complicated. The most powerful method is to introduce a functional dependence in
+ghost fields Ψ[X(σ), c(σ)] and to extend the BRST formalism to the string field, leading
+ultimately to the BV formalism.
+While the latter formalism is the most complete and
+ensures that the theory is consistent at the quantum level, it is difficult to characterize the
+interactions explicitly. Several constructions which exploit different properties of the theory
+have been proposed:
+• direct computation by reverse engineering of worldsheet amplitudes;
+• specific parametrization of the Riemann surfaces (hyperbolic, minimal area);
+• analogy with Chern–Simons and Wess–Zumino–Witten (WZW) theories;
+• exploitation of the L∞ and A∞ algebra structures.
+It can be shown that these constructions are all equivalent. For the superstring, the simplest
+strategy is to dress the bosonic interactions with data from the super-ghost sector, which
+motivates the study of the bosonic SFT by itself. The main difficulty in working with SFT is
+that only the first few interactions have been constructed explicitly. Finally, the advantage
+of the first formulation is that it provides a general formulation of SFT at the quantum
+level, from which the general structure can be studied.
+1.3.3
+Expression with spacetime fields
+To obtain a more intuitive picture and to make contact with the spacetime fields, the field
+is expanded in terms of 1-particle states in the momentum representation
+|Ψ⟩ =
+�
+n
+�
+dDk
+(2π)D ψα(k) |k, α⟩ ,
+(1.31)
+where α denotes collectively the discrete labels of the CFT eigenstates. The coefficients
+ψα(k) of the CFT states |k, α⟩ are spacetime fields, the first ones being the same as those
+found in the first-quantized picture (Section 1.2.1)
+ψα = {T, Gµν, Bµν, Φ, . . .}.
+(1.32)
+25
+
+Then, inserting this expansion in the action gives an expression like S[T, Gµν, . . .]. The exact
+expression of this action is out of reach and only the lowest terms are explicitly computable
+for a given CFT background. Nonetheless, examining the string field action indicates what
+is the generic form of the action in terms of the spacetime fields. One can then study the
+properties of such a general QFT: since it is more general than the SFT (expanded) action,
+any result derived for it will also be valid for SFT. This approach is very fruitful for studying
+properties related to consistency of QFT (unitarity, soft theorems. . . ) and this can provide
+helpful phenomenological models.
+In conclusion, SFT can be seen as a regular QFT with the following properties:
+• infinite number of fields;
+• non-local interaction (proportional to e−k2#);
+• the amplitudes agree with the worldsheet amplitudes (when the latter can be defined);
+• genuine (IR) divergences agree but can be handled with the usual QFT tools.
+1.3.4
+Applications
+The first aspect is the possibility to use standard QFT techniques (such as renormalization)
+to study – and to make sense of – string amplitudes. In this sense, SFT can be viewed as
+providing recipes for computing quantities in the worldsheet theory which are otherwise not
+defined. This program has been pushed quite far in the last years.
+Another reason to use SFT is gauge invariance: it is always easier to describe a sys-
+tem when its gauge invariance is manifest. We have explained that string theory contains
+Yang–Mills and graviton fields with the corresponding (spacetime) gauge invariances (non-
+Abelian gauge symmetry and diffeomorphisms). In fact, these symmetries are enhanced to
+an enormous gauge invariance when taking into account the higher-spin fields. This invari-
+ance is hidden in the standard formulation and cannot be exploited fully. On the other
+hand, the full gauge symmetry is manifest in string field theory.
+Finally, the worldvolume description of p-brane is difficult because there is no analogue
+of the Polyakov action. If one could find a first-principle description of SFT which does not
+rely on CFT and first-quantization, then one may hope to generalize it to build a brane field
+theory.
+We can summarize the general motivations for studying SFT:
+• field theory (second-quantization);
+• more rigorous and constructive formulation;
+• make gauge invariance explicit (L∞ algebras et al.);
+• use standard QFT techniques (renormalization, analyticity. . . )
+→ remove IR divergences, prove consistency (Cutkosky rules, unitarity, soft theorems,
+background independence. . . );
+• worldvolume theory ill-defined for (p > 1)-branes.
+Beyond these general ideas, SFT has been developed in order to address different questions:
+• worldsheet scattering amplitudes;
+• effective actions;
+• map of the consistent backgrounds (classical solutions, marginal deformations, RR
+fluxes. . . );
+26
+
+• collective, non-perturbative, thermal, dynamical effects;
+• symmetry breaking effects;
+• dynamics of compactification;
+• proof of dualities;
+• proof of the AdS/CFT correspondence.
+The last series of points is still out of reach within the current formulation of SFT. However,
+the last two decades have seen many important develoments developments:
+• construction of the open, closed and open-closed superstring field theories:
+– 1PI and BV actions and general properties [73, 165, 166, 213, 214, 216, 218, 220,
+222, 225, 226, 230],
+– dressing of bosonic products using the WZW construction and homotopy al-
+gebra [18, 19, 67–70, 74–76, 79, 80, 94, 111, 131, 134, 140–146, 180],
+– light-cone super-SFT [114–117],
+– supermoduli space [175, 241];
+• hyperbolic and minimal area constructions [38, 102, 103, 162–164, 183];
+• open string analytic solutions [77, 78];
+• level-truncation solutions [135–137];
+• field theory properties [34, 43, 150, 187, 221, 223, 224];
+• spacetime effective actions [65, 153, 154, 248];
+• defining worldsheet scattering amplitudes [184–186, 215, 216, 219, 227–229];
+• marginal and RR deformations [35, 229, 248].
+Recent reviews are [42, 71, 72].
+1.4
+Suggested readings
+For references about different aspects in this chapter:
+• Differences between the worldvolume and spacetime formalisms – and of the associated
+first- and second-quantization – for the particle and string [124, chap. 1, 265, chap. 11].
+• General properties of relativistic strings [92, 265].
+• Divergences in string theory [42, 217, 256, sec. 7.2].
+• Motivations for building a string field theory [192, sec. 4].
+27
+
+Part I
+Worldsheet theory
+28
+
+Chapter 2
+Worldsheet path integral:
+vacuum amplitudes
+Abstract
+In this chapter, we develop the path integral quantization for a generic closed
+string theory in worldsheet Euclidean signature. We focus on the vacuum amplitudes, leaving
+scattering amplitudes for the next chapter. This allows to focus on the definition and gauge
+fixing of the path integral measure.
+The exposition differs from most traditional textbooks in three ways: 1) we consider a
+general matter CFT, 2) we consider the most general treatment (for any genus) and 3) we
+don’t use complex coordinates but always a covariant parametrization.
+The derivation is technical and the reader is encouraged to not stop at this chapter
+in case of difficulties and to proceed forward: most concepts will be reintroduced from a
+different point of view later in other chapters of the book.
+2.1
+Worldsheet action and symmetries
+The string worldsheet is a Riemann surface W = Σg of genus g: the genus counts the
+number of holes or handles. Coordinates on the worldsheet are denoted by σa = (τ, σ).
+When there is no risk of confusion, σ denotes collectively both coordinates. Since closed
+strings are considered, the Riemann surface has locally the topology of a cylinder, with the
+spatial section being circles S1 with radius taken to be 1, such that
+σ ∈ [0, 2π),
+σ ∼ σ + 2π.
+(2.1)
+The string is embedded in the D-dimensional spacetime M with metric Gµν through maps
+Xµ(σa) : W → M with µ = 0, . . . , D − 1.
+The Nambu–Goto action is the starting point of the worldsheet description:
+SNG[Xµ] =
+1
+2πα′
+�
+d2σ
+�
+det Gµν(X)∂Xµ
+∂σa
+∂Xν
+∂σb ,
+(2.2)
+where α′ is the Regge slope (related to the string tension and string length).
+However,
+quantizing this action is difficult because it is highly non-linear. To solve this problem, a
+Lagrange multiplier is introduced to remove the squareroot. This auxiliary field corresponds
+to an intrinsic worldsheet metric gab(σ).
+The worldsheet dynamics is described by the
+Polyakov action:
+SP[g, Xµ] =
+1
+4πα′
+�
+d2σ√g gabGµν(X)∂Xµ
+∂σa
+∂Xν
+∂σb ,
+(2.3)
+29
+
+which is classically equivalent to the Nambu–Goto action (2.2). In this form, it is clear that
+the scalar fields Xµ(σ) (µ = 0, . . . D − 1) characterize the string theory under consideration
+in two ways.
+First, by specifying some properties of the spacetime in which the string
+propagates (for example, the number of dimensions is determined by the number of fields
+Xµ), second, by describing the internal degrees of freedom (vibration modes).1
+But, nothing prevents to consider a more general matter content in order to describe a
+different spacetime or different degrees of freedom. In Polyakov’s formalism, the worldsheet
+geometry is endowed with a metric gab(σ) together with a set of matter fields living on it.
+The scalar fields Xµ can be described by a general sigma model which encodes the embedding
+of the string in the D non-compact spacetime dimensions, and other fields can be added,
+for example to describe compactified dimensions or (spacetime) spin. Different sets of fields
+(and actions) correspond to different string theories.
+However, to describe precisely the
+different possibilities, we first have to understand the constraints on the worldsheet theories
+and to introduce conformal field theories (Part I). In this chapter (and in most of the book),
+the precise matter content is not important and we will denote the fields collectively as Ψ(σ).
+Before discussing the symmetries, let’s introduce a topological invariant which will be
+needed throughout the text: the Euler characteristics. It is computed by integrating the
+Riemann curvature R of the metric gab over the surface Σg:
+χg := χ(Σg) := 2 − 2g = 1
+4π
+�
+Σg
+d2σ√g R,
+(2.4)
+where g is the genus of the surface. Oriented Riemann surfaces without boundaries are
+completely classified (topologically or as complex manifolds) by their Euler characteristics
+χg, or equivalently by their genus g.
+In order to describe a proper string theory, the worldsheet metric gab(σ) should not
+be dynamical.
+This means that the worldsheet has no intrinsic dynamics and that no
+supplementary degrees of freedom are introduced when parametrizing the worldsheet with
+a metric. A solution to remove these degrees of freedom is to introduce gauge symmetries
+with as many gauge parameters as there are of degrees of freedom. The simplest symmetry
+is invariance under diffeomorphisms: indeed, the worldsheet theory is effectively a QFT
+coupled to gravity and it makes sense to require this invariance. Physically, this corresponds
+to the fact that the worldsheet spatial coordinate σ used along the string and worldsheet
+time are arbitrary. However, diffeomorphisms alone are not sufficient to completely fix the
+metric. Another natural candidate is Weyl invariance (local rescalings of the metric).
+A diffeomorphism f ∈ Diff(Σg) acts on the fields as
+σ′a = f a(σb),
+g′(σ′) = f ∗g(σ),
+Ψ′(σ′) = f ∗Ψ(σ),
+(2.5)
+where the star denotes the pullback by f: this corresponds simply to the standard coordinate
+transformation where each tensor index of the field receives a factor ∂σa/∂σ′b. In particular,
+the metric and scalar fields transform explicitly as
+g′
+ab(σ′) = ∂σc
+∂σ′a
+∂σd
+∂σ′b gcd(σ),
+X′µ(σ′) = Xµ(σ).
+(2.6)
+The index µ is inert since it is a target spacetime index: from the worldsheet point of view,
+it just labels a collection of worldsheet scalar fields. Infinitesimal variations are generated
+by vector fields on Σg:
+δξσa = ξa,
+δξΨ = LξΨ,
+δξgab = Lξgab,
+(2.7)
+1Obviously, the vibrational modes are also constrained by the spacetime geometry.
+30
+
+where Lξ is the Lie derivative2 with respect to the vector field ξ ∈ diff(Σg) ≃ TΣg. The Lie
+derivative of the metric is
+Lξgab = ξc∂cgab + gac∂bξc + gbc∂aξc = ∇aξb + ∇bξa.
+(2.8)
+The Lie algebra generates only transformations in the connected component Diff0(Σg) of
+the diffeomorphism group which contains the identity.
+Transformations not contained in Diff0(Σg) are called large diffeomorphisms: this in-
+cludes reflections, for example. The quotient of the two groups is called the modular group
+Γg (also mapping class group or MCG):
+Γg := π0
+�
+Diff(Σg)
+�
+= Diff(Σg)
+Diff0(Σg).
+(2.9)
+It depends only on the genus g of the Riemann surface, but not on the metric. It is an
+infinite discrete group for genus g ≥ 1 surfaces; in particular, Γ1 = SL(2, Z).
+A Weyl transformation e2ω ∈ Weyl(Σg) corresponds to a local rescaling of the metric
+and leaves the other fields unaffected3
+g′
+ab(σ) = e2ω(σ)gab(σ),
+Ψ′(σ) = Ψ(σ).
+(2.10)
+The exponential parametrization is generally more useful, but one should remember that it
+is e2ω and not ω which is an element of the group. The infinitesimal variation reads
+δωgab = 2ω gab,
+δωΨ = 0
+(2.11)
+where ω ∈ weyl(Σ) ≃ F(Σg) is a function on the manifold. Two metrics related in this way
+are said to be conformally equivalent. The conformal structure of the Riemann surface is
+defined by
+Conf(Σg) := Met(Σg)
+Weyl(Σg),
+(2.12)
+where Met(Σg) denotes the space of all metrics on Σg. Each element is a class of conformally
+equivalent metrics.
+Diffeomorphisms have two parameters ξa (vector field) and Weyl invariance has one,
+ω (function).
+Hence, this is sufficient to locally fix the three components of the metric
+(symmetric matrix) and the total gauge group of the theory is the semi-direct product
+G := Diff(Σg) ⋉ Weyl(Σg).
+(2.13)
+Similarly, the component connected of the identity is written as
+G0 := Diff0(Σg) ⋉ Weyl(Σg).
+(2.14)
+The semi-direct product arises because the Weyl parameter is not inert under diffeo-
+morphisms. Indeed, the combination of two transformations is
+g′ = f ∗�
+e2ωg
+�
+= e2f ∗ωf ∗g,
+(2.15)
+such that the diffeomorphism acts also on the conformal factor.
+2For our purpose here, it is sufficient to accept the definition of the Lie derivative as corresponding to
+the infinitesimal variation.
+3For simplicity, we consider only fields which do not transform under Weyl transformations, which
+excludes fermions.
+31
+
+The combination of transformations (2.15) can be chosen to fix the metric in a convenient
+gauge. For example, the conformal gauge reads
+gab(σ) = e2φ(σ)ˆgab(σ),
+(2.16)
+where ˆgab is some (fixed) background metric and φ(σ) is the conformal factor, also called
+the Liouville field. Fixing only diffeomorphisms amount to keep φ arbitrary: the latter can
+then be fixed with a Weyl transformation. For instance, one can adopt the conformally flat
+gauge
+ˆgab = δab,
+φ arbitrary
+(2.17)
+with a diffeomorphism, and then reach the flat gauge
+ˆgab = δab,
+φ = 0
+(2.18)
+with a Weyl transformation. Another common choice is the uniformization gauge where ˆg
+is taken to be the metric of constant curvature on the sphere (g = 0), on the plane (g = 1)
+or on the hyperbolic space (g > 1). All these gauges are covariant (both in spacetime and
+worldsheet).
+Remark 2.1 (Active and passive transformations) Usually, symmetries are described
+by active transformations, which means that the field is seen to be changed by the transform-
+ation. On the other hand, gauge fixing is seen as a passive transformation, where the field
+is expressed in terms of other fields (i.e. a different parametrization). These are mathem-
+atically equivalent since both cases correspond to inverse elements, and one can choose the
+most convenient representation. We will use indifferently the same name for the parameters
+to avoid introducing minus signs and inverse.
+Remark 2.2 (Topology and gauge choices) While it is always possible to adopt locally
+the flat gauge (2.18), it may not be possible to extend it globally. The can be seen intuitively
+from the fact that the sign of the curvature is given by the one of 1 − g, but the curvature of
+the flat metric is zero: curvature must then be localized somewhere and this prevents from
+using a single coordinate patch.
+The final step is to write an action Sm[g, Ψ] for the matter fields. According to the
+previous discussion, it must have the following properties:
+• local in the fields;
+• renormalizable;
+• non-linear sigma models for the scalar fields;
+• periodicity conditions;
+• invariant under diffeomorphisms (2.5);
+• invariant under Weyl transformations (2.10).
+The latter two conditions are summarized by
+Sm[f ∗g, f ∗Ψ] = Sm[g, Ψ],
+Sm[e2ωg, Ψ] = Sm[g, Ψ].
+(2.19)
+The invariance under diffeomorphisms is straightforward to enforce by using only covariant
+objects. Since the scalar fields represent embedding of the string in spacetime, the non-
+linear sigma model condition means that spacetime is identified with the target space of the
+sigma model, of which D dimensions are non-compact, and the spacetime metric appears
+32
+
+in the matter action as in (2.3).
+The isometries of the target manifold metric become
+global symmetries of Sm: while they are not needed in this chapter, they will have their
+importances in other chapters. Finally, to make the action consistent with the topology of
+the worldsheet, the fields must satisfy appropriate boundary conditions. For example, the
+scalar fields Xµ must be periodic for the closed string:
+Xµ(τ, σ) ∼ Xµ(τ, σ + 2π).
+(2.20)
+Remark 2.3 (2d gravity) The setup in two-dimensional gravity is exactly similar, except
+that the system is, in general, not invariant under Weyl transformations. As a consequence,
+one component of the metric (usually taken to be the Liouville mode) remains unconstrained:
+in the conformal gauge, (2.16) only ˆg is fixed.
+The symmetries (2.19) of the action have an important consequence: they imply that the
+matter action is conformally invariant on flat space gab = δab. A two-dimensional conformal
+field theory (CFT) is characterized by a central charge cm: roughly, it is a measure of the
+quantum degrees of freedom. The central charge is additive for decoupled sectors. In partic-
+ular, the scalar fields Xµ contribute as D, and it is useful to define the perpendicular CFT
+with central charge c⊥ as the matter which does not describe the non-compact dimensions:
+cm = D + c⊥.
+(2.21)
+This will be discussed in length in Part I. For this chapter and most of the book, it is
+sufficient to know that the matter is a CFT of central charge cm and includes D scalar fields
+Xµ:
+matter CFT parameters: D, cm.
+(2.22)
+The energy–momentum is defined by
+Tm,ab := − 4π
+√g
+δSm
+δgab .
+(2.23)
+The variation of the action under the transformations (2.7) vanishes on-shell if the energy–
+momentum tensor is conserved
+∇aTm,ab = 0
+(on-shell).
+(2.24)
+On the other hand, the variation under (2.11) vanishes off-shell (i.e. without using the
+equations of motion) if the energy–momentum tensor is traceless:
+gabTm,ab = 0
+(off-shell).
+(2.25)
+The conserved charges associated to the energy–momentum tensor generate worldsheet
+translations
+P a :=
+�
+dσ T 0a
+m .
+(2.26)
+The first component is identified with the worldsheet Hamiltonian P 0 = H and generates
+time translations, the second component generates spatial translations.
+Remark 2.4 (Tracelessness of the energy–momentum tensor) In fact, the trace can
+also be proportional to the curvature
+gabTm,ab ∝ R.
+(2.27)
+Then, the equations of motion are invariant since the integral of R is topological. The theory
+is invariant even if the action is not. Importantly, this happens for fields at the quantum
+level (Weyl anomaly), for the Weyl ghost field (Section 2.4) and for the Liouville theory
+(two-dimensional gravity coupled to conformal matter).
+33
+
+2.2
+Path integral
+The quantization of the system is achieved by considering the path integral, which yields
+the genus-g vacuum amplitude (or partition function):
+Zg :=
+�
+dggab
+Ωgauge[g] Zm[g],
+Zm[g] :=
+�
+dgΨ e−Sm[g,Ψ]
+(2.28)
+at fixed genus g (not to be confused with the metric). The integration over gab is performed
+over all metrics of the genus-g Riemann surface Σg: gab ∈ Met(Σg). The factor Ωgauge[g]
+is a normalization inserted in order to make the integral finite: it depends on the metric
+(but only through the moduli parameters, as we will show later) [53, p. 931], which explains
+why it is included after the integral sign. Its value will be determined in the next section by
+requiring the cancellation of the infinities due to the integration over the gauge parameters.
+This partition function corresponds to the g-loop vacuum amplitude: interactions and their
+associated scattering amplitudes are discussed in Section 3.1.
+In order to perform the gauge fixing and to manipulate the path integral (2.28), it is
+necessary to define the integration measure over the fields. Because the space is infinite-
+dimensional, this is a difficult task.
+One possibility is to define the measure implicitly
+through Gaussian integration over the field tangent space (see also Appendix C.1). A Gaus-
+sian integral involves a quadratic form, that is, an inner-product (or equivalently a metric)
+on the field space. The explanation is that a metric also defines a volume form, and thus
+a measure. To reduce the freedom in the definition of the inner-product, it is useful to
+introduce three natural assumptions:
+1. ultralocality: the measure is invariant under reparametrizations and defined point-wise,
+which implies that it can depend on the fields but not on their derivatives;
+2. invariant measure: the measure for the matter transforms trivially under any sym-
+metry of the matter theory by contracting indices with appropriate tensors;
+3. free-field measure: for fields other than the worldsheet metric and matter (like ghosts,
+Killing vectors, etc.), the measure is the one of a free field.
+This means that the inner-product is obtained by contracting the worldsheet indices of the
+fields with a tensor built only from the worldsheet metric, by contracting other indices (like
+spacetime) with some invariant tensor (like the spacetime metric), and finally by integrating
+over the worldsheet.
+We need to distinguish the matter fields from those appearing in the gauge fixing proced-
+ure. The matter fields live in the representation of some group under which the inner-product
+is invariant: this means that it is not possible to define each field measure independently
+if the exponential of inner-products does not factorize. As an example, on a curved back-
+ground: dX ̸= �
+µ dXµ. However, we will not need to write explicitly the partition function
+for performing the gauge fixing: it is sufficient to know that the matter is a CFT. In the
+gauge fixing procedure, different types of fields (including the metric) appear which don’t
+carry indices (beyond the worldsheet indices). Below, we focus on defining a measure for
+each of those single fields (and use free-field measures according to the third condition).
+Considering the finite elements δΦ1 and δΦ2 of tangent space at the point Φ of the state
+of fields, the inner-product (·, ·)g and its associated norm | · |g read
+(δΦ1, δΦ2)g :=
+�
+d2σ√g γg(δΦ1, δΦ2),
+|δΦ|2
+g := (δΦ, δΦ)g,
+(2.29)
+where γg is a metric on the δΦ space. It is taken to be flat for all fields except the metric itself,
+that is, independent of Φ. The dependence in the metric ensures that the inner-product is
+34
+
+diffeomorphism invariant, which in turns will lead to a metric-dependent but diffeomorphism
+invariant measure. The functional measure is then normalized by a Gaussian integral:
+�
+dgδΦ e− 1
+2 (δΦ,δΦ)g =
+1
+�
+det γg
+.
+(2.30)
+This, in turn, induces a measure on the field space itself:
+�
+dΦ
+�
+det γg
+(2.31)
+The determinant can be absorbed in the measure, such that
+�
+dgδΦ e− 1
+2 (δΦ,δΦ)g = 1.
+(2.32)
+In fact, this normalization and the definition of the inner-product is ambiguous, but the
+ultralocality condition allows to fix uniquely the final result (Section 2.3.4). Moreover, such
+a free-field measure is invariant under field translations
+Φ(σ) −→ Φ′(σ) = Φ(σ) + ε(σ).
+(2.33)
+The most natural inner-products for single scalar, vector and symmetric tensor fields are
+(δf, δf)g :=
+�
+d2σ√g δf 2
+(2.34a)
+(δV a, δV a)g :=
+�
+d2σ√g gabδV aδV b,
+(2.34b)
+(δTab, δTab)g :=
+�
+d2σ√g GabcdδTabδTcd,
+(2.34c)
+where the (DeWitt) metric for the symmetric tensor is
+Gabcd := Gabcd
+⊥
++ u gabgcd,
+Gabcd
+⊥
+:= gacgbd + gadgbc − gabgcd,
+(2.35)
+with u a constant. The first term G⊥ is the projector on the traceless component of the
+tensor. Indeed, consider a traceless tensor gabTab = 0 and a pure trace tensor Λgab, then we
+have:
+GabcdTcd = Gabcd
+⊥
+Tcd = 2Tab,
+Gabcd(Λgcd) = 2u (Λgab).
+(2.36)
+While all measures are invariant under diffeomorphisms, only the vector measure is
+invariant under Weyl transformations. This implies the existence of a quantum anomaly
+(the Weyl or conformal anomaly): the classical symmetry is broken by quantum effects
+because the path integral measure cannot respect all the classical symmetries. Hence, one
+can expect difficulties for imposing it at the quantum level and ensuring that the Liouville
+mode in (2.16) remains without dynamics.
+The metric variation (symmetric tensor) is decomposed in its trace and traceless parts
+δgab = gab δΛ + δg⊥
+ab,
+δΛ = 1
+2 gabδgab,
+gabδg⊥
+ab = 0.
+(2.37)
+In this decomposition, both terms are decoupled in the inner-product
+|δgab|2
+g = 4u|δΛ|2
+g + |δg⊥
+µν|
+2
+g,
+(2.38)
+35
+
+where the norm of δΛ is the one of a scalar field (2.34a). The norm for δg⊥
+ab is equivalent
+to (2.34c) with u = 0 (since it is traceless). Requiring positivity of the inner-product for a
+non-traceless tensor imposes the following constraint on u:
+u > 0.
+(2.39)
+One can absorb the coefficient with u in δΛ, which will just contribute as an overall factor:
+its precise value has no physical meaning. The simple choice u = 1/4 sets the coefficient of
+|δΛ|2
+g to 1 in (2.38) (another common choice is u = 1/2). Ultimately, this implies that the
+measure factorizes as
+dggab = dgΛ dgg⊥
+ab.
+(2.40)
+Computation – Equation (2.38)
+Gabcd δgabδgcd =
+�
+Gabcd
+⊥
++ u gabgcd��
+gab δΛ + δg⊥
+ab
+��
+gcd δΛ + δg⊥
+cd
+�
+=
+�
+2u gcd δΛ + Gabcd
+⊥
+δg⊥
+ab
+��
+gcd δΛ + δg⊥
+cd
+�
+= 4u (δΛ)2 + Gabcd
+⊥
+δg⊥
+abδg⊥
+cd
+= 4u δΛ2 + 2gacgbdδg⊥
+abδg⊥
+cd.
+Remark 2.5 Another common parametrization is
+Gabcd = gacgbd + c gabgcd.
+(2.41)
+It corresponds to (2.35) up to a factor 1/2 and setting u = 1 + 2c.
+Remark 2.6 (Matter and curved background measures) As explained previously, mat-
+ter fields carry a representation and the inner-product must yield an invariant combination.
+In particular, spacetime indices must be contracted with the spacetime metric Gµν(X) (which
+is the non-linear sigma model metric appearing in front of the kinetic term) for a general
+curved background. For example, the inner-product for the scalar fields Xµ is
+(δXµ, δXµ)g =
+�
+d2σ√g Gµν(X)δXµδXν.
+(2.42)
+It is not possible to normalize anymore the measure to set det G(X) = 1 like in (2.32) since
+it depends on the fields. On the other hand, this factor is not important for the manipu-
+lations performed in this chapter. Any ambiguity in the measure will again corresponds to
+a renormalization of the cosmological constant [53, p. 923]. Moreover, as explained above,
+it is not necessary to write explicitly the matter partition function as long as it describes a
+CFT.
+2.3
+Faddeev–Popov gauge fixing
+The naive integration over the space Met(Σg) of all metrics of Σg (note that the genus is
+fixed) leads to a divergence of the functional integral since equivalent configurations
+(f ∗g, f ∗Ψ) ∼ (g, Ψ),
+(e2ωg, Ψ) ∼ (g, Ψ)
+(2.43)
+gives the same contribution to the integral. This infinite redundancy causes the integral
+to diverge, and since the multiple counting is generated by the gauge group, the infinite
+contribution corresponds to the volume of the latter. The Faddeev–Popov procedure is a
+36
+
+means to extract this volume by separating the integration over the gauge and physical
+degrees of freedom
+d(fields) = Jacobian × d(gauge) × d(physical).
+(2.44)
+The space of fields (g, Ψ) is divided into equivalence classes and one integrates over only one
+representative of each class (gauge slice), see Figure 2.1. This change of variables introduces
+a Jacobian which can be represented by a partition function with ghost fields (fields with
+a wrong statistics). This program encounters some complications since G is a semi-direct
+product and is non-connected.
+Example 2.1 – Gauge redundancy
+A finite-dimensional integral which mimics the problem is
+Z =
+�
+R2 dx dy e−(x−y)2.
+(2.45)
+One can perform the change of variables
+r = x − y,
+y = a
+(2.46)
+such that
+Z =
+�
+R
+da
+� ∞
+0
+e−r2 =
+√π
+2 Vol(R),
+(2.47)
+and Vol(R) is to be interpreted as the volume of the gauge group (translation by a real
+number a).
+Remark 2.7 Mathematically, the Faddeev–Popov procedure consists in identifying the or-
+bits (class of equivalent metrics) under the gauge group G and to write the integral in terms
+of G-invariant objects (orbits instead of individual metrics). This can be done by decompos-
+ing the tangent space into variations generated by G and its complement. Then, one can
+define a foliation of the field space which equips it with a fibre bundle structure: the base
+is the push-forward of the complement and the fibre corresponds to the gauge orbits. The
+integral is then defined by selecting a section of this bundle.
+2.3.1
+Metrics on Riemann surfaces
+According to the above procedure, each metric gab ∈ Met(Σg) has to be expressed in terms
+of gauge parameters (ξ and ω) and of a metric ˆgab which contains the remaining gauge-
+independent degrees of freedom. As there are as many gauge parameters as metric com-
+ponents (Section 2.1), one could expect that there are no remaining physical parameters
+and then that ˆg is totally fixed. But, this is not the case and the metric ˆg depends on a
+finite number of parameters ti (moduli). The reason for this is topological: while locally it
+is always possible to completely fix the metric, topological obstructions may prevent doing
+it globally. This means that not all conformal classes in (2.12) can be (globally) related by
+a diffeomorphism.
+The quotient of the space of metrics by gauge transformations is called the moduli space
+Mg := Met(Σg)
+G
+.
+(2.48)
+Accordingly, its coordinates ti with i = 1, . . . , dimR Mg are called moduli parameters. The
+Teichmüller space Tg is obtained by taking the quotient of Met(Σg) with the component
+37
+
+Figure 2.1: The space of metrics decomposed in gauge orbits. Two metrics related by a
+gauge transformation lie on the same orbit. Choosing a gauge slice amounts to pick one
+metric in each orbit, and the projection gives the space of metric classes.
+connected to the identity
+Tg := Met(Σg)
+G0
+.
+(2.49)
+The space Tg is the covering space of Mg:
+Mg = Tg
+Γg
+,
+(2.50)
+where Γg is the modular group defined in (2.9). Both spaces can be endowed with a complex
+structure and are finite-dimensional [172]:
+Mg := dimR Mg = dimR Tg =
+�
+�
+�
+�
+�
+0
+g = 0,
+2
+g = 1,
+6g − 6
+g ≥ 2,
+(2.51)
+In particular, their volumes are related by
+�
+Mg
+dMgt =
+1
+ΩΓg
+�
+Tg
+dMgt
+(2.52)
+where ΩΓg is the volume of Γg.
+We will need to extract volumes of different groups, so it is useful to explain how they
+are defined. A natural measure on a connected group G is the Haar measure dg, which is
+the unique left-invariant measure on G. Integrating the measure gives the volume of the
+group
+ΩG :=
+�
+G
+dg =
+�
+G
+d(hg),
+(2.53)
+for any h ∈ G. Given the Lie algebra g of the group, a general element of the algebra is a
+linear combinations of the generators Ti with coefficients αi
+α = αiTi.
+(2.54)
+38
+
+Group elements can be parametrized in terms of α through the exponential map. Moreover,
+since a Lie group is a manifold, it is locally isomorphic to Rn: this motivates the use of a
+flat metric for the Lie algebra, such that
+ΩG =
+�
+dα :=
+� �
+i
+dαi.
+(2.55)
+Finally, it is possible to perform a change of coordinates from the Lie parameters to co-
+ordinates x on the group: the resulting Jacobian is the Haar measure for the coordinates
+x.
+Remark 2.8 While Tg is a manifold, this is not the case of Mg for g ≥ 2, which is an
+orbifold: the quotient by the modular group introduces singularities [173].
+Remark 2.9 (Moduli space and fundamental domain) Given a group acting on a space,
+a fundamental domain for a group is a subspace such that the full space is generated by act-
+ing with the group on the fundamental domain. Hence, one can view the moduli space Mg
+as a fundamental domain (sometimes denoted by Fg) for the group Γg and the space Tg.
+In the conformal gauge (2.16), the metric gab can be parametrized by
+gab = ˆg(f,φ)
+ab
+(t) := e2f ∗φf ∗ˆgab(t) = f ∗�
+e2φˆgab(t)
+�
+(2.56)
+where φ := ω and t denotes the dependence in the moduli parameters. To avoid surcharging
+the notations, we will continue to write g when there is no ambiguity. In coordinates, this
+is equivalent to:
+gab(σ) = ˆg(f,φ)
+ab
+(σ; t) := e2φ(σ)ˆg′
+ab(σ; t),
+ˆg′
+ab(σ; t) = ∂σ′c
+∂σa
+∂σ′d
+∂σb ˆgcd(σ′; t).
+(2.57)
+Remark 2.10 Strictly speaking, the matter fields also transform and one should write Ψ =
+Ψ(f) := f ∗ ˆΨ and include them in the change of integration measures of the following sections.
+But, this does not bring any particular benefits since these changes are trivial because the
+matter is decoupled from the metric.
+Remark 2.11 Although the metric cannot be completely gauge fixed, having just a finite-
+dimensional integral is much simpler than a functional integral. In higher dimensions, the
+gauge fixing does not reduce that much the degrees of freedom and a functional integral over
+ˆg remains (in similarity with Yang–Mills theories).
+The corresponding infinitesimal transformations are parametrized by (φ, ξ, δti).
+The
+variation of the metric (2.56) can be expressed as
+δgab = 2φ gab + ∇aξb + ∇bξa + δti∂igab,
+(2.58)
+which is decomposed in a reparametrization (2.7), a Weyl rescaling (2.11), and a contri-
+bution from the variations of the moduli parameters.
+The latter are called Teichmüller
+deformations and describe changes in the metric which cannot be written as a combination
+of diffeomorphism and Weyl transformation. Only the last term is written with a delta
+because the parameters ξ and φ are already infinitesimal. There is an implicit sum over i
+and we have defined
+∂i := ∂
+∂ti
+.
+(2.59)
+39
+
+According to the formula (2.55), the volumes ΩDiff0[g] and ΩWeyl[g] of the diffeomorph-
+isms connected to the identity and Weyl group are
+ΩDiff0[g] :=
+�
+dgξ,
+(2.60a)
+ΩWeyl[g] :=
+�
+dgφ.
+(2.60b)
+The full diffeomorphism group has one connected component for each element of the modular
+group Γg, according to (2.9): the volume ΩDiff[g] of the full group is the volume of the
+component connected to the identity times the volume ΩΓg
+ΩDiff[g] = ΩDiff0[g] ΩΓg.
+(2.60c)
+We have written that the volume depends on g: but, the metric itself is parametrized
+in terms of the integration variables, and thus the LHS of (2.60) cannot depend on the
+variable which is integrated over: ΩDiff0 can depend only on φ and ΩWeyl only on ξ. But, all
+measures (2.34b) are invariant under diffeomorphisms, and thus the result cannot depend on
+ξ. Moreover, the measure for vector is invariant under Weyl transformation, which means
+that ΩDiff0 does not depend on φ. This implies that the volumes depend only on the moduli
+parameters
+ΩDiff0[g] := ΩDiff0[e2φˆg] = ΩDiff0[ˆg],
+ΩWeyl[g] := ΩWeyl[Lξˆg] = ΩWeyl[ˆg].
+(2.61a)
+For this reason, it is also sufficient to take the normalization factor Ωgauge to have the same
+dependence:
+Ωgauge[g] := Ωgauge[ˆg].
+(2.61b)
+These volumes are also discussed in Section 2.3.4.
+Computation – Equation (2.61)
+ΩDiff0[e2φˆg] =
+�
+de2φLξˆgξ =
+�
+de2φˆgξ =
+�
+dˆgξ = ΩDiff0[ˆg],
+ΩWeyl[Lξˆg] =
+�
+de2φLξˆgφ =
+�
+de2φˆgφ = ΩWeyl[ˆg].
+Remark 2.12 (Free-field measure for the Liouville mode) The explicit measure (2.60b)
+of the Liouville mode is complicated since the inner-product contains an exponential of the
+field:
+|δφ|2 =
+�
+d2σ√g δφ2 =
+�
+d2σ
+�
+ˆg e2φδφ2.
+(2.62)
+It has been proposed by David–Distler–Kawai [40, 55], and later checked explicitly [50, 51,
+160], how to rewrite the measure in terms of a free measure weighted by an effective action.
+The latter is identified with the Liouville action (Section 2.3.3).
+In principle, we could follow the standard Faddeev–Popov procedure by inserting a delta
+function for the gauge fixing condition
+Fab := gab − ˆg(f,φ)
+ab
+(t),
+(2.63)
+with ˆg(f,φ)
+ab
+(t) defined in (2.56). However, we will take a detour to take the opportunity to
+study in details manipulations of path integrals and to understand several aspects of the
+40
+
+geometry of Riemann surfaces. In any case, several points are necessary even when going
+the short way, but less apparent.
+In order to make use of the factorization (2.40) of the integration measure, the variation
+(2.58) is decomposed into its trace (first term) and traceless parts (last two terms) (2.37)
+δgab = 2˜Λ gab + (P1ξ)ab + δti µiab,
+(2.64)
+where4
+(P1ξ)ab = ∇aξb + ∇bξa − gab∇cξc,
+(2.65a)
+µiab = ∂igab − 1
+2 gab gcd∂igcd,
+(2.65b)
+˜Λ = Λ + 1
+2 δti gab∂igab,
+Λ = φ + 1
+2 ∇cξc.
+(2.65c)
+The objects µi are called Beltrami differentials and correspond to traceless Teichmüller
+deformations (the factor of 1/2 comes from the symmetrization of the metric indices). The
+decomposition emphasizes which variations are independent from each other. In particular,
+changes to the trace of the metric due to a diffeomorphism generated by ξ or a modification
+of the moduli parameters can be compensated by a Weyl rescaling.
+One can use (2.40) to replace the integration over gab by one over the gauge parameters
+ξ and φ and over the moduli ti since they contain all the information about the metric:
+Zg =
+�
+dMgt dg ˜Λ dg(P1ξ) Ωgauge[g]−1 Zm[g].
+(2.66)
+It is tempting to perform the change of variables
+(P1ξ, ˜Λ) −→ (ξ, φ)
+(2.67)
+such that
+dg(P1ξ) dg ˜Λ
+?= dgξ dgφ ∆FP[g]
+(2.68)
+where ∆FP[g] is the Jacobian of the transformation
+∆FP[g] = det ∂(P1ξ, ˜Λ)
+∂(ξ, φ)
+= det
+�P1
+0
+⋆
+1
+�
+= det P1.
+(2.69)
+But, one needs to be more careful:
+1. The variations involving P1ξ and δti are not orthogonal and, as a consequence, the
+measure does not factorize.
+2. P1 has zero-modes, i.e. vectors such that P1ξ = 0, which causes the determinant to
+vanish, det P1 = 0.
+A rigorous analysis will be performed in Section 2.3.2 and will lead to additional factors in
+the path integral.
+Next, if the actions and measures were invariant under diffeomorphisms and Weyl trans-
+formations (which amounts to replace g by ˆg everywhere), it would be possible to factor out
+the integrations over the gauge parameters and to cancel the corresponding infinite factors
+thanks to the normalization Ωgauge[g]. A new problem arises because the measures are not
+Weyl invariant as explained above and one should be careful when replacing the metric
+(Section 2.3.3).
+4For comparison, Polchinski [193] defines P1 with an overall factor 1/2.
+41
+
+2.3.2
+Reparametrizations and analysis of P1
+The properties of the operator P1 are responsible for both problems preventing a direct
+factorization of the measure; for this reason, it is useful to study it in more details.
+The operator P1 is an object which takes a vector v to a symmetric traceless 2-tensor T,
+see (2.65a). Conversely, its adjoint P †
+1 can be defined from the scalar product (2.34c)
+(T, P1v)g = (P †
+1 T, v)g,
+(2.70)
+and takes symmetric traceless tensors to vectors. In components, one finds
+(P †
+1 T)a = −2∇bTab.
+(2.71)
+The Riemann–Roch theorem relates the dimension of the kernels of both operators [172]:
+dim ker P †
+1 − dim ker P1 = −3χg = 6g − 6.
+(2.72)
+Teichmüller deformations
+We first need to characterize Teichmüller deformations, the variations of moduli parameters
+which lead to transformations of the metric independent from diffeomorphisms and Weyl
+rescalings. This means that the different variations must be orthogonal for the inner-product
+(2.34).
+First, the deformations must be traceless, otherwise they can be compensated by a Weyl
+transformation. The traceless metric variations δg which cannot be generated by a vector
+field ξ are perpendicular to P1ξ (otherwise, the former would a linear combination of the
+latter):
+(δg, P1ξ)g = 0
+=⇒
+(P †
+1 δg, ξ)g = 0.
+(2.73)
+Since ξ is arbitrary, this means that the first argument vanishes
+P †
+1 δg = 0.
+(2.74)
+Metric variations induced by a change in the moduli ti are in the kernel of P †
+1
+δg ∈ ker P †
+1 .
+(2.75)
+Elements of ker P †
+1 are called quadratic differentials and a basis (not necessarily orthonor-
+mal) of ker P †
+1 is denoted as:
+ker P †
+1 = Span{φi},
+i = 1, . . . , dim ker P †
+1
+(2.76)
+(these should not be confused with the Liouville field). The dimension of ker P †
+1 is in fact
+equal to the dimension of the moduli space (2.51):
+dimR ker P †
+1 = Mg =
+�
+�
+�
+�
+�
+0
+g = 0,
+2
+g = 1,
+6g − 6
+g > 1.
+(2.77)
+The last two terms in the variation (2.64) of δgab are not orthogonal. Let’s introduce
+the projector on the complement space of ker P †
+1
+Π := P1
+1
+P †
+1 P1
+P †
+1 .
+(2.78)
+42
+
+The moduli variations can then be rewritten as
+δti µi = δti (1 − Π)µi + δti Πµi = δti (1 − Π)µi + δti P1ζi.
+(2.79)
+The ζi exist because Πµi ∈ Im P1, and they read
+ζi :=
+1
+P †
+1 P1
+P †
+1 µi.
+(2.80)
+The first term can be decomposed on the quadratic differential basis (2.76)
+(1 − Π)µi = φj(M −1)jk(φk, µi)g
+(2.81)
+where
+Mij := (φi, φj)g.
+(2.82)
+Ultimately, the variation (2.64) becomes
+δgab = (P1 ˜ξ)ab + 2˜Λ gab + Qiab δti.
+(2.83)
+where
+˜ξ = ξ + ζiδti,
+Qiab = φjab (M −1)jk(φk, µi)g.
+(2.84)
+Correspondingly, the norm of the variation splits in three terms since each variation is
+orthogonal to the others:
+|δg|2
+g = |δ˜Λ|
+2
+g + |P1 ˜ξ|
+2
+g + |Qiδti|2
+g.
+(2.85)
+Since the norm is decomposed as a sum, the measure factorizes:
+dggab = dg ˜Λ dg(P1 ˜ξ) dg(Qiδti).
+(2.86)
+One can then perform a change of coordinates
+(˜ξ, ˜Λ, Qiδti) −→ (ξ, Λ, δti),
+(2.87)
+where Λ was defined in (2.65c). The goal of this transformation is to remove the dependence
+in the moduli from the measures on the Weyl factor and vector fields, and to recover a finite-
+dimensional integral over the moduli:
+dg ˜Λ dg(P1 ˜ξ) dg(Qiδti) = dMgt dgΛ dg(P1ξ) det(φi, µj)g
+�
+det(φi, φj)g
+,
+(2.88)
+where the determinants correspond to the Jacobian. The role of the determinant in the
+denominator is to ensure a correct normalization when the basis is not orthonormal (in
+particular, it ensures that the Jacobian is independent of the basis). Plugging this result in
+(2.28) gives the partition function as
+Zg =
+�
+Tg
+dMgt
+1
+Ωgauge[ˆg]
+�
+dgΛ dg(P1ξ) det(φi, µj)g
+�
+det(φi, φj)g
+Zm[g].
+(2.89)
+The ti are integrated over the Teichmüller space Tg defined by (2.49) because the vectors ξ
+generate only reparametrizations connected to the identity, and thus the remaining freedom
+lies in Met(Σg)/G0. Next, we study how to perform the changes of variables to remove P1
+from the measure.
+43
+
+Conformal Killing vectors
+In this section, we focus on the dgΛ dg(P1ξ) part of the measure and we make contact with
+the rest at the end.
+Infinitesimal reparametrizations generated by a vector field ξa produce only transform-
+ations close to the identity. For this reason, integrating over all possible vector fields yields
+the volume (2.60a) of the component of the diffeomorphism group connected to the identity:
+�
+dgξ = ΩDiff0[ˆg].
+(2.90)
+Remember that the volume depends only on the moduli, but obviously not on ξ (integrated
+over) nor φ (the inner-product (2.34b) is invariant). But, due to the existence of zero-modes,
+one gets an integration over a subset of all vector fields, and this complicates the program,
+as we discuss now.
+Zero-modes ξ(0) of P1 are called conformal Killing vectors (CKV)
+ξ(0) ∈ Kg := ker P1
+(2.91)
+and satisfy the conformal Killing equation (see also Section 5.1):
+(P1ξ(0))ab = ∇aξ(0)
+b
++ ∇bξ(0)
+a
+− gab∇cξ(0)c = 0.
+(2.92)
+CKVs correspond to reparametrizations which can be absorbed by a change of the con-
+formal factor. They should be removed from the ξ integration in order to not double-count
+the corresponding metrics. The dimension of the zero-modes CKV space depends on the
+genus [172]:
+Kg := dimR Kg = dimR ker P1 =
+�
+�
+�
+�
+�
+6
+g = 0,
+2
+g = 1,
+0
+g > 1.
+(2.93)
+The associated transformations will be interpreted later (Chapter 5). The groups generated
+by the CKVs are
+g = 0 :
+K0 = SL(2, C),
+g = 1 :
+K1 = U(1) × U(1).
+(2.94)
+Note that the first group is non-compact while the second is compact.
+A general vector ξ can be separated into a zero-mode part and its orthogonal complement
+ξ′:
+ξ = ξ(0) + ξ′,
+(2.95)
+such that
+(ξ(0), ξ′)g = 0
+(2.96)
+for the inner-product (2.34b). Because zero-modes are annihilated by P1, the correct change
+of variables in the partition function (2.66) maps to ξ′ only:
+(P1ξ, Λ) −→ (ξ′, φ).
+(2.97)
+Integrating over ξ at this stage would double count the CKV (since they are already described
+by the φ integration). The appropriate Jacobian reads
+dgΛ dg(P1ξ) = dgφ dgξ′ ∆FP[g],
+(2.98)
+where the Faddeev–Popov determinant is
+∆FP[g] = det′ ∂(P1ξ, Λ)
+∂(ξ′, φ)
+= det′ P1 =
+�
+det′ P1P †
+1 ,
+(2.99)
+44
+
+the prime on the determinant indicating that the zero-modes are excluded. This brings the
+partition function (2.89) to the form
+Zg =
+�
+Tg
+dMgt Ωgauge[ˆg]−1
+�
+dgφ dgξ′
+det(φi, µj)g
+�
+det(φi, φj)g
+∆FP[g]Zm[g].
+(2.100)
+Computation – Equation (2.98)
+The Jacobian can be evaluated directly:
+∆FP[g] = det′ ∂(P1ξ, Λ)
+∂(ξ′, φ)
+= det′
+� P1
+0
+1
+2∇
+1
+�
+= det′ P1.
+(2.101)
+As a consequence of det′ P †
+1 = det′ P1, the Jacobian can be rewritten as:
+�
+det′ P †
+1 P1 = det′ P1.
+(2.102)
+It is instructive to derive this result also by manipulating the path integral. Con-
+sidering small variations of the fields, one has:
+1 =
+�
+dgδΛ dg(P1δξ) e−|δΛ|2
+g−|P1δξ′|2
+g
+= ∆FP[g]
+�
+dgδφ dgδξ′ e−|δφ+ 1
+2 ∇cδξc|2
+g−|P1δξ′|2
+g
+= ∆FP[g]
+�
+dgδφ dgδξ′ e−|δφ|2
+g−(δξ′,P †
+1 P1δξ′)g
+= ∆FP[g]
+�
+det′ P †
+1 P1
+�−1/2
+.
+That the expression is equal to 1 follows from the normalization of symmetric tensors
+and scalars (2.34) (the measures appearing in the path integral (2.89) arises without any
+factor). The third equality holds because the measure is invariant under translations
+of the fields, and we used the definition of the adjoint.
+The volume of the group generated by the vectors orthogonal to the CKV is denoted as
+Ω′
+Diff0[g] := Ω′
+Diff0[ˆg] =
+�
+dgξ′.
+(2.103)
+As explained in the beginning of this section, one should extract the volume of the full Diff0
+group, not only the volume Ω′
+Diff0[g]. Since the two sets of vectors are orthogonal, we can
+expect the measures, and thus the volumes, to factorize. However, a Jacobian can and does
+arise: its role it to take into account the normalization of the zero-modes. Denoting by ψi
+a basis (not necessarily orthonormal) for the zero-modes
+ker P1 = Span{ψi},
+i = 1, . . . , Kg,
+(2.104)
+the change of variables
+ξ′ −→ ξ
+(2.105)
+reads
+dgξ′ =
+1
+�
+det(ψi, ψj)g
+dgξ
+Ωckv[g],
+(2.106)
+45
+
+where Ωckv[g] is the volume of the CKV group. The determinant is necessary when the basis
+is not orthonormal. The relation between the gauge volumes is then
+ΩDiff0[g] =
+�
+det(ψi, ψj)g Ωckv[g] Ω′
+Diff0[g].
+(2.107)
+Note that the CKV volume is given in (2.111) and depends only on the topology but not on
+the metric. By using arguments similar to the ones which lead to (2.61), one can expect that
+each term is independently invariant under Weyl rescaling: this is indeed true (Section 2.3.3).
+Computation – Equation (2.106)
+Let’s expand ξ(0) on the zero-mode basis
+ξ(0) = αiψi,
+(2.108)
+where the αi are real numbers, such that one can write the changes of variables
+ξ −→ (ξ′, αi).
+(2.109)
+The Jacobian is computed from
+1 =
+�
+dξ e−|ξ|2
+g = J
+�
+dξ(0) dξ′ e−|ξ′|2
+g−|ξ(0)|
+2
+g
+= J
+� �
+i
+dαi e−αiαj(ψi,ψj)g
+�
+dξ′ e−|ξ′|2
+g
+= J (det(ψi, ψj)g)−1/2 .
+Note that the integration over the αi is a standard finite-dimensional integral. This
+gives
+dξ =
+�
+det(ψi, ψj)g dξ′ �
+i
+dαi.
+(2.110)
+Since nothing depends on the αi, they can be integrated over as in (2.53), giving the
+volume of the CKV group
+Ωckv[g] =
+� �
+i
+dαi.
+(2.111)
+Replacing the integration over ξ′ thanks to (2.106), the path integral becomes
+Zg =
+�
+Tg
+dMgt Ωgauge[ˆg]−1
+�
+dgφ dgξ
+det(φi, µj)g
+�
+det(φi, φj)g
+Ωckv[g]−1
+�
+det(ψi, ψj)g
+∆FP[g] Zm[g].
+(2.112)
+Since the matter action and measure, and the Liouville measure are invariant under
+reparametrizations, one can perform a change of variables
+(f ∗ˆg, f ∗φ, f ∗Ψ) −→ (ˆg, φ, Ψ)
+(2.113)
+such that everything becomes independent of f (or equivalently ξ). Since the measure for ξ
+is Weyl invariant, it is possible to separate it from the rest of the expression, which yields
+an overall factor of ΩDiff0[g]. This brings the partition function to the form
+Zg =
+�
+Tg
+dMgt ΩDiff0[ˆg]
+Ωgauge[ˆg]
+�
+dgφ det(φi, µj)g
+�
+det(φi, φj)g
+Ωckv[g]−1
+�
+det(ψi, ψj)g
+∆FP[g] Zm[g]
+(2.114)
+46
+
+where the same symbol is used for the metric
+gab := g(φ)
+ab = e2φˆgab.
+(2.115)
+Since the expression is invariant under the full diffeomorphism group Diff(Σg) and not
+just under its component Diff0(Σg), one needs to extract the volume of the full diffeomorph-
+ism group before cancelling it with the normalization factor. Otherwise, there is still an
+over-counting the configurations. Using the relation (2.60c) leads to:
+Zg =
+1
+ΩΓg
+�
+Tg
+dMgt ΩDiff[ˆg]
+Ωgauge[ˆg]
+�
+dgφ det(φi, µj)g
+�
+det(φi, φj)g
+Ωckv[g]−1
+�
+det(ψi, ψj)g
+∆FP[g] Zm[g].
+(2.116)
+The volume ΩΓg can be factorized outside the integral because it depends only on the genus
+and not on the metric. Finally, using the relation (2.52), one can replace the integration
+over the Teichmüller space by an integration over the moduli space
+Zg =
+�
+Mg
+dMgt ΩDiff[ˆg]
+Ωgauge[ˆg]
+�
+dgφ det(φi, µj)g
+�
+det(φi, φj)g
+Ωckv[g]−1
+�
+det(ψi, ψj)g
+∆FP[g] Zm[g].
+(2.117)
+2.3.3
+Weyl transformations and quantum anomalies
+The next question is whether the integrand depends on the Liouville mode φ such that
+the Weyl volume can be factorized out. While the matter action has been chosen to be
+Weyl invariant – see the condition (2.19) – the measures cannot be defined to be Weyl
+invariant. This means that there is a Weyl (or conformal) anomaly, i.e. a violation of the
+Weyl invariance due to quantum effects. Since the techniques needed to derive the results
+of this section are outside the scope of this book, we simply state the results.
+It is possible to show that the Weyl anomaly reads [53, p. 929]5
+∆FP[e2φˆg]
+�
+det(φi, φj)e2φˆg
+= e
+cgh
+6 SL[ˆg,φ]
+∆FP[ˆg]
+�
+det(ˆφi, ˆφj)ˆg
+(2.118a)
+Zm[e2φˆg] = e
+cm
+6 SL[ˆg,φ]Zm[ˆg],
+(2.118b)
+where SL is the Liouville action
+SL[ˆg, φ] := 1
+4π
+�
+d2σ
+�
+ˆg
+�
+ˆgab∂aφ∂bφ + ˆRφ
+�
+,
+(2.119)
+where ˆR is the Ricci scalar of the metric ˆgab. These relations require to introduce counter-
+terms, discussed further in Section 2.3.4. The coefficients cm and cgh are the central charges
+respectively of the matter and ghost systems, with:
+cgh = −26.
+(2.120)
+This value will be derived in Section 7.2.
+The inner-products between φi and µj, and between the ψi, and the CKV volume are
+independent of φ [172, sec. 14.2.2, 53, p. 931]
+det(φi, µj)e2φˆg = det(ˆφi, ˆµj)ˆg,
+det(ψi, ψj)e2φˆg = det(ψi, ψj)ˆg,
+Ωckv[e2φˆg] = Ωckv[ˆg].
+(2.121)
+5The relation is written for Zm since the action is invariant and is not affected by the anomaly.
+47
+
+Remark 2.13 (Weyl and gravitational anomalies) The Weyl anomaly translates into
+a non-zero trace of the quantum energy–momentum tensor
+⟨gµνTµν⟩ = c
+12 R,
+(2.122)
+where c is the central charge of the theory. The Weyl anomaly can be traded for a gravita-
+tional anomaly, which means that diffeomorphisms are broken at the quantum level [122].
+Inserting (2.118) in (2.117) yields
+Zg =
+�
+Mg
+dMgt ΩDiff[ˆg]
+Ωgauge[ˆg]
+det(φi, ˆµj)ˆg
+�
+det(φi, φj)ˆg
+Ωckv[ˆg]−1
+�
+det(ψi, ψj)ˆg
+∆FP[ˆg] Zm[ˆg]
+�
+dgφ e−
+cL
+6 SL[ˆg,φ],
+(2.123)
+with the Liouville central charge
+cL := 26 − cm.
+(2.124)
+The critical “dimension” is defined to be the value of the matter central charge cm such that
+the Liouville central charge cancels
+cL = 0
+=⇒
+cm = 26.
+(2.125)
+If the number of non-compact dimensions is D, it means that the central charge (2.21) of
+the transverse CFT satisfies
+c⊥ = 26 − D.
+(2.126)
+In this case, the integrand does not depend on the Liouville mode (because ΩDiff is
+invariant under Weyl transformations) and the integration over φ can be factored out and
+yields the volume of the Weyl group (2.60b)
+�
+dgφ = ΩWeyl[ˆg].
+(2.127)
+Then, taking
+Ωgauge[ˆg] = ΩDiff[ˆg] × ΩWeyl[ˆg]
+(2.128)
+removes the infinite gauge contributions and gives the partition function
+Zg =
+�
+Mg
+dMgt det(φi, ˆµj)ˆg
+�
+det(φi, φj)ˆg
+Ωckv[ˆg]−1
+�
+det(ψi, ψj)ˆg
+∆FP[ˆg] Zm[ˆg].
+(2.129)
+2.3.4
+Ambiguities, ultralocality and cosmological constant
+Different ambiguities remain in the previous computations, starting with the definitions of
+the measures (2.32) and (2.34), then in obtaining the volume of the diffeomorphism (2.60a)
+and Weyl (2.60b) groups, and finally in deriving the conformal anomaly (2.118).
+These different ambiguities can be removed by renormalizing the worldsheet cosmological
+constant. This implies that the action
+Sµ[g] =
+�
+d2σ√g
+(2.130)
+must be added to the classical Lagrangian, where µ0 is the bare cosmological constant. This
+means that Weyl invariance is explicitly broken at the classical level. After performing all
+the manipulations, µ0 is determined by removing all ambiguities and enforcing invariance
+under the Weyl symmetry at the quantum level.
+This amounts to set the renormalized
+cosmological constant to zero (since it breaks the Weyl symmetry).
+The possibility to
+48
+
+introduce a counter-term violating a classical symmetry arises because the symmetry itself
+is broken by a quantum anomaly, so there is no reason to enforce it in the classical action.
+We now review each issue separately. First, consider the inner-product of a single tensor
+(2.32): the determinant det γg depends on the metric and one should be more careful when
+fixing the gauge or integrating over all metrics.
+However, ultralocality implies that the
+determinant can only be of the form [53, pp. 923]
+�
+det γg = e−µγ Sµ[g],
+(2.131)
+for some µγ ∈ R, since Sµ is the only renormalizable covariant functional depending on the
+metric but not on its derivatives. The effect is just to redefine the cosmological constant.
+Second, the volume of the field space can be defined as the limit λ → 0 of a Gaussian
+integral [53, pp. 931]:
+ΩΦ = lim
+λ→0
+�
+dgΦ e−λ (Φ,Φ)g.
+(2.132)
+Due to ultralocality, the Gaussian integral should again be of the form
+�
+dgΦ e−λ (Φ,Φ)g = e−µ(λ) Sµ[g],
+(2.133)
+for some constant µ(λ). Hence, the limit λ → 0 gives
+ΩΦ =
+�
+dgΦ = e−µ(0) Sµ[g],
+(2.134)
+which can be absorbed in the cosmological constant. However, the situation is more com-
+plicated if Φ = ξ, φ since the integration variables also appear in the measure, as it was
+also discussed before (2.61). But, in that case, it cannot appear in the expression of the
+volume in the LHS. Moreover, invariances under diffeomorphisms for both measures, and
+under Weyl rescalings for the vector measure, imply that the LHS can only depend on the
+moduli through the background metric ˆg. The diffeomorphism and Weyl volumes can be
+written in terms of e−ˆµ Sµ[ˆg]: since there is no counter-term left (the cosmological constant
+counter-term is already fixed to cancel the coefficient of Sµ[g]), it is necessary to divide by
+Ωgauge to cancel the volumes.
+Finally, the computation of the Weyl anomaly (2.118) yields divergent terms of the form
+lim
+ϵ→0
+1
+ϵ
+�
+d2σ√g.
+(2.135)
+These divergences are canceled by the cosmological constant counter-term, see [54, app. 5.A]
+for more details.
+2.3.5
+Gauge-fixed path integral
+As a conclusion of this section, we found that the partition function (2.28) can be written
+as
+Zg =
+�
+Mg
+dMgt det(φi, ˆµj)ˆg
+�
+det(φi, φj)ˆg
+Ωckv[ˆg]−1
+�
+det(ψi, ψj)ˆg
+∆FP[ˆg] Zm[ˆg],
+(2.136a)
+=
+�
+Mg
+dMgt
+�
+det(φi, ˆµj)2
+ˆg
+det(φi, φj)ˆg
+det′ ˆP †
+1 ˆP1
+det(ψi, ψj)ˆg
+Zm[ˆg]
+Ωckv[ˆg].
+(2.136b)
+after gauge fixing of the worldsheet diffeomorphisms and Weyl rescalings. It is implicit that
+the factors for the CKV and moduli are respectively absent for g > 1 and g < 1. For g = 0
+the CKV group is non-compact and its volume is infinite. It looks like the partition vanishes,
+but there are subtleties which will be discussed in Section 3.1.3.
+49
+
+Remark 2.14 (Weil–Petersson metric) When the metric is chosen to be of constant
+curvature ˆR = −1, the moduli measure together with the determinants form the Weil–
+Petersson measure
+d(WP) =
+�
+Mg
+dMgt det(φi, ˆµj)ˆg
+�
+det(φi, φj)ˆg
+.
+(2.137)
+In (2.136), the background metric ˆgab is fixed. However, the derivation holds for any
+choice of ˆgab: as a consequence, it makes sense to relax the gauge fixing and allow it to vary
+while adding gauge symmetries. The first symmetry is background diffeomorphisms:
+σ′a = ˆf a(σb),
+ˆg′(σ′) = f ∗ˆg(σ),
+φ′(σ′) = f ∗φ(σ),
+Ψ′(σ′) = f ∗Ψ(σ).
+(2.138)
+This symmetry is automatic for Sm[ˆg, Ψ] since Sm[g, Ψ] was invariant under (2.5). Similarly,
+the integration measures are also invariant. A second symmetry is found by inspecting the
+decomposition (2.56)
+gab = f ∗�
+e2φˆgab(t)
+�
+,
+(2.139)
+which is left invariant under a background Weyl symmetry (also called emergent):
+g′
+ab(σ) = e2ω(σ)gab(σ),
+φ′(σ) = φ(σ) − ω(σ),
+Ψ′(σ) = Ψ(σ).
+(2.140)
+Let us stress that it is not related to the Weyl rescaling (2.10) of the metric gab.
+The
+background Weyl rescaling (2.140) is a symmetry even when the physical Weyl rescaling
+(2.10) is not. Together, the background diffeomorphisms and Weyl symmetry have three
+gauge parameters, which is sufficient to completely fix the background metric ˆg up to moduli.
+In fact, the combination of both symmetries is equivalent to invariance under the physical
+diffeomorphisms.
+To prove this statement, consider two metrics g and g′ related by a
+diffeomorphism F and both gauge fixed to pairs (f, φ, ˆg) and (f ′, φ′, ˆg′):
+g′
+ab = F ∗gab,
+g′
+ab = f ′∗�
+e2φ′ˆg′
+ab
+�
+,
+gab = f ∗�
+e2φˆgab
+�
+.
+(2.141)
+Then, the gauge fixing parametrizations are related by background symmetries ( ˆF, ω) as
+ˆF = f ′−1 ◦ F ◦ f,
+φ′ = ˆF ∗(φ − ω),
+ˆg′
+ab = ˆF ∗(e2ωˆgab).
+(2.142)
+Moreover, this also implies that there is a diffeomorphism ˜f = F ◦ f such that g′ is gauge
+fixed in terms of (φ, ˆg):
+g′
+ab = ˜f ∗�
+e2φˆgab
+�
+.
+(2.143)
+Computation – Equation (2.142)
+The functions F, f, f ′, φ, φ′ and the metrics gab, g′
+ab, ˆgab and ˆg′
+ab are all fixed and one
+must find ˆF and ω such that the relations (2.141) are compatible. First, one rewrites
+g′
+ab in terms of ˆgab and compare with the expression with ˆg′
+ab:
+g′
+ab = F ∗gab = F ∗�
+f ∗�
+e2φˆgab
+��
+= F ∗�
+f ∗�
+e2(φ−ω)e2ωˆgab
+��
+= f ′∗�
+e2φ′ˆg′
+ab
+�
+.
+In the third equality, we have introduced ω because ˆg′
+ab = ˆF ∗ˆgab is not true in general
+since there are 3 independent components but ˆF has only 2 parameters, so we cannot
+just define f ′ = F ◦ f and φ′ = φ. This explains the importance of the emergent Weyl
+symmetry.
+50
+
+Remark 2.15 (Gauge fixing and field redefinition) Although it looks like we are un-
+doing the gauge fixing, this is not exactly the case since the original metric is not used any-
+more. One can understand the procedure of this section as a field redefinition: the degrees of
+freedom in gab are repackaged into two fields (φ, ˆgab) adapted to make some properties of the
+system more salient. A new gauge symmetry is introduced to maintain the number of degrees
+of freedom. The latter helps to understand the structure of the action on the background.
+Finally, in this context, the Liouville action is understood as a Wess–Zumino action, which
+is defined as the difference between the effective actions evaluated in each metric. Another
+typical use of this point of view is to rewrite a massive vector field as a massless gauge field
+together with an axion [195].
+Remark 2.16 (Two-dimensional gravity) In 2d gravity, one does not work in the crit-
+ical dimension (2.125) and cL ̸= 0. Thus, the Liouville mode does not decouple: the con-
+formal anomaly breaks the Weyl symmetry at the quantum level which gives dynamics to
+gravity, even if it has no degree of freedom classically.
+As a consequence, one chooses
+Ωgauge = ΩDiff.
+Since the role of the classical Weyl symmetry is not as important as for string theory, it is
+even not necessary to impose it classically. This leads to consider non-conformal matter [21,
+22, 31, 82, 83]. Following the arguments from Section 2.1, the existence of the emergent
+Weyl symmetry (2.140) implies that the total action Sgrav[ˆg, φ] + Sm[ˆg, Ψ] must be a CFT
+for a flat background ˆg = δ, even if the two actions are not independently CFTs.
+2.4
+Ghost action
+2.4.1
+Actions and equations of motion
+It is well-known that a determinant can be represented with two anticommuting fields, called
+ghosts. The fields carry indices dictated by the map induced by the operator of the Faddeev–
+Popov determinant: one needs a symmetric and traceless anti-ghost bab and a vector ghost
+ca fields:
+∆FP[g] =
+�
+d′
+gb d′
+gc e−Sgh[g,b,c],
+(2.144)
+where the prime indicates that the ghost zero-modes are omitted. The ghost action is
+Sgh[g, b, c] := 1
+4π
+�
+d2σ√g gabgcdbac(P1c)bd
+(2.145a)
+= 1
+4π
+�
+d2σ√g gab�
+bac∇bcc + bbc∇acc − bab∇ccc�
+.
+(2.145b)
+The ghosts ca and anti-ghosts bab are associated respectively to the variations due to the
+diffeomorphisms ξa and to the variations perpendicular to the gauge slice. The normalization
+of 1/4π is conventional. In Minkowski signature, the action is multiplied by a factor i.
+Since bab is traceless, the last term of the action vanishes and could be removed. However,
+this implies to consider traceless variations of the bab when varying the action (to compute
+the equations of motion, the energy–momentum tensor, etc.). On the other hand, one can
+keep the term and consider unconstrained variation of bab (since the structure of the action
+will force the variation to have the correct symmetry), which is simpler. A last possibility is
+to introduce a Lagrange multiplier. These aspects are related to the question of introducing
+a ghost for the Weyl symmetry, which is described in Section 2.4.2.
+The equations of motion are
+(P1c)ab = ∇acb + ∇bca − gab∇ccc = 0,
+(P †
+1 b)a = −2∇bbab = 0.
+(2.146)
+51
+
+Hence, the classical solutions of b and c are respectively mapped to the zero-modes of the
+operators P †
+1 and P1, and they are thus associated to the CKV and Teichmüller parameters.
+The energy–momentum tensor is
+T gh
+ab = −bac∇bcc − bbc∇acc + cc∇cbab + gabbcd∇ccd.
+(2.147)
+Its trace vanishes off-shell (i.e. without using the b and c equations of motion)
+gabT gh
+ab = 0,
+(2.148)
+which shows that the action (2.145) is invariant under Weyl transformations
+Sgh[e2ωg, b, c] = Sgh[g, b, c].
+(2.149)
+The action (2.145) also has a U(1) global symmetry. The associated conserved charge is
+called the ghost number and counts the number of c ghosts minus the number of b ghosts,
+i.e.
+Ngh(b) = −1,
+Ngh(c) = 1.
+(2.150a)
+The matter fields are inert under this symmetry:
+Ngh(Ψ) = 0.
+(2.150b)
+In terms of actions, the path integral (2.136) can be rewritten as
+Zg =
+�
+Mg
+dMgt det(φi, ˆµj)ˆg
+�
+det(φi, φj)ˆg
+Ωckv[ˆg]−1
+�
+det(ψi, ψj)ˆg
+�
+dˆgΨ d′
+ˆgb d′
+ˆgc e−Sm[ˆg,Ψ]−Sgh[ˆg,b,c].
+(2.151)
+One can use (2.136) or (2.151) indifferently: the first is more appropriate when using spectral
+analysis to compute the determinant explicitly, while the second is more natural in the
+context of CFTs.
+2.4.2
+Weyl ghost
+Ghosts have been introduced for the reparametrizations (generated by ξa) and the traceless
+part of the metric (the gauge field associated to the transformation): one may wonder why
+there is not a ghost cw associated to the Weyl symmetry along with an antighost for the trace
+of the metric (i.e. the conformal factor). This can be understood from several viewpoints.
+First, the relation between a metric and its transformation – and the corresponding gauge
+fixing condition – does not involve any derivative: as such, the Jacobian is trivial. Second,
+one could choose
+F ⊥
+ab = √ggab −
+�
+ˆgˆgab = 0
+(2.152)
+as a gauge fixing condition instead of (2.63), and the trace component does not appear
+anywhere. Finally, a local Weyl symmetry is not independent from the diffeomorphisms.
+Remark 2.17 (Local Weyl symmetry) The topic of obtaining a local Weyl symmetry by
+gauging a global Weyl symmetry (dilatation) is very interesting [86, chap. 15, 113]. Under
+general conditions, one can express the new action in terms of the Ricci tensor (or of the
+curvature): this means that the Weyl gauge field and its curvature are composite fields.
+Moreover, one finds that local Weyl invariance leads to an off-shell condition while diffeo-
+morphisms give on-shell conditions. This explains why one imposes only Virasoro constraints
+(associated to reparametrizations) and no constraints for the Weyl symmetry in the covariant
+quantization.
+52
+
+However, it can be useful to introduce a ghost field cw for the Weyl symmetry nonetheless.
+In view of the previous discussion, this field should appear as a Lagrange multiplier which
+ensures that bab is traceless. Starting from the action (2.145), one finds
+S′
+gh[g, b, c, cw] = 1
+4π
+�
+d2σ√g gab�
+bac∇bcc + bbc∇acc + 2babcw
+�
+,
+(2.153)
+where bab is not traceless anymore. The ghost cw is not dynamical since the action does not
+contain derivatives of it, and it can be integrated out of the path integral to recover (2.145).
+The equations of motion for this modified action are
+∇acb + ∇bca + 2gabcw = 0,
+∇abab = 0,
+gabbab = 0.
+(2.154)
+Contracting the first equation with the metric gives
+cw = −1
+2 ∇aca,
+(2.155)
+and thus cw is nothing else than the divergence of the ca field: the Weyl ghost is a composite
+field (this makes connection with Remark 2.17) – see also (2.65c). The energy–momentum
+tensor of the ghosts with action (2.153) is
+T ′gh
+ab = −
+�
+bac∇bcc + bbc∇acc + 2babcw
+�
+− ∇c(babcc)
++ 1
+2 gabgcd�
+bce∇dce + bde∇cce + 2bcdcw
+�
+.
+(2.156)
+The trace of this tensor
+gabT ′gh
+ab = −gab∇c(babcc)
+(2.157)
+does not vanish off-shell, but it does on-shell since gabbab = 0. This implies that the theory
+is Weyl invariant even if the action is not. It is interesting to contrast this with the trace
+(2.148) when the Weyl ghost has been integrated out.
+The equations of motion (2.146) and energy–momentum tensor (2.147) for the action
+(2.145) can be easily derived by replacing cw by its solution in the previous formulas.
+Computation – Equation (2.156)
+The first parenthesis comes from varying gab, the second from the covariant derivatives,
+the last from the √g. The second term comes from
+gab�
+bacδ∇bcc + bbcδ∇acc�
+= 2gabbacδ∇bcc = 2gabbacδΓc
+bdcd
+= gabbacgce�
+∇bδgde + ∇dδgbe − ∇eδgbd
+�
+cd
+= bab�
+∇aδgbc + ∇cδgab − ∇bδgac
+�
+cc
+= bab∇cδgabcc,
+where two terms have cancelled due to the symmetry of bab. Integrating by part gives
+the term in the previous equation.
+Note that the integration on the Weyl ghost yields a delta function
+�
+dgcw e−(cw,gabbab)g = δ
+�
+gabbab
+�
+.
+(2.158)
+53
+
+2.4.3
+Zero-modes
+The path integral (2.151) excludes the zero-modes of the ghosts. One can expect them to
+be related to the determinants of elements of ker P1 and ker P †
+1 with Grassmann coefficients.
+They can be included after few simple manipulations (see also Appendix C.1.3).
+It is simpler to first focus on the b ghost (to avoid the problems related to the CKV).
+The path integral (2.151) can be rewritten as
+Zg =
+�
+Mg
+dMgt
+Ωckv[ˆg]−1
+�
+det(ψi, ψj)ˆg
+�
+dˆgΨ dˆgb d′
+ˆgc
+Mg
+�
+i=1
+(b, ˆµi)ˆg e−Sm[ˆg,Ψ]−Sgh[ˆg,b,c].
+(2.159)
+In this expression, c zero-modes are not integrated over, only the b zero-modes are. This is
+the standard starting point on Riemann surfaces with genus g ≥ 1. The inner-product reads
+explicitly
+(b, ˆµi)ˆg =
+�
+d2σ
+�
+ˆg Gabcd
+⊥
+babˆµi,cd =
+�
+d2σ
+�
+ˆg gacgbdbabˆµi,cd.
+(2.160)
+Computation – Equation (2.159)
+Since the zero-modes of b are in the kernel of P †
+1 , it means that the quadratic differentials
+(2.76) also provide a suitable basis:
+b = b0 + b′,
+b0 = b0iφi,
+where the b0i are Grassmann-odd coefficients. The first step is to find the Jacobian for
+the changes of variables b → (b′, b0i):
+1 =
+�
+dˆgb e−|b|2
+ˆg = J
+�
+dˆgb′ �
+i
+db0i e−|b′|2
+ˆg−|b0iφi|2 = J
+�
+det(φi, φj).
+Next, (2.151) has no zero-modes, so one must insert Mg of them at arbitrary positions
+σ0
+j to get a non-vanishing result when integrating over dMgb0i. The result of the integral
+is:
+�
+dMgb0i
+�
+j
+b0(σ0
+j ) =
+�
+dMgb0i
+�
+j
+�
+b0iφi(σ0
+j )
+�
+= det φi(σ0
+j ).
+The only combination of the φi which does not vanish is the determinant due to the
+anti-symmetry of the Grassmann numbers. Combining both results leads to:
+dˆgb′
+�
+det(φi, φj)ˆg
+=
+dˆgb
+det φi(σ0
+j )
+Mg
+�
+j=1
+b(σ0
+j ).
+(2.161)
+The locations positions σ0
+j are arbitrary (in particular, the RHS does not depend on
+them since the LHS does not either).
+Note that more details are provided in Ap-
+pendix C.1.3.
+An even simpler result can be obtained by combining the previous formula with the
+factor det(φi, ˆµj)ˆg:
+dˆgb′
+det(φi, ˆµj)ˆg
+�
+det(φi, φj)ˆg
+= dˆgb
+Mg
+�
+j=1
+(b, ˆµj)ˆg.
+(2.162)
+54
+
+This follows from
+Mg
+�
+j=1
+b(σ0
+j ) =
+Mg
+�
+j=1
+�
+b0iφi(σ0
+j )
+�
+= det φi(σ0
+j )
+Mg
+�
+j=1
+b0i,
+det(φi, ˆµj)ˆg
+Mg
+�
+j=1
+b0i =
+Mg
+�
+j=1
+�
+b0i(φi, ˆµj)ˆg
+�
+=
+Mg
+�
+j=1
+(b0iφi, ˆµj)ˆg =
+Mg
+�
+j=1
+(b, ˆµj)ˆg.
+Note that the previous manipulations are slightly formal: the symmetric traceless fields
+bab and φi,ab carry indices and there should be a product over the (two) independent
+components. This is a trivial extension and would just make the notations heavier.
+Similar manipulations lead to a new expression which includes also the c zero-mode (but
+which is not very illuminating):
+Zg =
+�
+Mg
+dMgt Ωckv[ˆg]−1
+det ψi(σ0
+j )
+�
+dˆgΨ dˆgb dˆgc
+Kc
+g
+�
+j=1
+ϵab
+2 ca(σ0
+j )cb(σ0
+j )
+×
+Mg
+�
+i=1
+(ˆµi, b)ˆg e−Sm[ˆg,Ψ]−Sgh[ˆg,b,c].
+(2.163)
+The σ0a
+j
+are Kc
+g = Kg/2 fixed positions and the integral does not depend on their values.
+Note that only Kc
+g positions are needed because the coordinate is 2-dimensional: fixing 3
+points with 2 components correctly gives 6 constraints. Then, ψi(σ0a
+j ) is a 6-dimensional
+matrix, with the rows indexed by i and the columns by the pair (a, j).
+The expression cannot be simplified further because the CKV factor is infinite for g = 0.
+This is connected to a fact mentioned previously: there is a remaining gauge symmetry
+which is not taken into account
+c −→ c + c0,
+P1c0 = 0.
+(2.164)
+A proper account requires to gauge fix this symmetry: the simplest possibility is to insert
+three or more vertex operators – this topic is discussed in Section 3.1.
+Finally, note that the same question arises for the b-ghost since one has the symmetry
+b −→ b + b0,
+P †
+1 b0 = 0.
+(2.165)
+That there is no problem in this case is related to the presence of the moduli.
+2.5
+Normalization
+In the previous sections, the closed string coupling constant gs did not appear in the ex-
+pressions. Loops in vacuum amplitudes are generated by splitting of closed strings. By
+inspecting the amplitudes, it seems that there are 2g such splittings (Figure 2.2), which
+would lead to a factor g2g
+s . However, this is not quite correct: this result holds for a 2-point
+function. Gluing the two external legs to get a partition function (that is, taking the trace)
+leads to an additional factor g−2
+s
+(to be determined later), such that the overall factor is
+g2g−2
+s
+. The fact that it is the appropriate power of the coupling constant can be more easily
+understood by considering n-point amplitudes (Section 3.1). The normalization of the path
+integral can be completely fixed by unitarity [193].
+The above factor has a nice geometrical interpretation. Defining
+Φ0 = ln gs
+(2.166)
+55
+
+and remembering the expression (2.4) of the Euler characteristics χg = 2 − 2g, the coupling
+factor can be rewritten as
+g2g−2
+s
+= e−Φ0χg = exp
+�
+−Φ0
+4π
+�
+d2σ√gR
+�
+= e−Φ0SEH[g],
+(2.167)
+where SEH is the Einstein–Hilbert action.
+This action is topological in two dimensions.
+Hence, the coupling constant can be inserted in the path integral simply by shifting the
+action by the above term.
+This shows that string theory on a flat target spacetime is
+completely equivalent to matter minimally coupled to Einstein–Hilbert gravity with a cos-
+mological constant (tuned to impose Weyl invariance at the quantum level). The advantage
+of describing the coupling power in this fashion is that it directly generalizes to scattering
+amplitudes and to open strings. The parameter Φ0 is interpreted as the expectation value of
+the dilaton. Replacing it by a general field Φ(Xµ) is a generalization of the matter non-linear
+sigma model, but this topic is beyond the scope of this book.
+Figure 2.2: g-loop partition function.
+2.6
+Summary
+In this chapter, we started with a fairly general matter CFT – containing at least D scalar
+fields Xµ – and explained under which condition it describes a string theory. The most
+important consequence is that the matter 2d QFT must in fact be a 2d CFT. We then
+continued by describing how to gauge fix the integration over the surfaces and we identified
+the remaining degrees of freedom – the moduli space Mg – up to some residual redundancy
+– the conformal Killing vector (CKV). Then, we showed how to rewrite the result in terms
+of ghosts and proved that they are also a CFT. This means that a string theory can be com-
+pletely described by two decoupled CFTs: a universal ghost CFT and a theory-dependent
+matter CFT describing the string spacetime embedding and the internal structure. The
+advantage is that one can forget the path integral formalism altogether and employ only
+CFT techniques to perform the computations. This point of view will be developed for off-
+shell amplitudes (Chapter 11) in order to provide an alternative description of how to build
+amplitudes. It is particularly fruitful because one can also consider matter CFTs which do
+not have a Lagrangian description. In the next chapter, we describe scattering amplitudes.
+2.7
+Suggested readings
+Numerous books have been published on the worldsheet string theory.
+Useful (but not
+required) complements to this chapter and subsequent ones are [151, 265] for introductory
+texts and [24, 47, 48, 128, 193] for more advanced aspects.
+• The definition of a field measure from a Gaussian integral and manipulations thereof
+can be found in [100, sec. 15.1, 22.1, 172, chap. 14, 191, 53].
+56
+
+• The most complete explanations of the gauge fixing procedure are [100, sec. 15.1, 22.1,
+24, sec. 3.4, 6.2, 193, chap. 5, 48, 124, chap. 5]. The original derivation can be found
+in [52, 161].
+• For the geometry of the moduli space, see [172, 173].
+• Ultralocality and its consequences are described in [53, 191] (see also [98, sec. 2.4]).
+• The use of a Weyl ghost is shown in [240, sec. 8, 258, sec. 9.2].
+57
+
+Chapter 3
+Worldsheet path integral:
+scattering amplitudes
+Abstract
+In this chapter, we generalize the worldsheet path integral to compute scattering
+amplitudes, which corresponds to insert vertex operators. The gauge fixing from the previous
+chapter is generalized to this case. In particular, we discuss the 2-point amplitude on the
+sphere. Finally, we introduce the BRST symmetry and motivate some properties of the
+BRST quantization, which will be performed in details later. The formulas in this chapter
+are all covariant: they will be rewritten in complex coordinates in the next chapter.
+3.1
+Scattering amplitudes on moduli space
+In this section, we describe the scattering of n strings. The momentum representation is
+more natural for describing interactions, especially in string theory. Therefore, each string is
+characterized by a state Vαi(ki) with momentum ki and some additional quantum numbers
+αi (i = 1, . . . , n). We start from the worldsheet path integral (2.28) before gauge fixing:
+Zg =
+�
+dggab
+Ωgauge[g] Zm[g],
+Zm[g] =
+�
+dgΨ e−Sm[g,Ψ].
+(3.1)
+3.1.1
+Vertex operators and path integral
+The external states are represented by infinite semi-tubes attached to the surfaces. Under a
+conformal mapping, the tubes can be mapped to points called punctures on the worldsheet.
+At g loops, the resulting space is a Riemann surface Σg,n of genus g with n punctures (or
+marked points). The external states are represented by integrated vertex operators
+Vα(ki) :=
+�
+d2σ
+�
+g(σ) Vα(k; σ).
+(3.2)
+The vertex operators Vα(k; σ) are built from the matter CFT operators and from the world-
+sheet metric gab. The functional dependence is omitted to not overload the notation, but
+one should read Vα(k; σ) := Vα[g, Ψ](k; σ). The integration over the state positions is neces-
+sary because the mapping of the tube to a point is arbitrary. Another viewpoint is that it
+is needed to obtain an expression invariant under worldsheet diffeomorphisms. The vertex
+operators described general states which not necessarily on-shell: this restriction will be
+found later when discussing the BRST invariance of scattering amplitudes (Section 3.2.2).
+58
+
+Following Section 2.3.5, the Einstein–Hilbert action with boundary term
+SEH[g] := 1
+4π
+�
+d2σ√g R + 1
+2π
+�
+ds k = χg,n.
+(3.3)
+is inserted in the path integral equals the Euler characteristics χg,n (the g in χg,n denotes
+the genus). On a surface with punctures, the latter is shifted by the number of punctures
+(which are equivalent to boundaries or disks) with respect to (2.4):
+χg,n := χ(Σg,n) = 2 − 2g − n.
+(3.4)
+This gives the normalization factor:
+g−χg,n
+s
+= e−Φ0SEH[g],
+Φ0 := ln gs.
+(3.5)
+The correctness factor can be verified by inspection of the Riemann surface for the scat-
+tering of n string at g loops. In particular, the string coupling constant is by definition
+the interaction strength for the scattering of 3 strings at tree-level. Moreover, the tree-level
+2-point amplitude contains no interaction and should have no power of gs. This factor can
+also be obtained by unitarity [193].
+By inserting these factors in (2.28), the g-loop n-point scattering amplitude is described
+by:
+Ag,n({ki}){αi} :=
+�
+dggab
+Ωgauge[g] dgΨ e−Sm[g,Ψ]−Φ0SEH[g]
+n
+�
+i=1
+��
+d2σi
+�
+g(σi) Vαi(ki; σi)
+�
+.
+(3.6)
+The σi dependence of each √g will be omitted from now on since no confusion is possible.
+The following equivalent notations will be used:
+Ag,n({ki}){αi} := Ag,n(k1, . . . , kn)α1,...,αn := Ag,n
+�
+Vα1(k1), . . . , Vαn(kn)
+�
+.
+(3.7)
+The complete (perturbative) amplitude is found by summing over all genus:
+An(k1, . . . , kn)α1,...,αn =
+∞
+�
+g=0
+Ag,n(k1, . . . , kn)α1,...,αn.
+(3.8)
+We omit a genus-dependent normalization which can be determined from unitarity [193].
+Sometimes, it is convenient to extract the factor e−Φ0χg,n of the amplitude Ag,n to display
+explicitly the genus expansion, but we will not follow this convention here. Since each term of
+the sum scales as Ag,n ∝ g2g+n−2
+s
+, this expression clearly shows that worldsheet amplitudes
+are perturbative by definition: this motivates the construction of a string field theory from
+which the full non-perturbative S-matrix can theoretically be computed.
+Finally, the amplitude (3.6) can be rewritten in terms of correlation functions of the
+matter QFT integrated over worldsheet metrics:
+Ag,n({ki}){αi} =
+�
+dggab
+Ωgauge[g] e−Φ0SEH[g]
+�
+n
+�
+i=1
+d2σi
+√g
+� n
+�
+i=1
+Vαi(ki; σi)
+�
+m,g
+.
+(3.9)
+The correlation function plays the same role as the partition function in (2.28). This shows
+that string expressions are integrals of CFT expressions over the space of worldsheet metrics
+(to be reduced to the moduli space).
+We address a last question before performing the gauge fixing: what does (3.6) computes
+exactly: on-shell or off-shell? Green functions or amplitudes? if amplitudes, the S-matrix
+59
+
+or just the interacting part T (amputated Green functions)? The first point is that a path
+integral over connected worldsheets will compute connected processes. We will prove later,
+when discussing the BRST quantization, that string states must be on-shell (Sections 3.2
+and 3.2.2) and that it corresponds to setting the Hamiltonian (2.26) to zero:
+H = 0.
+(3.10)
+From this fact, it follows that (2.28) must compute amplitudes since non-amputated Green
+functions diverge on-shell (due to external propagators). Finally, the question of whether it
+computes the S-matrix S = 1 + iT, or just the interacting part T is subtler. At tree-level,
+they agree for n ≥ 3, while T = 0 for n = 2 and S reduces to the identity. This difficulty
+(discussed further in Section 3.1.2) is thus related to the question of gauge-fixing tree-level
+2-point amplitude (Section 3.1.3). It has long been believed that (2.28) computes only the
+interacting part (amputated Green functions), but it has been understood recently that this
+is not correct and that (2.28) computes the S-matrix.
+Remark 3.1 (Scattering amplitudes in QFT) Remember that the S-matrix is separ-
+ated as:
+S = 1 + iT,
+(3.11)
+where 1 denotes the contribution where all particles propagate without interaction.
+The
+connected components of S and T are denoted by Sc and T c.
+The n-point (connected)
+scattering amplitudes An for n ≥ 3 can be computed from the Green functions Gn through
+the LSZ prescription (amputation of the external propagators):
+An(k1, . . . , kn) = Gn(k1, . . . , kn)
+n
+�
+i=1
+(k2
+i + m2
+i ).
+(3.12)
+The path integral computes the Green functions Gn; perturbatively, they are obtained from
+the Feynman rules. They include a D-dimensional delta function
+Gn(k1, . . . , kn) ∝ δ(D)(k1 + · · · + kn).
+(3.13)
+The 2-point amputated Green function T2 computed from the LSZ prescription vanishes
+on-shell. For example, considering a scalar field at tree-level, one finds:
+T2 = G2(k, k′) (k2 + m2)2 ∼ (k2 + m2) δ(D)(k + k′) −−−−−−→
+k2→−m2 0
+(3.14)
+since
+G2(k, k′) = δ(D)(k + k′)
+k2 + m2
+.
+(3.15)
+Hence, T2 = 0 and the S-matrix (3.11) reduces to the identity component Sc
+2 = 12 (which is
+a connected process). There are several way to understand this result:
+1. The recursive definition of the connected S-matrix Sc from the cluster decomposition
+principle requires a non-vanishing 2-point amplitude [121, sec. 5.1.5, 251, sec. 4.3, 63,
+sec. 6.1].
+2. The 2-point amplitude corresponds to the normalization of the 1-particle states (overlap
+of a particle state with itself, which is non-trivial) [250, eq. 4.1.4, 239, chap. 5].
+3. A single particle in the far past propagating to the far future without interacting is a
+connected and physical process [63, p. 133].
+4. It is required by the unitarity of the 2-point amplitude [66].
+60
+
+These points indicate that the 2-point amplitude is proportional to the identity in the mo-
+mentum representation [121, p. 212, 250, eq. 4.3.3 and 4.1.5]
+A2(k, k′) = 2k0 (2π)D−1δ(D−1)(k − k′).
+(3.16)
+The absence of interactions implies that the spatial momentum does not change (the on-
+shell condition implies that the energy is also conserved). This relation is consistent with
+the commutation relation of the operators with the Lorentz invariant measure1
+[a(k), a†(k′)] = 2k0 (2π)D−1δ(D−1)(k − k′).
+(3.17)
+That this holds for all particles at all loops can be proven using the Källen–Lehman repres-
+entation [121, p. 212].
+On the other hand, the identity part in (3.11) is absent for n ≥ 3 for connected amp-
+litudes: Sc
+n = T c
+n for n ≥ 3. This shows that the Feynman rules and the LSZ prescription
+compute only the interacting part T of the on-shell scattering amplitudes. The reason is that
+the derivation of the LSZ formula assumes that the incoming and outgoing states have no
+overlap, which is not the case for the 2-point function. A complete derivation of the S-matrix
+from the path integral is more involved [121, sec. 5.1.5, 260, sec. 6.7, 81] (see also [37]).
+The main idea is to consider a superposition of momentum states (here, in the holomorphic
+representation [260, sec. 5.1, 6.4])
+φ(α) =
+�
+dD−1k α(k)∗a†(k).
+(3.18)
+They contribute a quadratic piece to the connected S-matrix and, setting them to delta func-
+tions, one recovers the above result.
+3.1.2
+Gauge fixing: general case
+The Faddeev–Popov gauge fixing of the worldsheet diffeomorphisms and Weyl rescaling
+(2.15) goes through also in this case if the integrated vertex operators are diffeomorphism
+and Weyl invariant:
+δξVαi(ki) = δξ
+�
+d2σ√g Vαi(ki; σ) = 0,
+(3.19a)
+δωVαi(ki) = δω
+�
+d2σ√g Vαi(ki; σ) = 0,
+(3.19b)
+with the variations defined in (2.7) and (2.11). Diffeomorphism invariance is straightforward
+if the states are integrated worldsheet scalars. However, if the states are classically Weyl
+invariant, they are not necessary so at the quantum level: vertex operators are composite
+operators, which need to be renormalized to be well-defined at the quantum level. Renor-
+malization introduces a scale which breaks Weyl invariance. Enforcing it to be a symmetry
+of the vertex operators leads to constraints on the latter. We will not enter in the details
+since it depends on the matter CFT and we will assume that the operators Vαi(ki) are indeed
+Weyl invariant (see [193, sec. 3.6] for more details). In the rest of this book, we will use
+CFT techniques developed in Chapter 6. The Einstein–Hilbert action is clearly invariant
+under both symmetries since it is a topological quantity.
+1If the modes are defined as ˜a(k) = a(k)/
+√
+2k0 such that [˜a(k), ˜a†(k′)] = (2π)D−1δ(D−1)(k − k′), then
+one finds ˜
+A2(k, k′) = (2π)D−1δ(D−1)(k − k′).
+61
+
+Following the computations from Section 2.3 leads to a generalization of (2.136) with
+the vertex operators inserted for the amplitude (3.6):
+Ag,n({ki}){αi} = g−χg,n
+s
+�
+Mg
+dMgt det(φi, ˆµj)ˆg
+�
+det(φi, φj)ˆg
+Ωckv[ˆg]−1
+�
+det(ψi, ψj)ˆg
+×
+�
+n
+�
+i=1
+d2σi
+�
+ˆg
+� n
+�
+i=1
+ˆVαi(ki; σi)
+�
+m,ˆg
+.
+(3.20)
+The hat on the vertex operators indicates that they are evaluated in the background metric
+ˆg.
+The next step is to introduce the ghosts: following Section 2.4, the generalization of
+(2.159) is
+Ag,n({ki}){αi} = g−χg,n
+s
+�
+Mg
+dMgt
+Ωckv[ˆg]−1
+�
+det(ψi, ψj)ˆg
+�
+dˆgb d′
+ˆgc
+Mg
+�
+i=1
+(b, ˆµi)ˆg e−Sgh[ˆg,b,c]
+×
+�
+n
+�
+i=1
+d2σi
+�
+ˆg
+� n
+�
+i=1
+ˆVαi(ki; σi)
+�
+m,ˆg
+.
+(3.21)
+For the moment, only the b ghosts come with zero-modes.
+Then, c zero-modes can be
+introduced in (3.21)
+Ag,n = g−χg,n
+s
+�
+Mg
+dMgt Ωckv[ˆg]−1
+det ψi(σ0
+j )
+�
+dˆgb dˆgc
+Kc
+g
+�
+j=1
+ϵab
+2 ca(σ0
+j )cb(σ0
+j )
+Mg
+�
+i=1
+(ˆµi, b)ˆg e−Sgh[ˆg,b,c]
+×
+�
+n
+�
+i=1
+d2σi
+�
+ˆg
+� n
+�
+i=1
+ˆVαi(ki; σi)
+�
+m,ˆg
+,
+(3.22)
+by following the same derivation as (2.163). The formulas (3.21) and (3.22) are the correct
+starting point for all g and n. In particular, the c ghosts are not paired with any vertex
+(a condition often assumed or presented as mandatory). This fact will help resolve some
+difficulties for the 2-point function on the sphere.
+Remember that there is no CKV and no c zero-mode for g ≥ 2. For the sphere g = 0
+and the torus g = 1, there are CKVs, indicating that there is a residual symmetry in (3.21)
+and (3.22), which is the global conformal group of the worldsheet. It can be gauge fixed by
+imposing conditions on the vertex operators.2 The simplest gauge fixing condition amounts
+to fix the positions of Kc
+g vertex operators through the Faddeev–Popov trick:
+1 = ∆(σ0
+j )
+�
+dξ
+Kc
+g
+�
+j=1
+δ(2)(σj − σ0(ξ)
+j
+),
+σ0(ξ)
+j
+= σ0
+j + δξσ0
+j ,
+δξσ0
+j = ξ(σ0
+j ),
+(3.23)
+where ξ is a conformal Killing vector, and the variation of σ was given in (2.7). We find
+that
+∆(σ0
+j ) = det ψi(σ0
+j ).
+(3.24)
+A priori, the positions σ0
+j are not the same as the one appearing in (2.163) (since both sets
+are arbitrary): however, considering the same positions allows to cancel the factor (3.24)
+with the same one in (2.163).
+2In fact, it is only important to gauge fix for the sphere because the volume of the group is infinite. On
+the other hand, the volume of the CKV group for the torus is finite-dimensional such that dividing by Ωckv
+is not ambiguous.
+62
+
+Computation – Equation (3.24)
+The first step is to compute ∆ in (3.23). For this, we decompose the CKV ξ on the
+basis (2.104)
+ξ(σ0
+j ) = αiψi(σ0
+j )
+and write the Gaussian integral:
+1 =
+�
+Kc
+g
+�
+j=1
+d2δσj e
+−�
+j(δσj,δσj) = ∆
+�
+Kg
+�
+j=1
+dαi e
+−�
+j,i,i′(αiψi(σj),αi′ψi′(σj))
+= ∆
+�
+det ψi(σj)
+�−1.
+Again, we have reduced rigour in order to simplify the manipulations.
+After inserting the identity (3.23) into (3.22), one can integrate over Kc
+g vertex operator
+positions to remove the delta functions – at the condition that there are at least Kc
+g operators.
+As a consequence, we learn that the proposed gauge fixing works only for n ≥ 1 if g = 1 or
+n ≥ 3 if g = 0. This condition is equivalent to
+χg,n = 2 − 2g − n < 0.
+(3.25)
+In this case, the factors det ψi(σ0
+j ) cancel and (3.21) becomes
+Ag,n({ki}){αi} = g−χg,n
+s
+�
+Mg
+dMgt
+�
+dˆgb dˆgc
+Kc
+g
+�
+j=1
+ϵab
+2 ca(σ0
+j )cb(σ0
+j )
+Mg
+�
+i=1
+(ˆµi, b)ˆg e−Sgh[ˆg,b,c]
+×
+�
+n
+�
+i=Kc
+g+1
+d2σi
+�
+ˆg
+� Kc
+g
+�
+j=1
+ˆVαj(kj; σ0
+j )
+n
+�
+i=Kcg+1
+ˆVαi(ki; σi)
+�
+m,ˆg
+.
+(3.26)
+The result may be divided by a symmetry factor if the delta functions have solutions for
+several points [193, sec. 5.3]. Performing the gauge fixing for the other cases (in particular,
+g = 0, n = 2 and g = 1, n = 0) is more subtle (Section 3.1.3 and [193]).
+The amplitude can be rewritten in two different ways. First, the ghost insertions can be
+rewritten in terms of a ghost correlation functions
+Ag,n({ki}){αi} = g−χg,n
+s
+�
+Mg
+dMgt
+�
+n
+�
+i=Kc
+g+1
+d2σi
+�
+ˆg
+� Kc
+g
+�
+j=1
+ϵab
+2 ca(σ0
+j )cb(σ0
+j )
+Mg
+�
+i=1
+(ˆµi, b)ˆg
+�
+gh,ˆg
+×
+� Kc
+g
+�
+j=1
+ˆVαj(kj; σ0
+j )
+n
+�
+i=Kc
+g+1
+ˆVαi(ki; σi)
+�
+m,ˆg
+.
+(3.27)
+This form is particularly interesting because it shows that, before integration over the mod-
+uli, the amplitudes factorize. This is one of the main advantage of the conformal gauge, since
+the original complicated amplitude (3.6) for a QFT on a dynamical spacetime reduces to
+the product of two correlation functions of QFTs on a fixed curved background. In fact, the
+situation is even simpler when taking a flat background ˆg = δ since both the ghost and mat-
+ter sectors are CFTs and one can employ all the tools from two-dimensional CFT (Part I) to
+perform the computations and mostly forget about the path integral origin of these formulas.
+This approach is particularly fruitful for off-shell (Chapter 11) and superstring amplitudes
+(Chapter 17).
+63
+
+Remark 3.2 (Amplitudes in 2d gravity) The derivation of amplitudes for 2d gravity
+follows the same procedure, up to two differences: 1) there is an additional decoupled (before
+moduli and position integrations) gravitational sector described by the Liouville field, 2) the
+matter and gravitational action are not CFTs if the original matter was not.
+A second formula can be obtained by bringing the c-ghost on top of the matter vertex
+operators which are at the same positions
+Ag,n({ki}){αi} = g−χg,n
+s
+�
+Mg
+dMgt
+�
+n
+�
+i=Kc
+g+1
+d2σi
+�
+ˆg
+� Mg
+�
+i=1
+ˆBi
+Kc
+g
+�
+j=1
+ˆ
+Vαj(kj; σ0
+j )
+n
+�
+i=Kc
+g+1
+ˆVαi(ki; σi)
+�
+ˆg
+,
+(3.28)
+and where
+ˆ
+Vαj(kj; σ0
+j ) := ϵab
+2 ca(σ0
+j )cb(σ0
+j ) ˆVαj(kj; σ0
+j ),
+ˆBi := (ˆµi, b)ˆg.
+(3.29)
+The operators Vαi(ki; σ0
+j ) (a priori off-shell) are called unintegrated operators, by opposition
+to the integrated operators Vαi(ki). We will see that both are natural elements of the BRST
+cohomology.
+To stress that the ˆBi insertions are really an element of the measure, it is finally possible
+to rewrite the previous expression as
+Ag,n({ki}){αi} = g−χg,n
+s
+�
+Mg×Cn−Kcg
+� Mg
+�
+i=1
+ˆBi dti
+Kc
+g
+�
+j=1
+ˆ
+Vαi(ki; σ0
+j )
+n
+�
+i=Kc
+g+1
+ˆVαi(ki; σi) d2σi
+�
+ˆg
+�
+ˆg
+.
+(3.30)
+The result (3.28) suggests a last possibility for improving the expression of the amplitude.
+Indeed, the different vertex operators don’t appear symmetrically: some are integrated over
+and other come with c ghosts. Similarly, the two types of integrals have different roles: the
+moduli are related to geometry while the positions look like external data (vertex operators).
+However, punctures can obviously be interpreted as part of the geometry, and one may
+wonder if it is possible to unify the moduli and positions integrals. It is, in fact, possible to
+put all vertex operators and integrals on the same footing by considering the amplitude to
+be defined on the moduli space Mg,n of genus-g Riemann surfaces with n punctures instead
+of just Mg [193] (see also Section 11.3.1).
+3.1.3
+Gauge fixing: 2-point amplitude
+As discussed at the end of Section 3.1.1, it has long been believed that the tree-level 2-point
+amplitude vanishes. There were two main arguments: there are not sufficiently many vertex
+operators 1) to fix completely the SL(2, C) invariance or 2) to saturate the number of c-
+ghost zero-modes. Let’s review both points and then explain why they are incorrect. We
+will provide the simplest arguments, referring the reader to the literature [66, 208] for more
+general approaches.
+For simplicity, we consider the flat metric ˆg = δ and an orthonormal basis of CKV. The
+two weight-(1, 1) matter vertex operators are denoted as Vk(z, ¯z) and Vk′(z′, ¯z′) such that
+the 2-point correlation function on the sphere reads (see Chapters 6 and 7 for more details):
+⟨Vk(z, ¯z)Vk′(z′, ¯z′)⟩S2 = i (2π)Dδ(D)(k + k′)
+|z − z′|4
+.
+(3.31)
+The numerator comes from the zero-modes ei(k+k′)·x for a target spacetime with a Lorentzian
+signature [48, p. 866, 193] (required to make use of the on-shell condition).
+64
+
+Review of the problem
+The tree-level amplitude (3.20) for n = 2 reads:
+A0,2(k, k′) =
+CS2
+Vol K0,0
+�
+d2zd2z′ ⟨Vk(z, ¯z)Vk′(z′, ¯z′)⟩S2 ,
+(3.32)
+where K0,n is the CKV group of Σ0,n, the sphere with n punctures. In particular, the group
+of the sphere without puncture is K0,0 = PSL(2, C). The normalization of the amplitude is
+CS2 = 8πα′−1 for gs = 1 [193, 249]. Since there are two insertions, the symmetry can be
+partially gauge fixed by fixing the positions of the two punctures to z = 0 and z′ = ∞. In
+this case, the amplitude (3.32) becomes:
+A0,2(k, k′) =
+CS2
+Vol K0,2
+⟨Vk(∞, ∞)Vk′(0, 0)⟩S2 ,
+(3.33)
+where K0,2 = R∗
++×U(1) is the CKV group of the 2-punctured sphere – containing dilatations
+and rotations.3 Since the volume of this group is infinite Vol K0,2 = ∞, it looks like A0,2 = 0.
+However, this forgets that the 2-point correlation function (3.31) contains a D-dimensional
+delta function.
+The on-shell condition implies that the conservation of the momentum
+k + k′ = 0 is automatic for one component, such that the numerator in (3.33) contains a
+divergent factor δ(0):
+A0,2(k, k′) = (2π)D−1δ(D−1)(k + k′) CS2 2πi δ(0)
+Vol K0,2
+.
+(3.34)
+Hence, (3.33) is of the form A0,2 = ∞/∞ and one should be careful when evaluating it.
+The second argument relies on a loophole in the understanding of the gauge fixed amp-
+litude (3.28). The result (3.28) is often summarized by saying that one can go from (3.20)
+to (3.28) by replacing Kc
+g integrated vertices
+�
+V by unintegrated vertices c¯cV in order to
+saturate the ghost zero-modes and to obtain a non-zero result. For g = 0, this requires 3
+unintegrated vertices. But, since there are only two operators in (3.32), this is impossible
+and the result must be zero. However, this is also incorrect because it is always possible
+to insert 6 c zero-modes, as show the formulas (2.163) and (3.27). Indeed, they are part
+of how the path integral measure is defined and do not care of the matter operators. The
+question is whether they can be attached to vertex operators (for aesthetic reasons or more
+pragmatically to get natural states of the BRST cohomology). To find the correct result
+with ghosts requires to start with (3.27) and to see how this can be simplified when there
+are only two operators.
+Computation of the amplitude
+In this section, we compute the 2-point amplitude from (3.33):
+A0,2(k, k′) =
+CS2
+Vol K0,2
+⟨Vk(∞, ∞)Vk′(0, 0)⟩S2 .
+(3.35)
+The volume of K0,2 reads (by writing a measure invariant under rotations and dilatations,
+but not translations nor special conformal transformations) [53, 60]:
+Vol K0,2 =
+� d2z
+|z|2 = 2
+� 2π
+0
+dσ
+� ∞
+0
+dr
+r ,
+(3.36)
+3The subgroup and the associated measure depend on the locations of the two punctures.
+65
+
+by doing the change of variables z = reiσ. Since the volume is infinite, it must be regu-
+larized. A first possibility is to cut-off a small circle of radius ϵ around r = 0 and r = ∞
+(corresponding to removing the two punctures at z = 0, ∞). A second possibility consists
+in performing the change of variables r = eτ and to add an imaginary exponential:
+Vol K0,2 = 4π
+� ∞
+0
+dr
+r = 4π
+� ∞
+−∞
+dτ = 4π lim
+ε→0
+� ∞
+−∞
+dτ eiετ = 4π × 2π lim
+ε→0 δ(ε),
+(3.37)
+such that the regularized volume reads
+Volε K0,2 = 8π2 δ(ε).
+(3.38)
+In fact, τ can be interpreted as the Euclidean worldsheet time on the cylinder since r
+corresponds to the radial direction of the complex plane.
+Since the worldsheet is an embedding into the target spacetime, both must have the
+same signature. As a consequence, for the worldsheet to be also Lorentzian, the formula
+(3.37) must be analytically continued as ε = −iE and τ = it such that
+VolM,E K0,2 = 8π2i δ(E),
+(3.39)
+where the subscript M reminds that one considers the Lorentzian signature. Inserting this
+expression in (3.34) and taking the limit E → 0, it looks like the two δ(0) will cancel.
+However, we need to be careful about the dimensions. Indeed, the worldsheet time τ and
+energy E are dimensionless, while the spacetime time and energy are not. Thus, it is not
+quite correct to cancel directly both δ(0) since they don’t have the same dimensions. In order
+to find the correct relation between the integrals in (3.37) and of the zero-mode in (3.31),
+we can look at the mode expansion for the scalar field (removing the useless oscillators):
+X0(z, ¯z) = x0 + i
+2 α′k0 ln |z|2 = x0 + iα′k0τ,
+(3.40)
+where the second equality follows by setting z = eτ. After analytic continuation k0 = −ik0
+M,
+X0 = iX0
+M, x0 = ix0
+M and τ = it, we find [265, p. 186]:
+X0
+M = x0
+M + α′k0
+Mt.
+(3.41)
+This indicates that the measure of the worldsheet time in (3.39) must be rescaled by 1/α′k0
+M
+such that:
+VolM K0,2 −→ 8π2i δ(0)
+α′k0
+M
+= CS2 2πi δ(0)
+2k0
+M
+.
+(3.42)
+This is equivalent to rescale E by α′k0 and to use δ(ax) = a−1δ(x).
+Ultimately, the 2-point amplitude becomes (removing the subscript on k0):
+A0,2(k, k′) = 2k0(2π)D−1δ(D−1)(k + k′)
+(3.43)
+and matches the QFT formula (3.16). We see that taking into account the scale of the
+coordinates is important to reproduce this result.
+The computation displayed here presents some ambiguities because of the regularization.
+However, this ambiguity can be fixed from unitarity of the scattering amplitudes. A more
+general version of the Faddeev–Popov gauge fixing has been introduced in [66] to avoid
+dealing altogether with infinities.
+It is an interesting question whether these techniques
+can be extended to the compute the tree-level 1- and 0-point amplitudes on the sphere. In
+most cases, the 1-point amplitude is expected to vanish since 1-point correlation functions
+66
+
+of primary operators other than the identity vanish in unitary CFTs.4 The 0-point function
+corresponds to the sphere partition function: the saddle point approximation to leading
+order allows to relate it to the spacetime action evaluated on the classical solution φ0,
+Z0 ∼ e−S[φ0]/ℏ. Since the normalization is not known and because S[φ0] is expected to
+be infinite, only comparison between two spacetimes should be meaningful (à la Gibbons–
+Hawking–York [190, sec. 4.1]). In particular, for Minkowski spacetime we find naively
+Z0 ∼ δ(D)(0)
+Vol K0
+,
+(3.44)
+which is not well-defined. This question has no yet been investigated.
+Expression with ghosts
+There are different ways to rewrite the 2-point amplitude in terms of ghosts. In all cases, one
+correctly finds the 6 insertions necessary to get a non-vanishing result since, by definition,
+it is always possible to rewrite the Faddeev–Popov determinant in terms of ghosts. A first
+approach is to insert 1 =
+�
+d2z δ(2)(z) inside (3.32) to mimic the presence of a third operator.
+This is equivalent to use the identity
+⟨0| c−1¯c−1c0¯c0c1¯c1 |0⟩ = 1
+(3.45)
+inside (3.33), leading to:
+A0,2(k, k′) =
+CS2
+Vol K0,2
+⟨Vk(∞, ∞)c0¯c0 Vk′(0, 0)⟩S2 ,
+(3.46)
+where Vk(z, ¯z) = c¯cVk(z, ¯z). This shows that (3.16) can also be recovered using the correct
+insertions of ghosts. The presence of c0¯c0 can be expected from string field theory since they
+appear in the kinetic term (10.115).
+The disadvantage of this formula is to still contain the infinite volume of the dilatation
+group. It is also possible to introduce ghosts for the more general gauge fixing presented
+in [66]. An alternative approach has been proposed in [208].
+3.2
+BRST quantization
+The symmetries of a Lagrangian dictate the possible terms which can be considered. This
+continues to hold at the quantum level and the counter-terms introduced by renormalization
+are constrained by the symmetries. However, if the path integral is gauge fixed, the original
+symmetry is no more available for this purpose. Fortunately, one can show that there is
+a global symmetry (with anticommuting parameters) remnant of the local symmetry: the
+BRST symmetry. It ensures consistency of the quantum theory. It also provides a direct
+access to the physical spectrum.
+The goal of this section is to provide a general idea of the BRST quantization for the
+worldsheet path integral. A more detailed CFT analysis and the consequence for string
+theory are given in Chapter 8. The reader is assumed to have some familiarity with the
+BRST quantization in field theory – a summary is given in Appendix C.2.
+4The integral over the zero-mode gives a factor δ(D)(k) which implies k = 0.
+At zero momentum,
+the time scalar X0 is effectively described by unitary CFT. However, there can be some subtleties when
+considering marginal operator.
+67
+
+3.2.1
+BRST symmetry
+The partition function (2.159) is not the most suitable to display the BRST symmetry.
+The first step is to restore the dependence in the original metric gab by introducing a delta
+function
+Zg =
+�
+Mg
+dMgt
+Ωckv[g]
+�
+dggab dgΨ dgb d′
+gc δ
+�√ggab −
+�
+ˆgˆgab
+� Mg
+�
+i=1
+(φi, b)g e−Sm[g,Ψ]−Sgh[g,b,c].
+(3.47)
+Note that it is necessary to use the traceless gauge fixing condition (2.152) as it will become
+clear. The delta function is Fourier transformed in an exponential thanks to an auxiliary
+bosonic field:
+Zg =
+�
+Mg
+dMgt
+Ωckv[g]
+�
+dggab dgBab dgΨ dgb d′
+gc
+Mg
+�
+i=1
+(φi, b)g e−Sm[g,Ψ]−Sgf[g,ˆg,B]−Sgh[g,b,c]
+(3.48)
+where the gauge-fixing action reads:
+Sgf[g, ˆg, B] = − i
+4π
+�
+d2σ Bab�√ggab −
+�
+ˆgˆgab
+�
+.
+(3.49)
+Varying the action with respect to the auxiliary field Bab, called the Nakanish–Lautrup field,
+produces the gauge-fixing condition.
+The BRST transformations are
+δϵgab = iϵ Lcgab,
+δϵΨ = iϵ LcΨ,
+δϵca = iϵ Lcca,
+δϵbab = ϵ Bab,
+δϵBab = 0,
+(3.50)
+where ϵ is a Grassmann parameter (anticommuting number) independent of the position.
+If the traceless gauge fixing (2.152) is not used, then Bab is not traceless: in that case, the
+variation δϵbab will generate a trace, which is not consistent. Since the transformations act
+on the matter action Sm as a diffeomorphism with vector ϵca, it is obvious that it is invariant
+by itself. It is easy to show that the transformations (3.50) leave the total action invariant
+in (3.48). The invariance of the measure is given in [193].
+Remark 3.3 (BRST transformations with Weyl ghost) One can also consider the ac-
+tion (2.153) with the Weyl ghost.
+In this case, the transformation law of the metric is
+modified and the Weyl ghost transforms as a scalar:
+δϵgab = iϵ Lcgab + iϵ gabcw,
+δϵcw = iϵ Lccw.
+(3.51)
+The second term in δϵgab is a Weyl transformation with parameter ϵcw. Moreover, bab and
+Bab are not symmetric traceless.
+The equation of motion for the auxiliary field is
+Bab = i Tab := i
+�
+T m
+ab + T gh
+ab
+�
+,
+(3.52)
+where the RHS is the total energy–momentum tensor (matter plus ghosts). Integrating it
+out imposes the gauge condition gab = ˆgab and yields the modified BRST transformations
+δϵΨ = iϵ LcΨ,
+δϵca = iϵ Lcca,
+δϵbab = iϵ Tab.
+(3.53)
+Without starting with the path integral (3.48) with auxiliary field, it would have been
+difficult to guess the transformation of the b ghost. Since ca is a vector, one can also write
+δϵca = ϵ cb∂bca.
+(3.54)
+68
+
+Associated to this symmetry is the BRST current ja
+B and the associated conserved BRST
+charge QB
+QB =
+�
+dσ j0
+B.
+(3.55)
+The charge is nilpotent
+Q2
+B = 0,
+(3.56)
+and, through the presence of the c-ghost in the BRST transformation, the BRST charge has
+ghost number one
+Ngh(QB) = 1.
+(3.57)
+Variations of the matter fields can be written as
+δϵΨ = i [ϵQB, Ψ]±.
+(3.58)
+Note that the energy–momentum tensor is BRST exact
+Tab = [QB, bab].
+(3.59)
+3.2.2
+BRST cohomology and physical states
+Physical state |ψ⟩ are elements of the absolute cohomology of the BRST operator:
+|ψ⟩ ∈ H(QB) := ker QB
+Im QB
+,
+(3.60)
+or, more explicitly, closed but non-exact states:
+QB |ψ⟩ = 0,
+∄ |χ⟩ : |ψ⟩ = QB |χ⟩ .
+(3.61)
+The adjective “absolute” is used to distinguish it from two other cohomologies (relative
+and semi-relative) defined below. Two states of the cohomology differing by an exact state
+represent identical physical states:
+|ψ⟩ ∼ |ψ⟩ + QB |Λ⟩ .
+(3.62)
+This equivalence relation, translated in terms of spacetime fields, correspond to spacetime
+gauge transformations. In particular, it contains the (linearized) reparametrization invari-
+ance of the spacetime metric in the closed string sector, and, for the open string sector, it
+contains Yang–Mills symmetries. We will find that it corresponds to the gauge invariance
+of free string field theory (Chapter 10).
+However, physical states satisfy two additional constraints (remember that bab is traceless
+symmetric):
+�
+dσ bab |ψ⟩ = 0.
+(3.63)
+These conditions are central to string (field) theory, so they will appear regularly in this
+book. For this reason, it is useful to provide first some general motivations, and to refine
+the analysis later since the CFT language will be more appropriate. Moreover, these two
+conditions will naturally emerge in string field theory.
+In order to introduce some additional terminology, let’s define the following quantities:5
+b+ :=
+�
+dσ b00,
+b− :=
+�
+dσ b01.
+(3.64)
+5The objects b± are zero-modes of the b ghost fields. They correspond (up to a possible irrelevant factor)
+to the modes b±
+0 in the CFT formulation of the ghost system (7.132).
+69
+
+The semi-relative and relative cohomologies H−(QB) and H0(QB) are defined as6
+H−(QB) = H(QB) ∩ ker b−,
+H0(QB) = H−(QB) ∩ ker b+.
+(3.65)
+The first constraint arises as a consequence of the topology of the closed string worldsheet:
+the spatial direction is a circle, which implies that the theory must be invariant under
+translations along the σ direction (the circle is invariant under rotation). However, choosing
+a parametrization implies to fix an origin for the spatial direction: this is equivalent to
+a gauge fixing condition.
+As usual, this implies that the corresponding generator Pσ of
+worldsheet spatial translations (2.26) must annihilate the states:
+Pσ |ψ⟩ = 0.
+(3.66)
+This is called the level-matching condition. Using (3.59), this can be rewritten as
+Pσ |ψ⟩ =
+�
+dσ T01 |ψ⟩ =
+�
+dσ {QB, b01} |ψ⟩ = QB
+�
+dσ b01 |ψ⟩ ,
+(3.67)
+since QB |ψ⟩ = 0 for a state |ψ⟩ in the cohomology.
+The simplest way to enforce this
+condition is to set the state on which QB acts to zero:7
+b− |ψ⟩ = 0,
+(3.68)
+which is equivalent to one of the conditions in (3.63).
+The second condition does not follow as simply. The Hilbert space can be decomposed
+according to b+ as
+H− := H↓ ⊕ H↑,
+H↓ := H0 := H− ∩ ker b+.
+(3.69)
+Indeed, b+ is a Grassmann variable and generates a 2-state system. In the ghost sector, the
+two Hilbert spaces are generated from the ghost vacua | ↓⟩ and | ↑⟩ obeying
+b+ | ↓⟩ = 0,
+b+ | ↑⟩ = | ↓⟩ .
+(3.70)
+The action of the BRST charge on states |ψ↓⟩ ∈ H↓ and |ψ↑⟩ ∈ H↑ follow from these relations
+and from the commutation relation (3.59):
+QB |ψ↓⟩ = H |ψ↑⟩ ,
+QB |ψ↑⟩ = 0,
+(3.71)
+where H is the worldsheet Hamiltonian defined in (2.26). To prove this relation, start first
+with H |ψ↑⟩, then use (3.59)) to get the LHS of the first condition; then apply QB to get
+the second condition (using that QB commutes with H, and b+ with any other operators
+building the states). For H ̸= 0, the state |ψ↓⟩ is not in the cohomology and |ψ↑⟩ is exact.
+Thus, the exact and closed states are
+Im QB =
+�
+|ψ↑⟩ ∈ H↑ | H |ψ↑⟩ ̸= 0
+�
+,
+(3.72a)
+ker QB =
+�
+|ψ↑⟩ ∈ H↑
+�
+∪
+�
+|ψ↓⟩ ∈ H↓ | H |ψ↓⟩ = 0
+�
+.
+(3.72b)
+This implies that eigenstates of H in the cohomology satisfy the on-shell condition:
+H |ψ⟩ = 0.
+(3.73)
+6The BRST cohomologies described in this section are slightly different from the ones used in the rest
+of this book.
+To distinguish them, indices are written as superscripts in this section, and as subscripts
+otherwise.
+7The reverse is not true. We will see in Section 3.2.2 the relation between the two conditions in more
+details.
+70
+
+This is consistent with the fact that scattering amplitudes involve on-shell states. In this
+case, |ψ↑⟩ is not exact and is thus a member of the cohomology H(QB), as well as |ψ↓⟩ since
+it becomes close. But, the Hilbert space H↑ must be rejected for two reasons: there would
+be an apparent doubling of states and scattering amplitudes would behave badly. The first
+problem arises because one can show that the cohomological subspaces of each space are
+isomorphic: H↓(QB) ≃ H↑(QB). Hence, keeping both subspaces would lead to a doubling
+of the physical states. For the second problem, consider an amplitude where one of the
+external state is built from |ψ↑⟩: the amplitude vanishes if the states are off-shell since the
+state |ψ↑⟩ is exact, but it does not vanish on-shell [193, ch. 4]. This means that it must
+be proportional to δ(H). But, general properties in QFT forbid such dependence in the
+amplitude (only poles and cuts are allowed, except if D = 2). Projecting out the states in
+H↑ is equivalent to require
+b+ |ψ⟩ = 0
+(3.74)
+for physical states, which is the second condition in (3.63).
+In fact, this condition can be obtained very similarly as the b− = 0 condition: using the
+expression of H (2.26) and the commutation relation (3.59), (3.73) is equivalent to
+QB
+�
+dσ b00 |ψ⟩ = 0.
+(3.75)
+Hence, imposing (3.74) allows to automatically ensure that (3.73) holds.
+Since the on-shell character (3.73) of the BRST states and of the BRST symmetry are
+intimately related to the construction of the worldsheet integral, one can expect difficulty
+for going off-shell.
+3.3
+Summary
+In this chapter, we derived general formulas for string scattering amplitudes. The general
+BRST formalism has been summarized. Moreover, we gave general motivations for restrict-
+ing the absolute cohomology to the smaller relative cohomology.
+In Chapter 8, a more
+precise derivation of the BRST cohomology is worked out. It includes also a proof of the
+no-ghost theorem: the ghosts and the negative norm states (in Minkowski signature) are
+unphysical particles and should not be part of the physical states. This theorem asserts that
+it is indeed the case. It will also be the occasion to recover the details of the spectrum in
+various cases.
+3.4
+Suggested readings
+• The delta function approach to the gauge fixing is described in [193, sec. 3.3, 151,
+sec. 15.3.2], with a more direct computation is in [128].
+• The most complete references for scattering amplitudes in the path integral formalism
+are [53, 193].
+• Computation of the tree-level 2-point amplitude [66, 208] (for discussions of 2-point
+function, see [53, p. 936–7, 207, 60, 61, 48, p. 863–4]).
+• The BRST quantization of string theory is discussed in [155, 39, 193, chap. 4]. For
+a general discussion see [105, 247, 251]. The use of an auxiliary field is considered
+in [252, sec. 3.2].
+71
+
+Chapter 4
+Worldsheet path integral:
+complex coordinates
+Abstract
+In the two previous chapters, the amplitudes computed from the worldsheet
+path integrals have been written covariantly for a generic curved background metric. In
+this chapter, we start to use complex coordinates and finally take the background metric to
+be flat. This is the usual starting point for computing amplitudes since it allows to make
+contact with CFTs and to employ tools from complex analysis. We first recall few facts on
+2d complex manifolds before briefly describing how to rewrite the scattering amplitudes in
+complex coordinates.
+4.1
+Geometry of complex manifolds
+Choosing a flat background metric simplifies the computations. However, we have seen in
+Section 2.3 that there is a topological obstruction to get a globally flat metric. The solution
+is to work with coordinate patches (σ0, σ1) = (τ, σ) such that the background metric ˆgab is
+flat in each patch (conformally flat gauge):
+ds2 = gabdσadσb = e2φ(τ,σ)�
+dτ 2 + dσ2�
+,
+(4.1)
+or
+gab = e2φδab,
+ˆgab = δab.
+(4.2)
+To simplify the notations, we remove the dependence in the flat metric and the hat for
+quantities (like the vertex operators) expressed in the background metric when no confusion
+is possible.
+Introducing complex coordinates
+z = τ + iσ,
+¯z = τ − iσ,
+(4.3a)
+τ = z + ¯z
+2
+,
+σ = z − ¯z
+2i
+,
+(4.3b)
+the metric reads1
+ds2 = 2gz¯zdzd¯z = e2φ(z,¯z)|dz|2.
+(4.4)
+1In Section 6.1, we provide more details on the relation between the worldsheet (viewed as a cylinder or
+a sphere) and the complex plane.
+72
+
+The metric and its inverse can also be written in components:
+gz¯z = e2φ
+2 ,
+gzz = g¯z¯z = 0,
+(4.5a)
+gz¯z = 2e−2φ,
+gzz = g¯z¯z = 0.
+(4.5b)
+Equivalently, the non-zero components of the background metric are
+ˆgz¯z = 1
+2,
+ˆgz¯z = 2.
+(4.6)
+An oriented two-dimensional manifold is a complex manifold: this means that there exists
+a complex structure, such that the transition functions and changes of coordinates between
+different patches are holomorphic at the intersection of the two patches:
+w = w(z),
+¯w = ¯w(¯z).
+(4.7)
+For such a transformation, the Liouville mode transforms as
+e2φ(z,¯z) =
+����
+∂w
+∂z
+����
+2
+e2φ(w, ¯
+w)
+(4.8)
+such that
+ds2 = e2φ(w, ¯
+w)|dw|2.
+(4.9)
+This shows also that a conformal structure (2.12) induces a complex structure since the
+transformation law of φ is equivalent to a Weyl rescaling.
+The integration measures are related as
+d2σ := dτdσ = 1
+2 d2z,
+d2z := dzd¯z.
+(4.10)
+Due to the factor of 2 in the expression, the delta function δ(2)(z) also gets a factor of 2
+with respect to δ(2)(σ)
+δ(2)(z) = 1
+2 δ(2)(σ).
+(4.11)
+Then, one can check that
+�
+d2z δ(2)(z) =
+�
+d2σ δ(2)(σ) = 1.
+(4.12)
+The basis vectors (derivatives) and one-forms can be found using the chain rule:
+∂z = 1
+2 (∂τ − i∂σ),
+∂¯z = 1
+2 (∂τ + i∂σ),
+(4.13a)
+dz = dτ + idσ,
+d¯z = dτ − idσ.
+(4.13b)
+The Levi–Civita (completely antisymmetric) tensor is normalized by
+ϵ01 = ϵ01 = 1.
+(4.14a)
+ϵz¯z = i
+2,
+ϵz¯z = −2i,
+(4.14b)
+remembering that it transforms as a density. Integer indices run over local frame coordinates.
+The different tensors can be found from the tensor transformation law. For example, the
+components of a vector V a in both systems are related by
+V z = V 0 + iV 1,
+V ¯z = V 0 − iV 1
+(4.15)
+73
+
+such that
+V = V 0∂0 + V 1∂1 = V z∂z + V ¯z∂¯z.
+(4.16)
+For holomorphic coordinate transformations (4.7), the components of the vector do not mix:
+V w = ∂w
+∂z V z,
+V ¯
+w = ∂ ¯w
+∂¯z V ¯z.
+(4.17)
+This implies that the tangent space of the Riemann surface is decomposed into holomorphic
+and anti-holomorphic vectors:2
+TΣg ≃ TΣ+
+g ⊕ TΣ−
+g ,
+(4.18a)
+V z∂z ∈ TΣ+
+g ,
+V ¯z∂¯z ∈ TΣ−
+g ,
+(4.18b)
+as a consequence of the existence of a complex structure. Similarly, the components of a
+1-form ω – which is the only non-trivial form on Σg – can be written in terms of the real
+coordinates as:
+ωz = 1
+2 (ω0 − iω1),
+ω¯z = 1
+2 (ω0 + iω1)
+(4.19)
+such that
+ω = ω0dσ0 + ω1dσ1 = ωzdz + ω¯zd¯z.
+(4.20)
+Hence, a 1-form is decomposed into complex (1, 0)- and (0, 1)-forms:
+T ∗Σg ≃ Ω1,0(Σg) ⊕ Ω0,1(Σg),
+(4.21a)
+ωzdz ∈ Ω1,0(Σg),
+ω¯zd¯z ∈ Ω0,1(Σg),
+(4.21b)
+since both components will not mixed under holomorphic changes of coordinates (4.7). Fi-
+nally, the metric provides an isomorphism between TΣ+
+g and Ω0,1(Σg), and between TΣ−
+g
+and Ω1,0(Σg), since it can be used to lower/raise an index while converting it from holo-
+morphic to anti-holomorphic, or conversely:
+Vz = gz¯zV ¯z,
+V¯z = gz¯zV z.
+(4.22)
+This can be generalized further by considering components with more indices: all anti-
+holomorphic indices can be converted to holomorphic indices thanks to the metric:
+T
+q++p−
+����
+z···z
+z···z
+����
+p++q−
+= (gz¯z)p−(gz¯z)q−T
+q+
+����
+z···z
+q−
+����
+¯z···¯z
+z···z
+����
+p+
+¯z···¯z
+����
+p−
+.
+(4.23)
+Hence, it is sufficient to study (p, q)-tensors with p upper and q lower holomorphic indices.
+In this case, the transformation rule under (4.7) reads
+T
+q
+����
+w···w
+w···w
+����
+p
+=
+�∂w
+∂z
+�n
+T
+q
+����
+z···z
+z···z
+����
+p
+,
+n := q − p.
+(4.24)
+The number n ∈ Z is called the helicity or rank.3 The set of helicity-n tensors is denoted
+by T n.
+The first example is vectors (or equivalently 1-forms): V z ∈ T 1, Vz ∈ T −1. The second
+most useful case is traceless symmetric tensors, which are elements of T ±2. Consider a
+2However, at this stage, each component can still depend on both z and ¯z: V z = V z(z, ¯z) and V ¯z =
+V ¯z(z, ¯z).
+3In fact, it is even possible to consider n ∈ Z + 1/2 to describe spinors.
+74
+
+traceless symmetric tensor T ab = T ba and gabT ab = 0: this implies T 01 = T 10 and T 00 =
+−T 11 in real coordinates. The components in complex coordinates are:
+T zz = 2(T 00 + iT 01) ∈ T 2,
+T ¯z¯z = 2(T 00 − iT 01) ∈ T −2,
+T z
+z = 0.
+(4.25)
+Note that
+Tzz = gz¯zgz¯zT ¯z¯z = 1
+2(T 00 − iT 01),
+(4.26)
+and T z
+z = gz¯zT z¯z ∈ T 0 corresponds to the trace.
+Computation – Equation (4.25)
+T zz =
+�∂z
+∂τ
+�2
+T 00 +
+� ∂z
+∂σ
+�2
+T 11 + 2 ∂z
+∂τ
+∂z
+∂σ T 01 = T 00 − T 11 + 2i T 01.
+Stokes’ theorem in complex coordinates follows directly from (B.10):
+�
+d2z (∂zvz + ∂¯zv¯z) = −i
+� �
+dz v¯z − d¯zvz�
+= −2i
+�
+∂R
+(vzdz − v¯zd¯z),
+(4.27)
+where the integration contour is anti-clockwise. To obtain this formula, note that d2x = 1
+2d2z
+and ϵz¯z = i/2, such that the factor 1/2 cancels between both sides.
+4.2
+Complex representation of path integral
+In the previous section, we have found that tensors of a given rank are naturally decomposed
+into different subspaces thanks to the complex structure of the manifold.
+Accordingly,
+complex coordinates are natural and one can expect most objects in string theory to split
+similarly into holomorphic and anti-holomorphic sectors (or left- and right-moving). This
+will be particularly clear using the CFT language (Chapter 6). The main difficulty for this
+program is due to the matter zero-modes. In this section, we focus on the path integral
+measure and expression of the ghosts.
+There is, however, a subtlety in displaying explicitly the factorization: the notion of
+“holomorphicity” depends on the metric (because the complex structure must be compatible
+with the metric for an Hermitian manifold). Since the metric depends on the moduli which
+are integrated over in the path integral, it is not clear that there is a consistent holomorphic
+factorization. We will not push the question of achieving a global factorization further (but
+see Remark 4.1) to focus instead on the integrand. The latter is local (in moduli space) and
+there is no ambiguity.
+The results of the previous section indicate that the basis of Killing vectors (2.104) and
+quadratic differentials (2.76) split into holomorphic and anti-holomorphic components:
+ψi(z, ¯z) = ψz
+i ∂z + ψ¯z
+i ∂¯z,
+φi(z, ¯z) = φi,zz(dz)2 + φi,¯z¯z(d¯z)2.
+(4.28)
+Similarly, the operators P1 (2.65a) and P †
+1 (2.71) also split:
+(P1ξ)zz = 2∇zξz = ∂zξ¯z,
+(P1ξ)¯z¯z = 2∇¯zξ¯z = ∂¯zξz,
+(4.29a)
+(P †
+1 T)z = −2∇zTzz = −4 ∂¯zTzz,
+(P †
+1 T)¯z = −2∇¯zT¯z¯z = −4 ∂zT¯z¯z
+(4.29b)
+for arbitrary vector ξ and traceless symmetric tensor T (in the background metric). As a
+consequence, the components of Killing vectors and quadratic differentials are holomorphic
+or anti-holomorphic as a function of z:
+ψz = ψz(z),
+ψ¯z = ψ¯z(¯z),
+φzz = φzz(z),
+φ¯z¯z = φ¯z¯z(¯z),
+(4.30)
+75
+
+such that it makes sense to consider a complex basis instead of the previous real basis:
+ker P1 = Span{ψK(z)} ⊕ Span{ ¯ψK(¯z)},
+K = 1, . . . , Kc
+g,
+(4.31a)
+ker P †
+1 = Span{φI(z)} ⊕ Span{¯φI(¯z)},
+I = 1, . . . , Mc
+g.
+(4.31b)
+The last equation can inspire to search for a similar rewriting of the moduli parameters.
+In fact, the moduli space itself is a complex manifold and can be endowed with complex
+coordinates [173, 193]:
+mI = t2I−1 + it2I,
+¯mI = t2I−1 − it2I,
+I = 1, . . . , Mc
+g
+(4.32)
+with the integration measure
+dMgt = d2Mc
+gm.
+(4.33)
+The last ingredient to rewrite the vacuum amplitudes (2.136) is to obtain the determin-
+ants. The inner-products of vector and traceless symmetric fields also factorize:
+(T1, T2) = 2
+�
+d2σ
+�
+ˆg ˆgacgbdT1,abT2,cd = 4
+�
+d2z
+�
+T1,zzT2,¯z¯z + T1,¯z¯zT2,zz
+�
+,
+(4.34a)
+(ξ1, ξ2) =
+�
+d2σ
+�
+ˆg ˆgabξaξb = 1
+4
+�
+d2z
+�
+ξz
+1ξ¯z
+2 + ξ¯z
+1ξz
+2
+�
+.
+(4.34b)
+All inner-products are evaluated in the flat background metric. For (anti-)holomorphic fields,
+only one term survives in each integral: since each field appears twice in the determinants
+(φi, φj) and (φi, φj), the final expression is a square, which cancels against the squareroot
+in (2.136). The remaining determinant involves the Beltrami differential (2.65b):
+µizz = ∂i¯gzz,
+µi¯z¯z = ∂i¯g¯z¯z
+(4.35)
+(¯gzz = 0 in our coordinates system, but its variation under a shift of moduli is not zero).
+The basis can be changed to a complex basis such that the determinant of inner-products
+between Beltrami and quadratic differentials is a modulus squared. All together, the different
+formulas lead to the following rewriting of the vacuum amplitude :
+Zg =
+�
+Mg
+d2Mc
+gm | det(φI, µJ)|2
+| det(φI, ¯φJ)|
+det′ P †
+1 P1
+| det(ψI, ¯ψJ)|
+Zm[δ]
+Ωckv[δ],
+(4.36)
+where the absolute values are to be understood with respect to the basis of P1 and P †
+1 , for
+example |f(mI)|2 := f(mI)f( ¯mI).
+The same reasoning can be applied to the ghosts. The c and b ghosts are respectively
+a vector and a symmetric traceless tensor, both with two independent components: it is
+customary to define
+c := cz,
+¯c := c¯z,
+b := bzz,
+¯b := b¯z¯z.
+(4.37)
+In that case, the action (2.145) reads
+Sgh[g, b, c] = 1
+2π
+�
+d2z
+�
+b∂¯zc + ¯b∂z¯c
+�
+.
+(4.38)
+The action is the sum of two holomorphic and anti-holomorphic contributions and it is
+independent of φ(z, ¯z) as expected. In fact, the equations of motion are
+∂zc = 0,
+∂zb = 0,
+∂¯z¯c = 0,
+∂¯z¯b = 0,
+(4.39)
+76
+
+such that b and c (resp. ¯b and ¯c) are holomorphic (anti-holomorphic) functions. Then, the
+integration measure is simply
+Mg
+�
+i=1
+Bi dti =
+Mc
+g
+�
+I=1
+BI ¯BI dmI ∧ ¯mI,
+BI := (µI, b).
+(4.40)
+Note that BI does not contain ¯b(¯z), it is built only from b(z).
+Finally, the vacuum amplitude (2.163) reads
+Zg =
+�
+Mg
+d2Mc
+gm
+Ωckv[δ]−1
+| det ψI(z0
+j )|2
+�
+d(b,¯b) d(c, ¯c)
+Kc
+g
+�
+j=1
+c(z0
+j )¯c(¯z0
+j )
+Mc
+g
+�
+I=1
+|(µI, b)|2 e−Sgh[b,c] Zm[δ].
+(4.41)
+The c insertions are separated in holomorphic and anti-holomorphic components because,
+at the end, only the zero-modes contribute. The measures are written as d(b,¯b) and d(c, ¯c)
+because proving that they factorize is difficult (Remark 4.1).
+Remark 4.1 (Holomorphic factorization) It was proven in [15, 27, 33] (see [173, sec. 9,
+53, sec. VII, 237, sec. 3] for reviews) that the ghost and matter path integrals can be globally
+factorized, up to a factor due to zero-modes. Such a result is suggested by the factorization
+of the inner-products, which imply a factorization of the measures: the caveat is due to the
+zero-mode determinants and matter measure. Interestingly, the factorization is possible only
+in the critical dimension (2.125).
+4.3
+Summary
+In this chapter, we have introduced complex notations for the fields, path integral and
+moduli space.
+4.4
+Suggested readings
+• Good references for this chapter are [24, 53, 172, 173, 193].
+• Geometry of complex manifolds is discussed in [24, sec. 6.2, 172, chap. 14, 53].
+77
+
+Chapter 5
+Conformal symmetry in D
+dimensions
+Abstract
+Starting with this chapter, we discuss general properties of conformal field the-
+ories (CFT). The goal is not to be exhaustive, but to provide a short introduction and to
+gather the concepts and formulas that are needed for string theory. However, the subject is
+presented as a standalone topic such that it can be of interest for a more general public.
+The conformal group in any dimension is introduced in this chapter. The specific case
+D = 2, which is the most relevant for the current book, is developed in the following chapters.
+5.1
+CFT on a general manifold
+In this chapter and in the next one, we discuss CFTs as QFTs living on a spacetime M,
+independently from string theory (there is no reference to a target spacetime). As such, we
+will use spacetime notations together with some simplifications: coordinates are written as
+xµ with µ = 0, . . . , D − 1 and time is written as x0 = t (x0 = τ) in Lorentzian (Euclidean)
+signature.
+Given a metric gµν on a D-dimensional manifold M, the conformal group CISO(M) is
+the set of coordinate transformations (called conformal symmetries or isometries)
+xµ −→ x′µ = x′µ(x)
+(5.1)
+which leaves the metric invariant up to an overall scaling factor:
+gµν(x) −→ g′
+µν(x′) = ∂xρ
+∂x′µ
+∂xσ
+∂x′ν gρσ(x) = Ω(x′)2gµν(x′).
+(5.2)
+This means that angles between two vectors u and v are left invariant under the transform-
+ation:
+u · v
+|u| |v| = u′ · v′
+|u′| |v′|.
+(5.3)
+It is often convenient to parametrize the scale factor by an exponential
+Ω := eω.
+(5.4)
+Considering an infinitesimal transformation
+δxµ = ξµ,
+(5.5)
+78
+
+the condition (5.2) becomes the conformal Killing equation
+δgµν = Lξgµν = ∇µξν + ∇νξµ = 2
+d gµν∇ρξρ,
+(5.6)
+such that the scale factor is
+Ω2 = 1 + 2
+d ∇ρξρ.
+(5.7)
+The vector fields ξ satisfying this equation are called conformal Killing vectors (CKV). Con-
+formal transformations form a global subgroup of the diffeomorphism group: the generators
+of the transformations do depend on the coordinates, but the parameters do not (for an
+internal global symmetry, both the generators and the parameters don’t depend on the
+coordinates).
+The conformal group contains the isometry group ISO(M) of M as a subgroup, corres-
+ponding to the case Ω = 1:
+ISO(M) ⊂ CISO(M).
+(5.8)
+These transformations also preserve distances between points. The corresponding generators
+of infinitesimal transformations are called Killing vectors and satisfies the Killing equation
+δgµν = Lξgµν = ∇µξν + ∇νξµ = 0.
+(5.9)
+They form a subalgebra of the CKV algebra.
+An important point is to be made for the relation between infinitesimal and finite trans-
+formations: with spacetime symmetries it often happens that the first cannot be exponenti-
+ated into the second. The reason is that the (conformal) Killing vectors may be defined only
+locally, i.e. they are well-defined in a given domain but have singularities outside. When
+this happens, they do not lead to an invertible transformation, which cannot be an element
+of the group. These notions are sometimes confused in physics and the term of “group” is
+used instead of “algebra”. We shall be careful in distinguishing both concepts.
+Remark 5.1 (Isometries of M ⊂ Rp,q) In order to find the conformal isometries of a
+manifold M which is a subset of Rp,q defined in (5.10), it is sufficient to restrict the trans-
+formations of Rp,q to the subset M [206]. In the process, not all global transformations
+generically survive. On the other hand, the algebra of local (infinitesimal) transformations
+for M and Rp,q are identical since M is locally like Rp,q.
+5.2
+CFT on Minkowski space
+In this section, we consider the case where M = Rp,q (D = p + q) and where g = η is the
+flat metric with signature (p, q):
+η = diag(−1, . . . , −1
+�
+��
+�
+q
+, 1, . . . , 1
+� �� �
+p
+).
+(5.10)
+The conformal Killing equation becomes
+�
+ηµν∆ + (D − 2)∂µ∂ν
+�
+∂ · ϵ = 0,
+(5.11)
+where ∆ is the D-dimensional Beltrami–Laplace operator for the metric ηµν. The case D = 2
+is relegated to the next chapter. For D > 2, one finds the following transformations:
+translation:
+ξµ = aµ,
+(5.12a)
+rotation & boost:
+ξµ = ωµ
+νxν,
+(5.12b)
+dilatation:
+ξµ = λ xµ,
+(5.12c)
+SCT:
+ξµ = bµx2 − 2b · x xµ,
+(5.12d)
+79
+
+where ωµν is antisymmetric. The rotations include Lorentz transformations and SCT means
+“special conformal transformation”.
+All parameters {aµ, ωµν, λ, bµ} are constant. The generators are respectively denoted by
+{Pµ, Jµν, D, Kµ}. The finite translations and rotations form the Poincaré group SO(p, q),
+while the conformal group can be shown to be SO(p + 1, q + 1):
+ISO(Rp,q) = SO(p, q),
+CISO(Rp,q) = SO(p + 1, q + 1).
+(5.13)
+The dimension of this group is
+dim SO(p + 1, q + 1) = 1
+2 (p + q + 2)(p + q + 1).
+(5.14)
+5.3
+Suggested readings
+• References on higher-dimensional CFTs are [54, 196, 203, 206, 236].
+80
+
+Chapter 6
+Conformal field theory on the
+plane
+Abstract
+Starting with this chapter, we focus on two-dimensional Euclidean CFTs on the
+complex plane (or equivalently the sphere). We start by describing the geometry of the
+sphere and the relation to the complex plane and to the cylinder, in order to make contact
+with the string worldsheet. Then, we discuss classical CFTs and the Witt algebra obtained
+by classifying the conformal isometries of the complex plane. Then, we describe quantum
+CFTs and introduce the operator formalism. This last section is the most important for this
+book as it includes information on the operator product expansion, Hilbert space, Hermitian
+and BPZ conjugations.
+As described at the beginning of Chapter 5, we use spacetime notations for the coordin-
+ates, but follow otherwise the normalization for the worldsheet. In particular, integrals are
+normalized by 2π. However, the spatial coordinate on the cylinder is still written as σ to
+avoid confusions: xµ = (τ, σ).
+6.1
+The Riemann sphere
+6.1.1
+Map to the complex plane
+The Riemann sphere Σ0, which is diffeomorphic to the unit sphere S2, has genus g = 0 and
+is thus the simplest Riemann surface. Its most straightforward description is obtained by
+mapping it to the extended1 complex plane ¯C (also denoted ˆC), which is the complex plane
+z ∈ C to which the point at infinity z = ∞ is added:
+¯C = C ∪ {∞}.
+(6.1)
+One speaks about “the point at infinity” because all the points at infinity (i.e. the points z
+such that |z| → ∞)
+lim
+r→∞ r eiθ := ∞
+(6.2)
+are identified (the limit is independent of θ).
+The identification can be understood by mapping (say) the south pole to the origin of
+the plane and the north pole to infinity2 (Figure 6.1) through the stereographic projection
+z = eiφ cot θ
+2,
+(6.3)
+1This qualification will often be omitted.
+2Note that the points are distinguished in order to write the map, but they have nothing special by
+themselves (i.e. they are not punctures).
+81
+
+−−−−−−−−→
+Figure 6.1: Map from the Riemann sphere to the complex plane. The south and north poles
+are denoted by the letter S and N, and the equatorial circle by E.
+where (θ, φ) are angles on the sphere. Any circle on the sphere is mapped to a circle in the
+complex plane. Conversely, the Riemann sphere can be viewed as a compactification of the
+complex plane.
+Introducing Cartesian coordinates (x, y) related to the complex coordinates by3
+z = x + iy,
+¯z = x − iy,
+(6.4a)
+x = z + ¯z
+2
+,
+y = z − ¯z
+2i
+,
+(6.4b)
+the metric reads
+ds2 = dx2 + dy2 = dzd¯z.
+(6.5)
+The relations between the derivatives in the two coordinate systems are easily found:
+∂ := ∂z = 1
+2 (∂x − i∂y),
+¯∂ := ∂¯z = 1
+2 (∂x + i∂y).
+(6.6)
+The indexed form will be used when there is a risk of confusion. If the index is omitted then
+the derivative acts directly to the field next to it, for example
+∂φ(z1)∂φ(z2) := ∂z1∂z2φ(z1)φ(z2).
+(6.7)
+Generically, the meromorphic and anti-meromorphic parts of a object will be denoted
+without and with a bar, see (6.55) for an example.
+The extended complex plane ¯C can be covered by two coordinate patches z ∈ C and
+w ∈ C. In the first, the point at infinity (north pole) is removed, in the second, the origin
+(south pole) is removed. On the overlap, the transition function is
+w = 1
+z .
+(6.8)
+This description avoids to work with the infinity: studying the behaviour of f(z) at z = ∞
+is equivalent to study f(1/w) at w = 0.
+Since any two-dimensional metric is locally conformally equivalent to the flat metric, it
+is sufficient to work with this metric in each patch. This is particularly convenient for the
+Riemann sphere since one patch covers it completely except for one point.
+3General formulas can be found in Section 4.1 by replacing (τ, σ) with (x, y). In most cases, the conformal
+factor is set to zero (φ = 0) in this chapter.
+82
+
+6.1.2
+Relation to the cylinder – string theory
+The worldsheet of a closed string propagating in spacetime is locally topologically a cylinder
+R × S1 of circumference L. In this section, we show that the cylinder can also be mapped
+to the complex plane – and thus to the Riemann sphere – after removing two points. Since
+the cylinder has a clear physical interpretation in string theory, it is useful to know how to
+translate the results from the plane to the cylinder.
+It makes also sense to define two-dimensional models on the cylinder independently of
+a string theory interpretation since the compactification of the spatial direction from R to
+S1 regulates the infrared divergences. Moreover, it leads to a natural definition of a “time”
+and of an Hamiltonian on the Euclidean plane.
+Denoting the worldsheet coordinates in Lorentzian signature by (t, σ) with4
+t ∈ R,
+σ ∈ [0, L),
+σ ∼ σ + L,
+(6.9)
+the metric reads
+ds2 = −dt2 + dσ2 = −dσ+dσ−,
+(6.10)
+where the light-cone coordinates
+dσ± = dt ± dσ
+(6.11)
+have been introduced. It is natural to perform a Wick rotation from the Lorentzian time t
+to the Euclidean time
+τ = it,
+(6.12)
+and the metric becomes
+ds2 = dτ 2 + dσ2.
+(6.13)
+It is convenient to introduce the complex coordinates
+w = τ + iσ,
+¯w = τ − iσ
+(6.14)
+for which the metric is
+ds2 = dwd ¯w.
+(6.15)
+Note that the relation to Lorentzian light-cone coordinates are
+w = i(t + σ) = iσ+,
+¯w = i(t − σ) = iσ−.
+(6.16)
+Hence, an (anti-)holomorphic function of w ( ¯w) depends only on σ+ (σ−) before the Wick
+rotation: this leads to the identification of the left- and right-moving sectors with the holo-
+morphic and anti-holomorphic sectors of the theory.
+The cylinder can be mapped to the complex plane through
+z = e2πw/L,
+¯z = e2π ¯
+w/L,
+(6.17)
+and the corresponding metric is
+ds2 =
+� L
+2π
+�2 dzd¯z
+|z|2 .
+(6.18)
+A conformal transformation brings this metric to the flat metric (6.5). The conventions
+for the various coordinates and maps vary in the different textbooks. We have gathered in
+Table A.1 the three main conventions and which references use which.
+4Consistently with the comments at the beginning of Chapter 5, the Lorentzian worldsheet time is
+denoted by t instead of τM.
+83
+
+−−−−−→
+−−−−−→
+Figure 6.2: Map from the cylinder to the sphere with two tubes, to the 2-punctured sphere
+Σ0,2.
+The map from the cylinder to the plane is found by sending the bottom end (corres-
+ponding to the infinite past t → −∞) to the origin of the plane, and the top end (infinite
+future t → ∞) to the infinity. Since the cylinder has two boundaries (its two ends) the map
+excludes the point z = 0 and z = ∞ and one really obtains the space ¯C − {0, ∞} = C∗.
+This space can, in turn, be mapped to the 2-punctured Riemann sphere Σ0,2.
+The physical interpretation for the difference between Σ0 and Σ0,2 is simple: since one
+considers the propagation of a string, it means that the worldsheet corresponds to an amp-
+litude with two external states, which are the mapped to the sphere as punctures (Figure 6.2,
+Section 3.1.1). Removing the external states (yielding the tree-level vacuum amplitude) cor-
+responds to gluing half-sphere (caps) at each end of the cylinder (Figure 6.3). Then, it can
+be mapped to the Riemann sphere without punctures. As a consequence, the properties of
+tree-level string theory are found by studying the matter and ghost CFTs on the Riemann
+sphere. Scattering amplitudes are computed through correlation functions of appropriate op-
+erators on the sphere. This picture generalizes to higher-genus Riemann surfaces. Moreover,
+since local properties of the CFT (e.g. the spectrum of operators) are determined by the
+conformal algebra, they will be common to all surfaces.
+Mathematically, a difference between Σ0 and Σ0,2 had to be expected since the sphere
+has a positive curvature (and χ = −2) but the cylinder is flat (with χ = 0). Punctures
+contribute negatively to the curvature (and thus positively to the Euler characteristics).
+Remark 6.1 The coordinate z is always used as a coordinate on the complex plane, but the
+corresponding metric may be different – compare (6.5) and (6.18). As explained previously,
+this does not matter since the theory is insensitive to the conformal factor.
+6.2
+Classical CFTs
+In this section, we consider an action S[Ψ] which is conformally invariant. We first identify
+and discuss the properties of the conformal algebra and group, before explaining how a CFT
+is defined.
+84
+
+−−−−−−−−→
+Figure 6.3: Map from the cylinder with two caps (half-spheres) to the Riemann sphere Σ0.
+6.2.1
+Witt conformal algebra
+Since the Riemann sphere is identified with the complex plane, they share the same conformal
+group and algebra. Consider the metric (6.5)
+ds2 = dzd¯z,
+(6.19)
+then, any meromorphic change of coordinates
+z −→ z′ = f(z),
+¯z −→ ¯z′ = ¯f(¯z)
+(6.20)
+is a conformal transformation since the metric becomes
+ds2 = dz′d¯z′ =
+����
+df
+dz
+����
+2
+dzd¯z.
+(6.21)
+However, only holomorphic functions which are globally defined on ¯C are elements of
+the group. At the algebra level, any holomorphic function f(z) regular in a domain D gives
+a well-defined transformation in this domain D. Hence, the algebra is infinite-dimensional.
+On the other hand, f(z) is only meromorphic on C generically: it cannot be exponentiated
+to a group element. We first characterize the algebra and then obtain the conditions to
+promote the local transformations to global ones.
+Since the transformations are defined only locally, it is sufficient to consider an infinites-
+imal transformation
+δz = v(z),
+δ¯z = ¯v(¯z),
+(6.22)
+where v(z) is a meromorphic vector field on the Riemann sphere. Indeed, the conformal
+Killing equation (5.6) in D = 2 is equivalent to the Cauchy–Riemann equations:
+¯∂v = 0,
+∂¯v = 0.
+(6.23)
+The vector field admits a Laurent series
+v(z) =
+�
+n∈Z
+vnzn+1,
+¯v(¯z) =
+�
+n∈Z
+¯vn¯zn+1,
+(6.24)
+and the vn and ¯vn are to be interpreted as the parameters of the transformation. A basis of
+vectors (generators) is:
+ℓn = −zn+1∂z,
+¯ℓn = −¯zn+1∂¯z,
+n ∈ Z.
+(6.25)
+One can check that each set of generators satisfies the Witt algebra
+[ℓm, ℓn] = (m − n)ℓm+n,
+[¯ℓm, ¯ℓn] = (m − n)¯ℓm+n,
+[ℓm, ¯ℓn] = 0.
+(6.26)
+85
+
+Since there are two commuting copies of the Witt algebra, it is natural to extend the
+ranges of the coordinates from C to C2 and to consider z and ¯z as independent variables.
+In particular, this gives a natural action of the product algebra over C2. This procedure
+will be further motivated when studying CFTs since the holomorphic and anti-holomorphic
+parts will generally split, and it makes sense to study them separately. Ultimately, phys-
+ical quantities can be extracted by imposing the condition ¯z = z∗ at the end (the star is
+always reserved for the complex conjugation, the bar will generically denote an independent
+variable). In that case, the two algebras are also related by complex conjugation.
+Note that the variation of the metric (B.6) under a meromorphic change of coordinates
+(6.22) becomes
+δgz¯z = ∂v + ¯∂¯v,
+δgzz = δg¯z¯z = 0.
+(6.27)
+6.2.2
+PSL(2, C) conformal group
+The next step is to determine the globally defined vectors and to study the associated group.
+First, the conditions for a vector v(z) to be well-defined at z = 0 are
+lim
+|z|→0 v(z) < ∞
+=⇒
+∀n < −1 :
+vn = 0.
+(6.28)
+The behaviour at z = ∞ can be investigated thanks to the map z = 1/w
+v(1/w) = dz
+dw
+�
+n
+vnw−n−1,
+(6.29)
+where the additional derivative arises because v is a vector. Then, the regularity conditions
+at z = ∞ are
+lim
+|z|→∞ v(z) = lim
+|w|→0
+dz
+dw v(1/w) = − lim
+|w|→0
+v(1/w)
+w2
+< ∞
+=⇒
+∀n > 1 :
+vn = 0.
+(6.30)
+As a result, the globally defined generators are
+{ℓ−1, ℓ0, ℓ1} ∪ {¯ℓ−1, ¯ℓ0, ¯ℓ1}
+(6.31)
+where
+ℓ−1 = −∂z,
+ℓ0 = −z∂z,
+ℓ1 = −z2∂z.
+(6.32)
+It is straightforward to check that they form two copies of the sl(2, C) algebra
+[ℓ0, ℓ±1] = ∓ℓ±1,
+[ℓ1, ℓ−1] = 2ℓ0.
+(6.33)
+The global conformal group is sometimes called Möbius group:
+PSL(2, C) := SL(2, C)/Z2 ∼ SO(3, 1),
+(6.34)
+where the additional division by Z2 is clearer when studying an explicit representation. It
+corresponds with ker P1 defined in (2.91):
+K0 = PSL(2, C).
+(6.35)
+A matrix representation of SL(2, C) is
+g =
+�a
+b
+c
+d
+�
+,
+a, b, c, d ∈ C,
+det g = ad − bc = 1,
+(6.36)
+86
+
+which shows that this group has six real parameters
+K0 := dim SL(2, C) = 6.
+(6.37)
+The associated transformation on the complex plane reads
+fg(z) = az + b
+cz + d.
+(6.38)
+The quotient by Z2 is required since changing the sign of all parameters does not change the
+transformation. These transformations have received different names: Möbius, projective,
+homographic, linear fractional transformations. . .
+Holomorphic vector fields are then of the form
+v(z) = β + 2αz + γz2,
+¯v(¯z) = ¯β + 2¯α¯z + ¯γ¯z2,
+(6.39)
+where
+a = 1 + α,
+b = β,
+c = −γ,
+d = 1 − α.
+(6.40)
+The finite transformations associated to (5.12) are:
+translation:
+fg(z) = z + a,
+a ∈ C,
+(6.41a)
+rotation:
+fg(z) = ζ z,
+|ζ| = 1,
+(6.41b)
+dilatation:
+fg(z) = λ z,
+λ ∈ R,
+(6.41c)
+SCT:
+fg(z) =
+z
+cz + 1,
+c ∈ C.
+(6.41d)
+Investigation leads to the following association between the generators and transformations:
+• translation: ℓ−1 and ¯ℓ−1;
+• dilatation (or radial translation): (ℓ0 + ¯ℓ0);
+• rotation (or angular translation): i(ℓ0 − ¯ℓ0);
+• special conformal transformation: ℓ1 and ¯ℓ1.
+The inversion defined by
+inversion:
+I+(z) := I(z) := 1
+z
+(6.42a)
+is not an element of SL(2, C). However, the inversion with a minus sign
+I−(z) := −I(z) = I(−z) = −1
+z
+(6.42b)
+is a SL(2, C) transformation.
+A useful transformation is the circular permutation of (0, 1, ∞):
+g∞,0,1(z) =
+1
+1 − z .
+(6.43)
+87
+
+6.2.3
+Definition of a CFT
+A CFT is characterized by its set of (composite) fields (also called operators) O(z, ¯z) which
+correspond to any local expression constructed from the fields Ψ appearing in the Lagrangian
+and of their derivatives.5 For example, in a scalar field theory, the simplest operators are of
+the form ∂mφn.
+Among the operators, two particular categories are distinguished according to their trans-
+formation laws:
+• primary operator:
+∀f meromorphic :
+O(z, ¯z) =
+�df
+dz
+�h �d ¯f
+d¯z
+�¯h
+O′�
+f(z), ¯f(¯z)
+�
+,
+(6.44)
+• quasi-primary (or SL(2, C) primary) operator:
+∀f ∈ PSL(2, C) :
+O(z, ¯z) =
+�df
+dz
+�h �d ¯f
+d¯z
+�¯h
+O′�
+f(z), ¯f(¯z)
+�
+.
+(6.45)
+The parameters (h, ¯h) are the conformal weights of the operator O (both are independent
+from each other), and combinations of them give the conformal dimension ∆ and spin s:
+∆ := h + ¯h,
+s := h − ¯h.
+(6.46)
+The conformal weights correspond to the charges of the operator under ℓ0 and ¯ℓ0. We will
+use “(h, ¯h) (quasi-)primary” as a synonym of “(quasi-)primary field with conformal weight
+(h, ¯h)”.
+Remark 6.2 (Complex conformal weights) While we consider h, ¯h ∈ R, and more spe-
+cifically h, ¯h ≥ 0 for a unitary theory (which is the case of string theory except for the re-
+parametrization ghosts), theories with h, ¯h ∈ C make perfectly sense. One example is the
+Liouville theory with complex central charge c ∈ C [200, 202] (central charges are defined
+below, see (6.58)).
+Primaries and quasi-primaries are hence operators which have nice transformations re-
+spectively under the algebra and group. Obviously, a primary is also a quasi-primary. These
+transformations are similar to those of a tensor with h holomorphic and ¯h anti-holomorphic
+indices (Section 4.1). Another point of view is that the object
+O(z, ¯z) dzhd¯z
+¯h
+(6.47)
+is invariant under local / global conformal transformations.
+The notation f ◦ O indicates the complete change of coordinates, including the tensor
+transformation law and the possible corrections if the operator is not primary.6 For a primary
+field, we have:
+f ◦ O(z, ¯z) := f ′(z)h ¯f ′(¯z)
+¯h O′�
+f(z), ¯f(¯z)
+�
+.
+(6.48)
+We stress that it does not correspond to function composition.
+Under an infinitesimal transformations
+δz = v(z),
+δ¯z = ¯v(¯z),
+(6.49)
+5Not all CFTs admit a Lagrangian description. But, since we are mostly interested in string theories
+defined from Polyakov’s path integral, it is sufficient to study CFTs with a Lagrangian.
+6In fact, one has f ◦ O := f∗O in the notations of Chapter 2.
+88
+
+a primary operator changes as
+δO(z, ¯z) = (h ∂v + v ∂)O(z, ¯z) + (¯h ¯∂¯v + ¯v ¯∂)O(z, ¯z).
+(6.50)
+The transformation of a non-primary field contains additional terms, see for example (6.89).
+Remark 6.3 (Higher-genus Riemann surfaces) According to Remark 5.1, all Riemann
+surfaces Σg share the same conformal algebra since locally they are all subsets of R2. On
+the other hand, one finds that no global transformations are defined for g > 1, and only the
+subgroup U(1) × U(1) survives for the torus.
+The most important operator in a CFT is the energy–momentum tensor Tµν, if it exists
+as a local operator. According to Section 2.1, this tensor is conserved and traceless
+∇νTµν = 0,
+gµνTµν = 0.
+(6.51)
+The traceless equation in components reads
+gµνTµν = 4 Tz¯z = Txx + Tyy = 0
+(6.52)
+which implies that the off-diagonal component vanishes in complex coordinates
+Tz¯z = 0.
+(6.53)
+Then, the conservation equation yields
+∂zT¯z¯z = 0,
+∂¯zTzz = 0,
+(6.54)
+such that the non-vanishing components Tzz and T¯z¯z are respectively holomorphic and anti-
+holomorphic. This motivates the introduction of the notations:
+T(z) := Tzz(z),
+¯T(¯z) := T¯z¯z(¯z).
+(6.55)
+This is an example of the factorization between the holomorphic and anti-holomorphic sec-
+tors.
+Currents are local objects and thus one expects to be able to write an infinite number of
+such currents associated to the Witt algebra. Applying the Noether procedure gives
+Jv(z) := J ¯z
+v (z) = −T(z)v(z),
+¯Jv(¯z) := Jz
+v (¯z) = − ¯T(¯z)¯v(¯z).
+(6.56)
+6.3
+Quantum CFTs
+The previous section was purely classical. The quantum theory is first defined through the
+path integral
+Z =
+�
+dΨ e−S[Ψ].
+(6.57)
+We will also develop an operator formalism. The latter is more general than the path integral
+and allows to work without reference to path integrals and Lagrangians. This is particularly
+fruitful as it extends the class of theories and parameter ranges (e.g. Remark 6.2) which can
+be studied.
+89
+
+6.3.1
+Virasoro algebra
+As discussed in Section 2.3.3, field measures in path integrals display a conformal anom-
+aly, meaning that they cannot be defined without introducing a scale. This anomaly can
+be traded for a gravitational anomaly by introducing counter-terms in the action [87, 95,
+sec. 3.2, 96, 101, 122, 129]. As a consequence, the Witt algebra (6.26) is modified to its
+central extension, the Virasoro algebra.7
+The generators in both sectors are denoted by
+{Ln} and {¯Ln} and are called Virasoro operators (or modes). The algebra is given by:
+[Lm, Ln] = (m − n)Lm+n + c
+12 m(m − 1)(m + 1)δm+n,
+(6.58a)
+[¯Lm, ¯Ln] = (m − n)¯Lm+n + ¯c
+12 m(m − 1)(m + 1)δm+n,
+(6.58b)
+[Lm, ¯Ln] = 0,
+[c, Lm] = 0,
+[¯c, ¯Lm] = 0,
+(6.58c)
+where c, ¯c ∈ C are the holomorphic and anti-holomorphic central charges. Consistency of
+the theory on a curved space implies ¯c = c, but there is otherwise no constraint on the
+plane [95, 246].
+The sl(2, C) subalgebra is not modified by the central extension. This means that states
+are still classified by eigenvalues of (h, ¯h) of (L0, ¯L0).
+Remark 6.4 In most models relevant for string theory, one finds that the central charges
+are real, c, ¯c ∈ R.
+Moreover, unitarity requires them to be positive c, ¯c > 0, and only
+reparametrization ghosts do not satisfy this condition. On the other hand, it makes perfect
+sense to discuss general CFTs for c, ¯c ∈ C (the Liouville theory is such an example [200,
+202]).
+6.3.2
+Correlation functions
+A n-point correlation function is defined by
+� n
+�
+i=1
+Oi(zi, ¯zi)
+�
+=
+�
+dΨ e−S[Ψ]
+n
+�
+i=1
+Oi(zi, ¯zi),
+(6.59)
+choosing a normalization such that ⟨1⟩ = 1. The path integral defines the time-ordered
+product (on the cylinder) of the corresponding operators.
+Invariance under global transformations leads to strong constraints on the correlation
+functions. For quasi-primary fields, they transform under SL(2, C) as
+� n
+�
+i=1
+Oi(zi, ¯zi)
+�
+=
+n
+�
+i=1
+�df
+dz (zi)
+�hi �df
+d¯z (¯zi)
+�¯hi
+×
+� n
+�
+i=1
+Oi
+�
+f(zi), ¯f(¯zi)
+�
+�
+.
+(6.60)
+Considering an infinitesimal variation (6.50) yields a differential equation for the n-point
+function
+δ
+� n
+�
+i=1
+Oi(zi, ¯zi)
+�
+=
+n
+�
+i=1
+�
+hi∂iv(zi) + v(zi)∂i + c.c.
+�
+� n
+�
+i=1
+Oi(zi, ¯zi)
+�
+= 0,
+(6.61)
+where ∂i := ∂zi and v is a vector (6.39) of sl(2, C). These equations are sufficient to determine
+7That the central charge in the Virasoro algebra indicates a diffeomorphism anomaly can be understood
+from the fact that
+90
+
+completely the forms of the 1-, 2- and 3-point functions of quasi-primaries:
+⟨Oi(zi, ¯zi)⟩ = δhi,0δ¯hi,0,
+(6.62a)
+⟨Oi(zi, ¯zi)Oj(zj, ¯zj)⟩ = δhi,hjδ¯hi,¯hj
+gij
+z2hi
+ij ¯z2¯hi
+ij
+,
+(6.62b)
+⟨Oi(zi, ¯zi)Oj(zj, ¯zj)Ok(zk, ¯zk)⟩ =
+Cijk
+zhi+hj−hk
+ij
+zhj+hk−hi
+jk
+zhi+hk−hj
+ki
+×
+1
+¯z
+¯hi+¯hj−¯hk
+ij
+¯z
+¯hj+¯hk−¯hi
+jk
+¯z
+¯hi+¯hk−¯hj
+ki
+,
+(6.62c)
+where we have defined
+zij = zi − zj.
+(6.63)
+The coefficients Cijk are called structure constants and the matrix gij defines a metric
+(Zamolodchikov metric) on the space of fields. The metric is often taken to be diagonal
+gij = δij, which amounts to use an orthonormal eigenbasis of L0 and ¯L0. The vanishing of
+the 1-point function of a non-primary quasi-primary holds only on the plane: for example
+the value on the cylinder can be non-zero since the map is not globally defined – see in
+particular (6.167).
+Remark 6.5 (Logarithmic CFTs) Logarithmic CFTs display a set of unusual proper-
+ties [84, 85, 90, 97, 130].
+In particular, the correlation functions are not of the form
+displayed above. The most striking feature of those theories is that the L0 operator is non-
+diagonalisable (but it can be set in the Jordan normal form).
+Remark 6.6 (Fake identity) Usually, the only primary operator with h = ¯h = 0 is the
+identity 1. While this is always true for unitary theories, there are non-unitary theories
+(c ≤ 1 Liouville theory, SLE, loop models) where there is another field (called the indicator,
+marking operator, or also fake identity) with h = ¯h = 0 [13, 46, 99, 112, 182, 200, 202].
+The main difference between both fields is that the identity is a degenerate field (it has a
+null descendant), whereas the other operator with h = ¯h = 0 is not. Such theories will not
+be considered in this book. Operators with h = ℏ = 0 can also be built by comining several
+CFTs, and they play a very important role in string theory since they describe on-shell states.
+Finally, the 4-point function is determined up to a function of a single variable x and its
+complex conjugate:
+� 4
+�
+i=1
+Oi(zi, ¯zi)
+�
+= f(x, ¯x)
+�
+i<j
+1
+z(hi+hj)−h/3
+ij
+× c.c.
+(6.64)
+where
+h :=
+4
+�
+i=1
+hi,
+¯h :=
+4
+�
+i=1
+¯hi.
+(6.65)
+The cross-ratio x is SL(2, C) invariant and reads
+x := z12z34
+z13z24
+.
+(6.66)
+The interpretation is that the SL(2, C) invariance allows to fix 3 of the points to an arbitrary
+value, and the final result does not depend on this choice.
+91
+
+6.4
+Operator formalism and radial quantization
+Radial quantization is a convenient description of a CFT on the plane in terms of operators.
+It relies on the maps given in Section 6.1.2:
+z = eτ+iσ = x + iy.
+(6.67)
+Taking the physical spacetime to be the cylinder, every question is rephrased on the com-
+plex plane in order to exploit the powerful tools from complex analysis. The term “radial
+quantization” comes from the fact that time translation of the cylinder
+τ −→ τ + T
+(6.68)
+corresponds to dilatation on the plane
+z −→ eT z.
+(6.69)
+Thus, time evolution on the cylinder and radial evolution (from the origin to the complex
+infinity) are identified. In particular, the Hamiltonian of the system of the plane is
+H = 2π
+L (L0 + ¯L0),
+(6.70)
+since the RHS is the dilatation operator. The cylinder length L was defined in (6.9). The
+theory is quantized according to this Hamiltonian. In the string theory language, a state
+with H = 0 is said to be on-shell:
+on-shell state:
+h + ¯h = 0.
+(6.71)
+6.4.1
+Radial ordering and commutators
+Time-ordering in τ becomes radial ordering in the plane:
+R
+�
+A(z)B(w)
+�
+=
+�
+A(z)B(w)
+|z| > |w|,
+(−1)F B(w)A(z)
+|w| > |z|,
+(6.72)
+where F = 0 (F = 1) for bosonic (fermionic) operators. Radial ordering will often be kept
+implicit.
+The equal-time (anti-)commutator becomes an equal radius commutator defined by
+point-splitting:
+[A(z), B(w)]±,|z|=|w| = lim
+δ→0
+�
+A(z)B(w)||z|=|w|+δ ± B(w)A(z)||z|=|w|−δ
+�
+.
+(6.73)
+If A and B are two operators which can be written as the contour integrals of a(z) and b(z)
+(corresponding to integral over closed curves on the cylinder)
+A =
+�
+C0
+dz
+2πi a(z),
+B =
+�
+C0
+dz
+2πi b(z),
+(6.74)
+then one finds the following commutators:
+[A, B]± =
+�
+C0
+dw
+2πi
+�
+Cw
+dz
+2πi a(z)b(w),
+(6.75a)
+[A, b(w)]± =
+�
+Cw
+dz
+2πi a(z)b(w).
+(6.75b)
+92
+
+Figure 6.4: Graphical proof of (6.75).
+The contours C0 and Cw are respectively centered around the points 0 and w. For a proof,
+see Figure 6.4. Since these are contour integrals in the complex plane, the Cauchy–Riemann
+formula (B.1) can be used to write the result as soon as one knows the poles of the above
+expression (ultimately, this amounts to pick the sum of residues). In CFTs, the poles of such
+expressions are given by operator product expansions (OPE), defined below (Section 6.4.2).
+Given a conserved current jµ
+∂µjµ = ∂jz + ¯∂j ¯z = 2(∂j¯z + ¯∂jz) = 0,
+(6.76)
+the associated conserved charge is defined by
+Q =
+1
+2πi
+�
+C0
+(jzdz − j¯zd¯z),
+(6.77)
+where C0 denotes the anti-clockwise contour around z = 0 (equivalently the interior of the
+contour is located to the left). The difference of sign in the second term follows directly
+from Stokes’ theorem (B.14g) (and can be understood as a conjugation of the contour).
+The additional factor of 1/2π is consistent with the normalization of spatial integrals in two
+dimensions. The current components are not necessarily holomorphic and anti-holomorphic
+at this level, but in practice this will often be the case (and each component is independently
+conserved), and one writes
+j(z) := jz(z),
+¯ȷ(¯z) := j¯z(¯z).
+(6.78)
+In this case, the charge also splits into a holomorphic and an anti-holomorphic (left- and
+right-moving8) contributions
+Q = QL + QR,
+QL :=
+1
+2πi
+�
+C0
+j(z)dz,
+QR := − 1
+2πi
+�
+C0
+¯ȷ(¯z)d¯z.
+(6.79)
+The infinitesimal variation of a field under the symmetry generated by Q reads
+δϵO(z, ¯z) = −[ϵQ, O(z, ¯z)] = −ϵ
+�
+Cz
+dw
+2πi j(w)O(z, ¯z) + ϵ
+�
+Cz
+d ¯w
+2πi ¯ȷ( ¯w)O(z, ¯z).
+(6.80)
+The contour integrals are easily evaluated once the OPE between the current and the oper-
+ator is known. This formula gives the infinitesimal variation under the transformation for
+any field, not only for primaries.
+8For charges, we use subscript L and R to distinguish both sectors to avoid introducing a new symbol for
+the total charge. However, since Q = QL in the holomorphic sector, it is often not necessary to distinguish
+between the two symbols when acting on an operator or a state (however, this is useful for writing mode
+expansions). We do not write a bar on QR because the charges don’t depend on the position.
+93
+
+Computation – Equation (6.77)
+In real coordinates, the charge is defined by integrating the time component of the
+current jµ over space for fixed time (A.23):
+Q = 1
+2π
+�
+dσ j0.
+The first step is to rewrite this formula covariantly. Since the time is fixed on the slice,
+dτ = 0 and one can write
+Q = 1
+2π
+�
+(dσ j0 − dτ j1) = − 1
+2π
+�
+ϵµνjµ dxν.
+The last formula is valid for any contour. Moreover, it can be evaluated for complex
+coordinates:
+Q = − 1
+2π
+�
+ϵz¯z
+�
+jz d¯z − j ¯z dz
+�
+= − i
+4π
+� �
+jz d¯z − j ¯z dz
+�
+= − 1
+2πi
+� �
+jz dz − j¯z d¯z
+�
+.
+One finds a contour integral because τ = cst circles of the cylinder are mapped to
+|z| = cst contours.
+6.4.2
+Operator product expansions
+The operator product expansion (OPE) is a tool used frequently in CFT: it means that
+when two local operators come close to each other, it is possible to replace their product by
+a sum of local operators
+Oi(zi, ¯zi)Oj(zj, ¯zj) =
+�
+k
+ck
+ij
+zhi+hj−hk
+ij
+¯z
+¯hi+¯hj−¯hk
+ij
+Ok(zj, ¯zj),
+(6.81)
+where the OPE coefficients ck
+ij are some constants and the sum runs over all operators.
+When Ok is primary, the coefficients ck
+ij are related to the structure constants and the field
+metric by
+Cijk = gkℓcℓ
+ij.
+(6.82)
+The radius of convergence for the OPE is given by the distance to the nearest operators in
+the correlation function. The OPE defines an associative algebra (commutative for bosonic
+operators), and the holomorphic sector forms a subalgebra (called the chiral algebra).
+Example 6.1 – OPE with the identity
+The OPE of a field φ(z) with the identity 1 is found by a direct series expansion
+φ(z)1 =
+�
+n∈N
+(z − w)n
+n!
+∂nφ(w).
+(6.83)
+Obviously there are no singular terms.
+Starting from this point we consider only the holomorphic sector except when stated
+otherwise. The formula for the OPE (6.81) can be rewritten as
+A(z)B(w) :=
+N
+�
+n=−∞
+{AB}n(z)
+(z − w)n
+(6.84)
+94
+
+to simplify the manipulations.
+N is an integer and there are singular terms if N > 0.
+Generally, only the terms singular as w → z are necessary in the computations (for example,
+to use the Cauchy–Riemann formula (B.1)): equality up to non-singular terms is denoted
+by a tilde
+A(z)B(w) ∼
+N
+�
+n=1
+{AB}n(z)
+(z − w)n =: A(z)B(w).
+(6.85)
+The RHS of this expression defines the contraction of the operators A and B.
+While, most of the time, only singular terms are kept
+φi(zi)φj(zj) ∼
+�
+k
+θ(hi + hj − hk)
+ck
+ij
+(z − w)hi+hj−hk φk(w)
+(6.86)
+(with θ(x) the Heaviside step function), it can happen that one keeps also non-singular terms
+(the product of two OPE have singular terms coming from non-singular terms multiplying
+singular terms). Explicit contractions of operators through the OPE is also denoted by a
+bracket when there are other operators.
+For a primary field φ(z), one finds the OPE with the energy–momentum tensor to be
+T(z)φ(w) ∼
+h φ(w)
+(z − w)2 + ∂φ(w)
+z − w ,
+(6.87)
+where h is the conformal weight of the field. This OPE together with (6.80) for j(z) =
+−v(z)T(z) correctly reproduces (6.50).
+Computation – Equation (6.50)
+δφ(z) =
+�
+Cz
+dw
+2πi v(w)T(w)φ(z) ∼
+�
+Cz
+dw
+2πi v(w)
+� h φ(z)
+(w − z)2 + ∂φ(z)
+w − z
+�
+= h ∂v(z) φ(z) + v(z)∂φ(z).
+For a non-primary operator, the OPE becomes more complicated (as it is reflected by
+the transformation property), but the conformal weight can still be identified at the term
+in z−2. The most important example is the energy–momentum tensor: the central charge is
+found as the coefficient of the z−4 term its OPE with itself:
+T(z)T(w) ∼
+c/2
+(z − w)4 +
+2T(w)
+(z − w)2 + ∂T(w)
+z − w .
+(6.88)
+The OPE indicates that the conformal weight of T is h = 2.
+Using (6.80) for j(z) =
+−v(z)T(z), one finds the infinitesimal variation
+δT = 2 ∂v T + v ∂T + c
+12 ∂3v,
+(6.89)
+The last term vanishes for global transformations: this translates the fact that T is only a
+quasi-primary. The finite form of this transformation is
+T ′(w) =
+� dz
+dw
+�−2 �
+T(z) − c
+12 S(w, z)
+�
+=
+� dz
+dw
+�−2
+T(z) + c
+12 S(z, w)
+(6.90)
+where S(w, z) is the Schwarzian derivative
+S(w, z) = w(3)
+w′ − 3
+2
+�w′′
+w′
+�2
+,
+(6.91)
+95
+
+where the derivatives of w are with respect to z. This vanishes if the transformation is in
+SL(2, C), and it transforms as
+S(u, z) = S(w, z) +
+�dw
+dz
+�2
+S(u, w)
+(6.92)
+under successive changes of coordinates.
+Computation – Equation (6.89)
+δT(z) =
+�
+Cz
+dw
+2πi v(w)T(w)T(z) ∼
+�
+Cz
+dw
+2πi v(w)
+�
+c/2
+(z − w)4 +
+2T(w)
+(z − w)2 + ∂T(w)
+z − w
+�
+=
+c
+2 × 3! ∂3v(z) + 2∂v(z) T(z) + v(z)∂T(z).
+6.4.3
+Hermitian and BPZ conjugation
+In this section, we introduce two different notions of conjugations: one is adapted for amp-
+litudes because it defines a unitary Euclidean time evolution, while the second is more
+natural as an inner product of CFT states. Both can be interpreted as providing a map
+from in-states to out-states on the cylinder.
+Given an operator O, we need to define an operation O‡ – called Euclidean adjoint (or
+simply adjoint) – which, after Wick rotation from Euclidean to Lorentzian signature, can
+be interpreted as the Hermitian adjoint.9 This is necessary in order to define a Hermitian
+inner-product and to impose reality conditions.
+To motivate the definition, consider first the cylinder in Lorentzian signature.
+Since
+Hermitian conjugation does not affect the Lorentzian coordinates, the Euclidean time must
+reverse its sign:
+t† = −iτ † = t
+=⇒
+τ † = −τ.
+(6.93)
+Hence, an appropriate definition of the Euclidean adjoint is an Hermitian conjugation to-
+gether with time reversal.10
+Another point of view is that the time evolution operator
+U(τ) := e−τH is not unitary when H is Hermitian H† = H: the solution is to define a new
+Euclidean adjoint U(τ)‡ := U(−τ)† such that U(τ) is unitary for it.
+Time reversal on the cylinder corresponds to inversion and complex conjugation on the
+complex plane:
+z
+τ→−τ
+−−−−→ e−τ+iσ = 1
+z∗ = I(¯z),
+(6.94)
+where I(z) = 1/z is the inversion (6.42).11 On the real surface12 ¯z = z∗, which leads to the
+definition of the Euclidean adjoint as follows:
+O(z, ¯z)‡ :=
+�¯I ◦ O(z, ¯z)
+�†,
+(6.95)
+9In [193], it is denoted by a bar on top of the operator: we avoid this notation since the bar already
+denotes the anti-holomorphic sector. In [262], it is indicated by a subscript hc. Otherwise, in most of the
+literature, it has no specific symbol since one directly works with the modes.
+10The Euclidean adjoint can be used to define an inner product: positive-definiteness of the latter is
+called reflection positivity or OS-positive and is a central axiom of constructive QFT.
+11We do not write “z†” because this notation is confusing as one should not complex conjugate the factor
+of i in the exponential (Section 6.2.1).
+12Remember that ¯z is not the complex conjugate of z but an independent variable.
+96
+
+where ¯I(z) := 1/¯z. If O is quasi-primary, we have:
+O(z, ¯z)‡ =
+�
+1
+¯z2hz2¯h O
+�1
+¯z , 1
+z
+��†
+=
+1
+z2h¯z2¯h O†
+�1
+z , 1
+¯z
+�
+,
+(6.96)
+The last equality shows that Euclidean conjugation is equivalent to take the conjugate of all
+factors of i but otherwise leaves z and ¯z unaffected. The Euclidean adjoint acts by complex
+conjugation of any c-number and reverses the order of the operators (acting as a transpose):
+(λ O1 · · · On)‡ = λ∗ O‡
+n · · · O‡
+1,
+λ ∈ C,
+(6.97)
+without any sign.
+A second operation, called the BPZ conjugation, is useful.
+It can be defined in two
+different ways:
+O(z, ¯z)t := I± ◦ O(z, ¯z) = (∓1)h+¯h
+z2h¯z2¯h O
+�
+±1
+z , ±1
+¯z
+�
+,
+(6.98)
+where I±(z) = ±1/z is the inversion (6.42). The minus and plus signs are respectively more
+convenient when working with the open and closed strings.13 The BPZ conjugation does
+not complex conjugate c-number nor changes the order of the operators:14
+(λ O1 · · · On)t = λ Ot
+1 · · · Ot
+n,
+λ ∈ C.
+(6.99)
+The identity is invariant under both conjugation
+1‡ = 1t = 1.
+(6.100)
+6.4.4
+Mode expansion
+Any field of weight (h, ¯h) can be expanded in terms of modes Om,n
+O(z, ¯z) =
+�
+m,n
+Om,n
+zm+h¯zn+¯h .
+(6.101)
+Note that the modes Om,n themselves are operators. The ranges of the two indices are such
+that
+m + h ∈ Z + ν,
+n + ¯h ∈ Z + ¯ν,
+ν, ¯ν =
+�
+0
+periodic,
+1/2
+anti-periodic.
+(6.102)
+The values of ν and ¯ν depend on whether the fields satisfies periodic or anti-periodic bound-
+ary conditions on the plane (for half-integer weights, the periodicity is reversed on the
+cylinder):
+O(e2πiz, ¯z) = e2πiνO(z, ¯z),
+O(z, e2πi¯z) = e2πi¯νO(z, ¯z).
+(6.103)
+Depending on whether the weights are integers or half-integers, additional terminology is
+introduced:
+13The index t should not be confused with the matrix transpose: it is used in opposition with ‡ and † to
+indicate that no complex conjugation is involved.
+14However, the fields become anti-radially ordered after a BPZ conjugation since it sends z to 1/z. The
+radial ordering can be restored by (anti-)commuting the fields, which can introduce additional signs [220].
+This problem does not arise when working in terms of the modes.
+97
+
+• If h ∈ Z + 1/2, then one can choose anti-periodic (Neveu–Schwarz or NS) or periodic
+(Ramond or R) boundary conditions on the cylinder (reversed for the plane):
+ν, ¯ν =
+�
+0
+NS
+1/2
+R
+(6.104)
+The indices are half-integers (resp. integers) for the NS (R) sector.
+• If h ∈ Z, periodic (or untwisted) boundary conditions are more natural, but anti-
+periodic boundary conditions may also be considered:
+ν, ¯ν =
+�
+0
+untwisted
+1/2
+twisted
+(6.105)
+The modes of untwisted (resp. twisted) fields have integer (half-integers) indices.
+The mode expansions have no branch cut (fractional power of z or ¯z) for periodic fields
+(bosonic untwisted or fermionic twisted). We will see explicit examples of such operators
+later in this book.
+Under Euclidean conjugation (6.95), the modes are related by
+(O‡)−m,−n = (Om,n)†.
+(6.106)
+In particular, if the operator is Hermitian (under the Euclidean adjoint), the reality condition
+on the modes relates the negative modes with the conjugated positive modes
+O‡ = O
+=⇒
+(Om,n)† = O−m,−n.
+(6.107)
+When no confusion is possible (for Hermitian operators), we will write O†
+m,n instead of
+(Om,n)†.
+For a holomorphic field φ(z), the above expansion becomes
+φ(z) =
+�
+n∈Z+h+ν
+φn
+zn+h .
+(6.108)
+Conversely, the modes are recovered from the field through
+φn =
+�
+C0
+dz
+2πi zn+h−1φ(z),
+(6.109)
+where the integration is counter-clockwise around the origin.
+If the field is Hermitian, then
+φ‡ = φ
+=⇒
+(φn)† = φ−n.
+(6.110)
+The operators φn have a conformal weight of −n (since the weight of z is −1). The BPZ
+conjugate of the modes is
+φt
+n = (I± ◦ φ)n = (−1)h(±1)nφ−n.
+(6.111)
+98
+
+Computation – Equation (6.111)
+φt
+n = (I± ◦ φ)n =
+�
+dz
+2πi zn+h−1I± ◦ φ(z)
+=
+�
+dz
+2πi zn+h−1
+�
+∓ 1
+z2
+�h
+φ
+�
+±1
+z
+�
+= (∓1)h
+�
+dz
+2πi zn−h−1φ
+�
+±1
+z
+�
+= (∓1)h
+� dw
+2πi
+�
+± 1
+w
+�n−h
+w−1φ(w)
+= (∓1)h(±1)n−h
+� dw
+2πi w−n+h−1φ(w),
+where we have set w = ±1/z such that
+dz
+z = ∓ dw
+w2z = −dw
+w ,
+(6.112)
+and the minus sign disappears upon reversing the contour orientation.
+The mode expansion of the energy–momentum tensor is
+T(z) =
+�
+n∈Z
+Ln
+zn+2 ,
+Ln =
+�
+dz
+2πi T(z)zn+1,
+(6.113)
+where one recognizes the Virasoro operators as the modes. In most situations, the Virasoro
+operators are Hermitian
+L†
+n = L−n.
+(6.114)
+The OPE (6.88) and (6.87) together with (6.75a) help to reconstruct the Virasoro algebra
+(6.58) and the commutation relations between the Lm and the modes φn of a weight h
+primary:
+[Lm, φn] =
+�
+m(h − 1) − n
+�
+φm+n.
+(6.115)
+This easily gives the commutation relation for the complete field:
+[Lm, φ(z)] = zm�
+z∂ + (n + 1)h
+�
+φ(z).
+(6.116)
+We will often use (6.58) and (6.115) for m = 0:
+[L0, L−n] = nL−n,
+[L0, φ−n] = nφ−n.
+(6.117)
+This means that both φn and Ln act as raising operators for L0 if n < 0, and as lowering
+operators if n > 0 (remember that L0 is the Hamiltonian in the holomorphic sector). When
+both the holomorphic and anti-holomorphic sectors enter, it is convenient to introduce the
+combinations
+L±
+n = Ln ± ¯Ln,
+(6.118)
+such that L+
+0 is the Hamiltonian.
+Finally, every holomorphic current j(z) has a conformal weight h = 1 and can be expan-
+ded as
+j(z) =
+�
+n
+jn
+zn+1 .
+(6.119)
+By definition, the zero-mode is equal to the holomorphic charge
+QL = j0.
+(6.120)
+99
+
+6.4.5
+Hilbert space
+The Hilbert space of the CFT is denoted by H. The SL(2, C) (or conformal) vacuum15 |0⟩
+is defined by the state which is invariant under the global conformal transformations:
+L0 |0⟩ = 0,
+L±1 |0⟩ = 0.
+(6.121)
+Expectation value of an operator O in the SL(2, C) vacuum is denoted as:
+⟨O⟩ :=⟨0| O |0⟩ .
+(6.122)
+If the fields are expressed in terms of creation and annihilation operators (which happens
+e.g. for free scalars, free fermions and ghosts), then the Hilbert space has the structure of a
+Fock space.
+State–operator correspondence
+The state–operator correspondence identifies every state |O⟩ of the CFT Hilbert space with
+an operator O(z, ¯z) through
+|O⟩ = lim
+z,¯z→0 O(z, ¯z) |0⟩ = O(0, 0) |0⟩ .
+(6.123)
+Such a state can be interpreted as an “in” state since it is located at τ → −∞ on the
+cylinder. Focusing now on a holomorphic field φ(z), the state is defined as
+|φ⟩ = lim
+z→0 φ(z) |0⟩ = φ(0) |0⟩ .
+(6.124)
+For this to make sense, the modes which diverge as z → 0 must annihilate the vacuum. In
+particular, for a weight h field φ(z), one finds:
+∀n ≥ −h + 1 :
+φn |0⟩ = 0.
+(6.125)
+Thus, the φn for n ≥ −h + 1 are annihilation operators for the vacuum |0⟩, and conversely
+the states φn with n < −h + 1 are creation operators. As a consequence, the state |φ⟩ is
+found by applying the mode n = −h to the vacuum:
+|φ⟩ = φ−h |0⟩ =
+�
+dz
+2πi
+φ(z)
+z
+|0⟩ .
+(6.126)
+Since L−1 is the generator of translations on the plane, one finds
+φ(z) |0⟩ = ezL−1φ(0)e−zL−1 |0⟩ = ezL−1 |φ⟩ .
+(6.127)
+The vacuum |0⟩ is the state associated to the identity 1. Translating the conditions (6.125)
+to the energy–momentum tensor gives
+∀n ≥ −1 :
+Ln |0⟩ = 0.
+(6.128)
+This is consistent with the definition (6.121) since it includes the sl(2, C) subalgebra.
+If h < 0, some of the modes with n > 0 do not annihilate the vacuum: (6.117) implies that
+some states have an energy lower than the one of |0⟩. The state |Ω⟩ (possibly degenerate)
+with the lowest energy is called the energy vacuum
+∀ |φ⟩ ∈ H :
+⟨Ω| L0 |Ω⟩ ≤⟨φ| L0 |φ⟩ .
+(6.129)
+15There are different notions of “vacuum”, see (6.129). However, the SL(2, C) vacuum is unique. Indeed,
+it is mapped to the unique identity operator under the state–operator correspondence (however, there can
+be other states of weight 0, see Remark 6.6).
+100
+
+It is obtained by acting repetitively with the modes φn>0.
+This vacuum defines a new
+partition of the non-zero-modes operators into annihilation and creation operators. If there
+are zero-modes, i.e. n = 0 modes, then the vacuum is degenerate since they commute
+with the Hamiltonian, [L0, φ0] = 0 according to (6.117). The partition of the zero-modes
+into creation and annihilation operators depends on the specific state chosen among the
+degenerate vacua.
+The energy aΩ of |Ω⟩, which is also its L0 eigenvalue
+L0 |Ω⟩ := aΩ |Ω⟩ ,
+(6.130)
+is called zero-point energy. Bosonic operators with negative h are dangerous because they
+lead to an infinite negative energy together with an infinite degeneracy (from the zero-mode).
+The conjugate vacuum is defined by BPZ or Hermitian conjugation
+⟨0| = |0⟩‡ = |0⟩t
+(6.131)
+since both leave the identity invariant. It is also annihilated by the sl(2, C) subalgebra:
+⟨0| L0 = 0,
+⟨0| L±1 = 0.
+(6.132)
+Since there are two kinds of conjugation, two different conjugated states can be defined. They
+are also called “out” states since they are located at τ → ∞ on the cylinder (Figure 6.2).
+Euclidean and BPZ conjugations and inner products
+The Euclidean adjoint ⟨O‡| of the state |O⟩ is defined as
+⟨O‡| =
+lim
+w, ¯
+w→0⟨0| O(w, ¯w)‡ =
+lim
+w, ¯
+w→0
+1
+w2h ¯w2¯h ⟨0| O
+� 1
+¯w, 1
+w
+�†
+(6.133a)
+=
+lim
+z,¯z→∞ z2h¯z2¯h⟨0| O†(z, ¯z)
+(6.133b)
+=⟨0| I ◦ O†(0, 0),
+(6.133c)
+where the two coordinate systems are related by w = 1/¯z. From this formula, the definition
+of the adjoint of a holomorphic operator φ follows
+⟨φ‡| = lim
+¯
+w→0⟨0| φ(w)‡ = lim
+¯
+w→0
+1
+w2h ⟨0| φ†
+� 1
+w
+�
+(6.134a)
+= lim
+z→∞ z2h⟨0| φ†(z)
+(6.134b)
+=⟨0| I ◦ φ†(0).
+(6.134c)
+Then, expanding the field in terms of the modes gives
+⟨φ‡| =⟨0| (φ†)h.
+(6.135)
+The BPZ conjugated state is
+⟨φ| := lim
+w→0⟨0| φ(w)t
+(6.136a)
+= (±1)h lim
+z→∞ z2h⟨0| φ(z)
+(6.136b)
+=⟨0| I± ◦ φ(0).
+(6.136c)
+In terms of the modes, one has
+⟨φ| = (±1)h⟨0| φh.
+(6.137)
+101
+
+If φ is Hermitian, then the relation between both conjugated states corresponds to a reality
+condition:
+⟨φ‡| = (±1)h⟨φ| .
+(6.138)
+Taking the BPZ conjugation of the conditions (6.125) tells which modes must annihilate
+the conjugate vacuum:
+∀n ≤ h − 1 :
+⟨0| φn = 0,
+(6.139)
+and one finds more particularly for the Virasoro operators
+∀n ≤ 1 :
+⟨0| Ln = 0.
+(6.140)
+This can also be derived directly from (6.136) by requiring that applying an operator on the
+conjugate vacuum ⟨0| is well-defined.
+All conditions taken together mean that the expectation value of the energy–momentum
+tensor in the conformal vacuum vanishes:
+⟨0| T(z) |0⟩ = 0.
+(6.141)
+In particular, this means that the energy vacuum |Ω⟩, if different from |0⟩, has a negative
+energy.
+The Hermitian16 and BPZ inner products are respectively defined by:
+⟨φ‡
+i|φj⟩ =⟨0| ¯I ◦ φj(0)φi(0) |0⟩ = lim
+z→∞
+w→0
+z2hi⟨0| φ†
+i(z)φj(w) |0⟩ ,
+(6.142a)
+⟨φi|φj⟩ =⟨0| I ◦ φj(0)φi(0) |0⟩ = (±1)hi lim
+z→∞
+w→0
+z2hi⟨0| φi(z)φj(w) |0⟩ .
+(6.142b)
+These products can be recast as 2-point correlation functions (6.62b) on the sphere:
+⟨φi|φj⟩ = ⟨I ◦ φi(0)φj(0)⟩,
+⟨φ‡
+i|φj⟩ = ⟨I ◦ φ†
+i(0)φj(0)⟩.
+(6.143)
+From the state–operator correspondence, the action of one operator on the in-state can be
+reinterpreted as the matrix element of this operator using the two external states, or also as
+a 3-point function:
+⟨φi| φj(z) |φk⟩ = (±1)hi lim
+w→∞ w2hi⟨φi(w)φj(z)φk(0)⟩.
+(6.144)
+Given a basis of states {φi} (i can run over both discrete and continuous indices), the
+conjugate or dual states {φc
+i} are defined by:
+⟨φc
+i|φj⟩ = δij
+(6.145)
+(the delta function is discrete and/or continuous according to the indices).
+Verma modules
+If φ(z) is a weight h primary, then the associated state |φ⟩ satisfies:
+L0 |φ⟩ = h |φ⟩ ,
+∀n ≥ 1 :
+Ln |φ⟩ = 0.
+(6.146)
+Such a state is also called a highest-weight state. The descendant states are defined by all
+possible states of the form
+|φ{ni}⟩ :=
+�
+i
+L−ni |φ⟩ ,
+(6.147)
+where the same L−ni can appear multiple times and ni > 0. The set of states φ{ni} is called
+a Verma module V (h, c). One finds that the L0 eigenvalues of this state is
+L0 = h +
+�
+i
+ni.
+(6.148)
+16Depending on the normalization, it can also be anti-Hermitian.
+102
+
+Normal ordering
+The normal ordering of an operator with respect to a vacuum corresponds to placing all cre-
+ation (resp. annihilation) operators of this vacuum on the left (resp. right). From this defin-
+ition, the expectation value of a normal ordered operator in the vacuum vanishes identically.
+The main reason for normal ordering is to remove singularities in expectation values.
+Given an operator φ(z), we define two normal orderings:
+• The conformal normal order (CNO) :O: is defined with respect to the conformal va-
+cuum (6.121):
+⟨0| :O: |0⟩ = 0.
+(6.149)
+• The energy normal order (ENO)
+⋆
+⋆ O
+⋆
+⋆ is defined with respect to the energy vacuum
+(6.129):
+⟨Ω|
+⋆
+⋆O
+⋆
+⋆ |Ω⟩ = 0.
+(6.150)
+We first discuss the conformal normal ordering before explaining how to relate it to the
+energy normal ordering.
+Given two operators A and B, the simplest normal ordering amounts to subtract the
+expectation value:
+:A(z)B(w):
+?= A(z)B(w) − ⟨A(z)B(w)⟩.
+(6.151)
+This is equivalent to defining the products of two operators at coincident points via point-
+splitting:
+:A(z)B(z):
+?= lim
+w→z
+�
+A(z)B(w) − ⟨A(z)B(w)⟩
+�
+.
+(6.152)
+While this works well for free fields, this does not generalize for composite or interacting
+fields.
+The reason is that this procedure removes only the highest singularity in the product: it
+does not work if the OPE has more than one singular term. An appropriate definition is
+:A(z)B(w): := A(z)B(w) − A(z)B(w) =
+�
+n∈N
+(z − w)n{AB}−n(z),
+(6.153)
+where the contraction between A and B is defined in (6.85), and the second equality comes
+from (6.84).
+Then, the product evaluated at coincident points is found by taking the limit (in this
+case the argument is often indicated only at the end of the product)
+:AB(z): := :A(z)B(z): := lim
+w→z :A(z)B(w): = {AB}0(z).
+(6.154)
+Indeed, since all powers of (z − w) are positive in the RHS of (6.153), all terms but the first
+one disappear. The form of (6.154) shows that the normal order can also be computed with
+the contour integral
+:AB(z): =
+�
+Cz
+dw
+2πi
+A(z)B(w)
+z − w
+.
+(6.155)
+It is common to remove the colons of normal ordering when there is no ambiguity and, in
+particular, to write:
+AB(z) := :AB(z):.
+(6.156)
+In terms of modes, one has
+:AB(z): =
+�
+m
+:AB:m
+zm+hA+hB ,
+(6.157a)
+:AB:m =
+�
+n≤−hA
+AnBm−n +
+�
+n>−hA
+Bm−nAn.
+(6.157b)
+103
+
+This expression makes explicit that normal ordering is non-commutative and non-associative:
+:AB(z): ̸= :BA(z):,
+:A(BC)(z): ̸= :(AB)C(z):.
+(6.158)
+The product of normal ordered operators can then be computed using Wick theorem. In
+fact, one is more interested in the contraction of two such operators in order to recover the
+OPE between these operators: the product is then derived with (6.153).
+If Ai (i = 1, 2, 3) are free fields, one has
+A1(z) :A2A3(w): = :A1(z)A2A3(w): + A1(z) :A2 A3(w):,
+A1(z) :A2 A3(w): = A1(z)A2(w) :A3(w): + A1(z)A3(w) :A2(w):.
+(6.159)
+If the fields are not free, then the contraction cannot be extracted from the normal ordering.
+Similarly if there are more fields, then one needs to perform all the possible contractions.
+Given two free fields A and B, one has the following identities:
+A(z) :B(w)n: = n A(z)B(w) :B(w)n−1:,
+(6.160a)
+A(z) :eB(w): = A(z)B(w) :eB(w):,
+(6.160b)
+:eA(z): :eB(w): = exp
+�
+A(z)B(w)
+�
+:eA(z)eB(w):.
+(6.160c)
+The last relation generalizes for a set of n fields Ai:
+n
+�
+i=1
+:eAi: = : exp
+� n
+�
+i=1
+Ai
+�
+: exp
+�
+i<j
+⟨AiAj⟩,
+(6.161a)
+� n
+�
+i=1
+:eAi:
+�
+= exp
+�
+i<j
+⟨AiAj⟩.
+(6.161b)
+Computation – Equation (6.160b)
+A(z) :eB(w): = A(z)
+�
+n
+1
+n! :B(w)n: = A(z)B(w)
+�
+n
+1
+(n − 1)! :B(w)n−1:.
+Computation – Equation (6.160c)
+:eA(z): :eB(w): =
+�
+m,n
+1
+m!n! :A(z)m: :B(w)n:
+=
+�
+m,n,k
+k!
+m!n!
+�m
+k
+��n
+k
+��
+A(z)B(w)
+�k:A(z)m−k: :B(w)n−k:
+=
+�
+m,n,k
+1
+k!(m − k)!(n − k)!
+�
+A(z)B(w)
+�k:A(z)m−k: :B(w)n−k:.
+The factorial k! counts the number of possible ways to contract the two operators.
+104
+
+The general properties of normal ordered expressions are identical for both vacua: what
+differs is the precise computation in terms of the operators (or modes). Hence, the energy
+normal ordering can be defined in parallel with (6.157), but changing the definitions of
+creation and annihilation operators:
+⋆
+⋆AB(z)
+⋆
+⋆ =
+�
+m
+⋆
+⋆AB(z)
+⋆
+⋆n
+zm+hA+hB ,
+(6.162a)
+⋆
+⋆AB
+⋆
+⋆m =
+�
+n≤0
+AnBm−n +
+�
+n>0
+Bm−nAn.
+(6.162b)
+To simplify the definition we assume that A0 is a creation operator and it is thus included
+in the first sum (this must be adapted in function of which vacuum state is chosen if the
+latter is degenerate).
+The relation between the normal ordered modes is
+:AB:m =
+⋆
+⋆AB
+⋆
+⋆m +
+hA−1
+�
+n=0
+[Bm+n, A−n].
+(6.163)
+Computation – Equation (6.163)
+:AB:m =
+�
+n≤−hA
+AnBm−n +
+�
+n>−hA
+Bm−nAn
+=
+�
+n≥hA
+A−nBm+n +
+�
+n>0
+Bm−nAn +
+hA−1
+�
+n=0
+Bm+nA−n
+=
+�
+n≥0
+A−nBm+n +
+�
+n>0
+Bm−nAn +
+hA−1
+�
+n=0
+[Bm+n, A−n]
+=
+⋆
+⋆AB
+⋆
+⋆m +
+hA−1
+�
+n=0
+[Bm+n, A−n].
+The choice of the normal ordering for the operators is related to the ordering ambiguity
+when quantizing the system: when the product of two non-commuting modes appears in the
+classical composite field, the corresponding quantum operator is ambiguous (generally up to
+a constant). In practice, one starts with the conformal ordering since it is invariant under
+conformal transformations and because one can compute with contour integrals. Then, the
+expression can be translated in the energy ordering using (6.163). But, knowing how the
+conformal and energy vacua are related, it is often simpler to find the difference between
+the two orderings by applying the operator on the vacua.
+6.4.6
+CFT on the cylinder
+According to (6.44), the relation between the field on the cylinder and on the plane is
+φ(z) =
+� L
+2π
+�h
+z−hφcyl(w)
+(6.164)
+(quantities without indices are on the plane by definition). The mode expansion on the
+cylinder is
+φcyl =
+�2π
+L
+�h �
+n∈Z
+φne− 2π
+L w =
+�2π
+L
+�h �
+n∈Z
+φn
+zn .
+(6.165)
+105
+
+Using the finite transformation (6.90) for the energy–momentum tensor T, one finds the
+relation
+Tcyl(w) =
+�2π
+L
+�2 �
+T(z)z2 − c
+24
+�
+.
+(6.166)
+For the L0 mode, one finds
+(L0)cyl = L0 − c
+24,
+(6.167)
+and thus the Hamiltonian is
+H = (L0)cyl + (¯L0)cyl = L0 + ¯L0 − c + ¯c
+24 .
+(6.168)
+6.5
+Suggested readings
+• The most complete reference on CFTs is [54] but it lacks some recent developments.
+Two excellent complementary books are [25, 206].
+String theory books generally dedicate a fair amount of pages to CFTs: particularly
+good summaries can be found in [24, 128, 193, 194].
+Finally, a modern and fully algebraic approach can be found in [200, 201]. Other good
+reviews are [196, 259].
+• There are various other books [104, 120, 126, 171] and reviews [32, 89, 91, 204, 243].
+• The maps from the sphere and the cylinder to the complex plane are discussed in [193,
+sec. 2.6, 6.1].
+• Normal ordering is discussed in details in [54, chap. 6] (see also [24, sec. 4.2, 193,
+sec. 2.2]).
+• Euclidean conjugation is discussed in [54, sec. 6.1.1, 193, p. 202–3]. For a comparison
+of Euclidean and BPZ conjugations, see [262, sec. 2.2, 220, p. 11].
+• Normal ordering and difference between the different definitions are described in [193,
+chap. 2, 54, sec. 6.5].
+106
+
+Chapter 7
+CFT systems
+Abstract
+This chapter summarizes the properties of some CFT systems. We focus on
+the free scalar field and on the first-order bc system (which generalizes the reparametriz-
+ation ghosts). For the different systems, we first provide an analysis on a general curved
+background before focusing on the complex plane. This is sufficient to describe the local
+properties on all Riemann surfaces g ≥ 0.
+7.1
+Free scalar
+7.1.1
+Covariant action
+The Euclidean action of a free scalar X on a curved background gµν is
+S =
+ϵ
+4πℓ2
+�
+d2x√g gµν∂µX∂νX,
+(7.1)
+where ℓ is a length scale1 and
+ϵ :=
+�
++1
+spacelike
+−1
+timelike ,
+√ϵ :=
+�
++1
+spacelike
+i
+timelike
+(7.2)
+denotes the signature of the kinetic term. The field is periodic along σ
+X(τ, σ) ∼ X(τ, σ + 2π).
+(7.3)
+The energy–momentum tensor reads
+Tµν = − ϵ
+ℓ2
+�
+∂µX∂νX − 1
+2 gµν(∂X)2
+�
+,
+(7.4)
+and it is traceless
+T µ
+µ = 0.
+(7.5)
+The equation of motion is
+∆X = 0,
+(7.6)
+where ∆ is the Laplacian (A.28).
+1To be identified with the string scale, such that α′ = ℓ2.
+107
+
+The simplest method for finding the propagator in flat space is by using the identity
+(assuming that there is no boundary term)
+0 =
+�
+dX
+δ
+δX(σ)
+�
+e−S[X]X(σ′)
+�
+,
+(7.7)
+which yields a differential equation for the propagator:
+⟨∂2X(σ)X(σ′)⟩ = −2πϵℓ2 δ(2)(σ − σ′).
+(7.8)
+This is easily integrated to
+⟨X(σ)X(σ′)⟩ = −ϵℓ2
+2
+ln |σ − σ′|2.
+(7.9)
+Computation – Equation (7.9)
+By translation and rotation invariance, one has
+⟨X(σ)X(σ′)⟩ = G(r),
+r = |σ − σ′|.
+(7.10)
+In polar coordinates, the Laplacian reads
+∆G(r) = 1
+r ∂r(rG′(r)).
+(7.11)
+Integrating the differential equation (7.8) over d2σ = rdrdθ yields
+−2πϵℓ2 = 2π
+� r
+0
+dr′ r′ × 1
+r′ ∂r′(r′G′(r′)) = 2πrG′(r).
+(7.12)
+The solution is
+G′(r) = −ϵℓ2 ln r
+(7.13)
+and the form (7.9) follows by writing
+ln r = 1
+2 ln r2 = 1
+2 ln |σ − σ′|2.
+(7.14)
+The action (7.1) is obviously invariant under constant translations of X:
+X −→ X + a,
+a ∈ R.
+(7.15)
+The associated U(1) current2 is conserved and reads
+Jµ := 2πiϵ
+∂L
+∂(∂µX) = i
+ℓ2 gµν∂νX,
+∇µJµ = 0.
+(7.16)
+On flat space, the charge follows from (A.23):
+p = 1
+2π
+�
+dσ J0 =
+i
+2πℓ2
+�
+dσ ∂0X.
+(7.17)
+This charge is called momentum because it corresponds to the spacetime momentum in
+string theory.
+2The group is R but the algebra is u(1) (since locally there is no difference between the real line and the
+circle).
+108
+
+Moreover, there is a another topological current
+�Jµ := −i ϵµνJν = 1
+ℓ2 ϵµν∂νX,
+(7.18)
+which is identically conserved:
+∇µ �Jµ ∝ ϵµν[∇µ, ∇ν]X = 0
+(7.19)
+since [∇µ, ∇ν] = 0 when acting on a scalar field. Note that ˜Jµ is the Hodge dual of Jµ. The
+conserved charge is called the winding number and reads on flat space:
+w = 1
+2π
+�
+dσ �J0 =
+1
+2πℓ2
+� 2π
+0
+dσ ∂1X =
+1
+2πℓ2
+�
+X(τ, 2π) − X(τ, 0)
+�
+.
+(7.20)
+Remark 7.1 (Normalization of the current) The definition of the current (7.16) may
+look confusing. The factor of i is due to the Euclidean signature, see (A.25a), and the factor
+of 2π comes from the normalization of the spatial integral. We have inserted ϵ in order to
+interpret the conserved charge p as a component of the momentum contravariant vector in
+string theory.
+To make contact with string theory, consider D scalar fields Xa(xµ). Then, the current
+becomes
+Jµ
+a =
+i
+2πℓ2 ηab∂µXb,
+(7.21)
+where the position of the indices is in agreement with the standard form of Noether’s formula
+(A.25a) (a current has indices in opposite locations as the parameters and fields). Since we
+have η00 = −1 = ϵX0, we find that J0µ = ϵX0Jµ
+0 has no epsilon after replacing the expression
+(7.17) of Jµ
+0 .
+The transformation Xa → Xa +ca is a global translation in target spacetime: the charge
+pa is identified with the spacetime momentum.
+The factor of i indicates that pa is the
+Euclidean contravariant momentum vector by comparison with (A.7).
+The convention of this section is to always work with quantities which will become con-
+travariant vector to avoid ambiguity.
+7.1.2
+Action on the complex plane
+In complex coordinates, the action on flat space reads
+S =
+ϵ
+2πℓ2
+�
+dzd¯z ∂zX∂¯zX,
+(7.22)
+giving the equation of motion:
+∂z∂¯zX = 0.
+(7.23)
+This indicates that ∂zX and ∂¯zX are respectively holomorphic and anti-holomorphic such
+that
+X(z, ¯z) = XL(z) + XR(¯z),
+(7.24)
+and we will remove the subscripts when there is no ambiguity (for example, when the position
+dependence is written):
+X(z) := XL(z),
+X(¯z) := XR(¯z).
+(7.25)
+It looks like XL(z) and XR(¯z) are unrelated, but this is not the case because of the zero-
+mode, as we will see below.
+109
+
+The U(1) current is written as
+J := Jz = i
+ℓ2 ∂zX,
+¯J := J¯z = i
+ℓ2 ∂¯zX,
+(7.26)
+where we used the relations Jz = J ¯z/2 and J¯z = Jz/2. The equation of motion implies that
+the current J is holomorphic, and ¯J is anti-holomorphic:
+¯∂J = 0,
+∂ ¯J = 0.
+(7.27)
+The momentum splits into left- and right-moving parts:
+p = pL + pR,
+pL =
+1
+2πi
+�
+dz J,
+pR = − 1
+2πi
+�
+d¯z ¯J.
+(7.28)
+The components of the topological current (7.18) are related to the ones of the U(1)
+current:
+�Jz = i
+ℓ2 ∂zX = J,
+�J¯z = − i
+ℓ2 ∂¯zX = − ¯J.
+(7.29)
+As a consequence, the winding number is
+w = pL − pR.
+(7.30)
+Note that we have the relations
+pL = p + w
+2
+,
+pR = p − w
+2
+,
+(7.31a)
+p2 + w2 = p2
+L + p2
+R,
+2pw = p2
+L − p2
+R.
+(7.31b)
+The energy–momentum tensor is
+T := Tzz = − ϵ
+ℓ2 ∂zX∂zX,
+¯T := T¯z¯z = − ϵ
+ℓ2 ∂¯zX∂¯zX,
+Tz¯z = 0.
+(7.32)
+Since the ∂zX (∂¯zX) is (anti-)holomorphic, so is T(z) ( ¯T(¯z)). Since the energy–momentum
+tensor, the current and the field itself (up to zero-modes) split in holomorphic and anti-
+holomorphic components in a symmetric way, it is sufficient to focus on one of the sectors,
+say the holomorphic one.
+The other primary operators of the theory are given by the vertex operators Vk(z):3
+Vk(z, ¯z) := :eiϵkX(z,¯z):.
+(7.33)
+Remark 7.2 In fact, it is possible to introduce more general vertex operators
+VkL,kR(z, ¯z) := :e2iϵ
+�
+kLX(z)+kRX(¯z)
+�
+:,
+(7.34)
+but we will not consider them in this book.
+Remark 7.3 (Plane and cylinder coordinates) The action in w-coordinate (cylinder)
+takes the same form as a result of the conformal invariance of the scalar field, which in
+practice results from the cancellation between the determinant and inverse metric. As a
+consequence, every quantity derived from the classical action (equation of motion, energy–
+momentum tensor. . . ) will have the same form in both coordinate systems: we will focus
+on the z-coordinate, writing the w-coordinate expression when it is insightful to compare.
+This is not anymore the case at the quantum level: anomalies may translate into differences
+between quantities: to differentiate between the plane and cylinder quantities an index “cyl”
+will be added when necessary (by convention, all quantities without qualification are on the
+plane).
+3The ϵ in the exponential is consistent with interpreting X and k as a contravariant vector.
+110
+
+7.1.3
+OPE
+The OPE between X and itself is directly found from the propagator:
+X(z)X(w) ∼ −ϵℓ2
+2
+ln(z − w).
+(7.35)
+By successive derivations, one finds the OPE between X and ∂X
+∂X(z)X(w) ∼ −ϵℓ2
+2
+1
+z − w,
+(7.36)
+and between ∂X with itself
+∂X(z)∂X(w) ∼ −ϵℓ2
+2
+1
+(z − w)2 .
+(7.37)
+The invariance under the permutation of z and w reflects the fact that X is bosonic and
+that both operators in (7.37) are identical.
+The OPE between ∂X and T allows to verify that the field ∂X is primary with h = 1:
+T(z)∂X(w) ∼ ∂X(w)
+(z − w)2 + ∂
+�
+∂X(w)
+�
+z − w
+.
+(7.38)
+The OPE of T with itself gives
+T(z)T(w) ∼ 1
+2
+1
+(z − w)4 +
+2T(w)
+(z − w)2 + ∂T(w)
+z − w
+(7.39)
+which shows that the central charge is
+c = 1.
+(7.40)
+One finds that the operator ∂nX has conformal weight
+h = n
+(7.41)
+since the OPE with T is
+T(z)∂nX(w) ∼ · · · + n ∂nX(w)
+(z − w)2 + ∂(∂nX(w))
+z − w
+(7.42)
+where the dots indicate higher negative powers of (z − w). These states are not primary for
+n ≥ 2. Explicitly, for n = 2, one finds
+T(z)∂2X(w) ∼ 2 ∂X(w)
+(z − w)3 +
+2 ∂2X
+(z − w)2 + ∂(∂2X(w))
+z − w
+.
+(7.43)
+The OPE of a vertex operator with the current J is
+J(z)Vk(w, ¯w) ∼ ℓ2k
+2
+Vk(w, ¯w)
+z − w
+.
+(7.44)
+This shows that the vertex operators Vk are eigenstates of the U(1) holomorphic current
+with the eigenvalue given by the momentum (with a normalization of ℓ2). Then, the OPE
+with T:
+T(z)Vk(w, ¯w) ∼ hk Vk(w, ¯w)
+(z − w)2
++ ∂Vk(w, ¯w)
+z − w
+(7.45)
+111
+
+together with its anti-holomorphic counterpart show that the Vk are primary operators with
+weight
+(hk, ¯hk) =
+�ϵℓ2k2
+4
+, ϵℓ2k2
+4
+�
+,
+∆k = ϵℓ2k2
+2
+,
+sk = 0.
+(7.46)
+Note that classically hk = 0 since ℓ ∼ ℏ [246, p. 81]. The weight is invariant under k → −k.
+Finally, the OPE between two vertex operators is
+Vk(z, ¯z)Vk′(w, , ¯w) ∼
+Vk+k′(w, ¯w)
+(z − w)−ϵkk′ℓ2/2 ,
+(7.47)
+where only the leading term (non-necessarily singular) is displayed. In particular, correlation
+functions should be computed for ϵkk′ < 0 in order to avoid exponential growth.
+Computation – Equation (7.38)
+T(z)∂X(w) = − ϵ
+ℓ2 :∂X(z)∂X(z): ∂X(w) ∼ −2ϵ
+ℓ2 :∂X(z)∂X(z): ∂X(w) ∼
+∂X(z)
+(z − w)2 .
+The result (7.38) follows by Taylor expanding the numerator.
+Computation – Equation (7.39)
+T(z)∂X(w) = 1
+ℓ4 :∂X(z)∂X(z): :∂X(w)∂X(w):
+∼ 1
+ℓ4
+�
+:∂X(z)∂X(z): :∂X(w)∂X(w): + :∂X(z)∂X(z): :∂X(w)∂X(w):
++ :∂X(z)∂X(z): :∂X(w)∂X(w): + perms
+�
+∼ 2 × 1
+4
+1
+(z − w)4 − 4 × 1
+2ℓ2
+1
+(z − w)2 :∂X(z)∂X(w):
+∼ 1
+2
+1
+(z − w)4 − 2
+ℓ2
+1
+(z − w)2
+�
+:∂X(w)∂X(w): + (z − w) :∂2X(w)∂X(w):
+�
+.
+Computation – Equation (7.42)
+T(z)∂nX(w) ∼ ∂n−1
+w
+∂X(z)
+(z − w)2
+∼ n!
+∂X(z)
+(z − w)n+1
+∼
+n!
+(z − w)n+1
+�
+· · · +
+1
+(n − 1)! (z − w)n−1∂n−1(∂X(w))
++ 1
+n! (z − w)n∂n(∂X(w))
+�
+.
+112
+
+Computation – Equation (7.44)
+Using (6.160b), one has:
+∂X(z)Vk(w, ¯w) ∼ iϵk ∂X(z)X(w) Vk(w, ¯w) ∼ iϵk
+�
+−ϵℓ2
+2
+1
+z − w
+�
+Vk(w, ¯w).
+Computation – Equation (7.45)
+T(z)Vk(w, ¯w) ∼ − ϵ
+ℓ2 :∂X(z)∂X(z): :eiϵkX(w, ¯
+w):
+∼ iϵk
+2
+1
+z − w ∂X(z) :eiϵkX(w, ¯
+w): − ϵ
+ℓ2 ∂X(z) :∂X(z)eiϵkX(w, ¯
+w):
+∼ iϵk
+2
+1
+z − w
+�
+:∂X(z) eiϵkX(w, ¯
+w): + ∂X(z) :eiϵkX(w, ¯
+w):
+�
++ iϵk
+2
+:∂X(z)eiϵkX(w, ¯
+w):
+z − w
+∼ ϵk2ℓ2
+4
+Vk(w, ¯w)
+(z − w)2 + iϵk :∂X(w)eiϵkX(w, ¯
+w):
+z − w
+.
+In the first line, we consider a single contraction (hence, there is no factor of 2): the
+reason is that considering the contractions symmetrically and not successively counts
+twice the first term of the last line. Indeed, there is only one way to generate this term.
+It is also possible to achieve the same result by expanding the exponential.
+Computation – Equation (7.47)
+Using (6.160c) and keeping only the leading term, one has:
+Vk(z, ¯z)Vk′(w, ¯w) ∼ exp
+�
+− kk′ X(z, ¯z)X(w, ¯w)
+�
+:eiϵkX(z,¯z)eiϵk′X(w, ¯
+w):
+∼ (z − w)ϵkk′ℓ2/2 Vk+k′(w, ¯w).
+7.1.4
+Mode expansions
+Since ∂X is holomorphic and of weight h = 1, it can be expanded as:4
+∂X = −i
+�
+ℓ2
+2
+�
+n∈Z
+αn z−n−1,
+¯∂X = −i
+�
+ℓ2
+2
+�
+n∈Z
+¯αn ¯z−n−1,
+(7.48)
+where an individual mode can be extracted with a contour integral:
+αn = i
+�
+dz
+2πi zn−1∂X(z),
+¯αn = i
+�
+dz
+2πi zn−1 ¯∂X(z).
+(7.49)
+4The Fourier expansion is taken to be identical for ϵ = ±1 fields since ∂X is contravariant in target
+space. The difference between the two cases will appear in the commutators.
+113
+
+Integrating this formula gives:
+X(z) = xL
+2 − i
+�
+ℓ2
+2 α0 ln z + i
+�
+ℓ2
+2
+�
+n̸=0
+αn
+n z−n,
+X(¯z) = xR
+2 − i
+�
+ℓ2
+2 ¯α0 ln ¯z + i
+�
+ℓ2
+2
+�
+n̸=0
+¯αn
+n ¯z−n.
+(7.50)
+The zero-modes are respectively α0 and ¯α0 for ∂X and ¯∂X, and xL and xR for XL and
+XR. The meaning of the modes will become clearer in Section 7.1.5 where we study the
+commutation relations.
+First, we relate the zero-modes α0 and ¯α0 to the conserved charges pL and pR (7.28) of
+the U(1) current:
+pL =
+α0
+√
+2ℓ2 ,
+pR =
+¯α0
+√
+2ℓ2
+(7.51)
+such that
+X(z) = xL
+2 − iℓ2 pL ln z + i
+�
+ℓ2
+2
+�
+n̸=0
+αn
+n z−n,
+(7.52)
+Then, the relations (7.28) and (7.30) allow to rewrite this result in terms of the momentum
+p and winding w:
+p =
+1
+√
+2ℓ2
+�
+α0 + ¯α0
+�
+,
+w =
+1
+√
+2ℓ2
+�
+α0 − ¯α0
+�
+.
+(7.53)
+These relations can be inverted as
+α0 =
+�
+ℓ2
+2 (p + w),
+¯α0 =
+�
+ℓ2
+2 (p − w).
+(7.54)
+In the same sense that there are two momenta pL and pR conjugated to xL and xR,
+it makes sense to introduce two coordinates x and q conjugated to p and w. From string
+theory, the operator x is called the center of mass. The expression (7.54) suggests to write:
+xL = x + q,
+xR = x − q,
+(7.55)
+and conversely:
+x = 1
+2 (xL + xR),
+q = 1
+2 (xL − xR).
+(7.56)
+In terms of these new variables, the expansion of the full X(z, ¯z) reads:
+X(z, ¯z) = x − i ℓ2
+2
+�
+p ln |z|2 + w ln z
+¯z
+�
++ i
+�
+ℓ2
+2
+�
+n̸=0
+1
+n
+�
+αn z−n + ¯αn ¯z−n�
+.
+(7.57)
+In terms of the coordinates on the cylinder, the part without oscillations becomes:
+X(τ, σ) = x − i ℓ2 pτ + ℓ2 wσ + · · ·
+(7.58)
+Note how the presence of ℓ2 gives the correct scale to the second term. The mode q does not
+appear at all, and x is the zero-mode of the complete field X(z, ¯z). As it is well-known, the
+physical interpretation of x and p is as the position and momentum of the centre-of-mass
+of the string.5 If there is a compact dimension, then w counts the number of times the
+string winds around it, and q can be understood as the position of the centre-of-mass after
+a T-duality.6
+5In worldsheet Lorentzian signature, this becomes X(τ, σ) = x + ℓ2 pt + ℓ2 wσ as expected.
+6T-duality and compact bosons fall outside the scope of this book and we refer the reader to [265,
+chap. 17, 193, chap. 8] for more details.
+114
+
+Computation – Equation (7.51)
+pL =
+1
+2πi
+�
+dz J = i
+ℓ2
+1
+2πi
+�
+dz ∂X = i
+ℓ2
+1
+2πi
+�
+dz ∂X
+=
+1
+√
+2ℓ2
+1
+2πi
+�
+dz
+�
+n
+αn z−n−1 =
+1
+√
+2ℓ2 α0.
+The computation gives pR after replacing α0 by ¯α0.
+If the scalar field is non-compact but periodic on the cylinder, the periodicity condition
+X(τ, σ + 2π) ∼ X(τ, σ)
+(7.59)
+translates as
+X(e2πiz, e−2πi¯z) ∼ X(z, ¯z).
+(7.60)
+Evaluating the LHS from (7.50) gives a constraint on the zero-modes:
+X(e2πiz, e−2πi¯z) = X(z, ¯z) − i
+�
+ℓ2
+2 (α0 − ¯α0),
+(7.61)
+which implies
+α0 = ¯α0
+=⇒
+pL = pR = p
+2,
+w = 0.
+(7.62)
+The other cases will not be discussed in this book, but we still use the general notation to
+make the contact with the literature easier. This also implies that XL and XR cannot be
+periodic independently. Hence, the zero-mode couples the holomorphic and anti-holomorphic
+sectors together.
+The number operators Nn ¯Nn at level n > 0 are defined by:
+Nn = ϵ
+n α−nαn,
+¯Nn = ϵ
+n ¯α−n¯αn.
+(7.63)
+The modes have been normal ordered. They count the number of excitations at the level n:
+the factor n−1 is necessary because the modes are not canonically normalized. Then, one
+can build the level operators
+N =
+�
+n>0
+n Nn.
+(7.64)
+They count the number of excitations at level n weighted by the level itself. This corresponds
+to the total energy due to the oscillations (the higher the level, the more energy it needs to
+be excited).
+The Virasoro operators are
+Lm = ϵ
+2
+�
+n
+:αnαm−n:
+(7.65)
+For m ̸= 0, we have
+m ̸= 0 :
+Lm = ϵ
+2
+�
+n̸=0,m
+:αnαm−n: + ϵ α0αm,
+(7.66)
+there is no ordering ambiguity and the normal order can be removed. In the case of the
+zero-mode, one finds
+L0 = ϵ
+2
+�
+n
+:αnα−n: = N + ϵ
+2 α2
+0 = N + ϵℓ2 p2
+L,
+(7.67)
+115
+
+using (7.64) and (7.51). It is also useful to define �L0 which corresponds to L0 stripped from
+the zero-mode contribution:
+�L0 := N.
+(7.68)
+Similarly, the anti-holomorphic zero-mode is
+¯L0 = ¯N + ϵℓ2 p2
+R,
+�¯L0 := ¯N,
+(7.69)
+such that
+L+
+0 = N + ¯N + ϵℓ2 (p2
+L + p2
+R) = N + ¯N + ϵℓ2
+2 (p2 + w2),
+(7.70a)
+L−
+0 = N − ¯N + ϵℓ2 (p2
+L − p2
+R) = N − ¯N + ϵℓ2 wp,
+(7.70b)
+where L±
+0 := L0 ± ¯L0 as defined in (6.118). The last equality of each line follows from
+(7.31b). The expression of L+
+0 for N = ¯N = 0 matches the weights (7.46) of the vertex
+operators for pL = pR = p/2 (no winding), which will be interpreted below. It is a good
+place to stress that pL, pR, p and w are operators, while k is a number.
+7.1.5
+Commutators
+The commutators can be computed from (6.75a) knowing the OPE (7.37). The modes of
+∂X and ¯∂X satisfy
+[αm, αn] = ϵ m δm+n,0,
+[¯αm, ¯αn] = ϵ m δm+n,0,
+[αm, ¯αn] = 0
+(7.71)
+for all m, n ∈ Z (including the zero-modes). The appearance of the factor m in the RHS
+explains the normalization of the number operator (7.63).
+From the commutators of the zero-modes, we directly find the ones for the momentum
+and winding:
+[p, w] = [p, p] = [w, w] = 0,
+[p, αn] = [p, ¯αn] = [w, αn] = [w, ¯αn] = 0.
+(7.72)
+The OPE (7.36) yields
+[xL, pL] = iϵ,
+[xR, pR] = iϵ,
+(7.73)
+which can be used to determine the commutators of x and q:
+[x, p] = [q, w] = iϵ,
+[x, w] = [q, p] = 0.
+(7.74)
+This shows that (x, p) and (q, w) are pairs of conjugate variables. Interestingly, the winding
+number w commutes will all other modes except q, but the latter disappears from the
+description. Hence, it can be interpreted as a number which labels different representations:
+if no other principle (like periodicity) forbids w ̸= 0, then one can except to have states with
+all possible w in the spectrum, each value of w forming a different sector. There are other
+interpretations from the point of view of T-duality and double field theory [107, 110, 189,
+265].
+The commutator of the modes with the Virasoro operators is
+[Lm, αn] = −n αm+n.
+(7.75)
+as expected from (6.115). For m = 0, this reduces to
+[L0, α−n] = n α−n,
+(7.76)
+which shows that negative modes increase the energy.
+The commutator of the creation
+modes α−n with the number operators is
+[Nm, α−n] = α−mδm,n.
+(7.77)
+116
+
+7.1.6
+Hilbert space
+The Hilbert space of the free scalar has the structure of a Fock space.
+From (7.76), the momentum p commutes with the Hamiltonian L+
+0 such that it is a
+good quantum number to label the states:7 this translates the fact the action (7.1) does not
+depend on the conjugate variable x. As a consequence, there exists a family of vacua |k⟩.
+The vacua |k⟩ are the states related to the vertex operators (7.33) through the state-
+operator correspondence:
+|k⟩ := lim
+z,¯z→0 Vk(z, ¯z) |0⟩ = eiϵkx |0⟩ ,
+(7.78)
+where |0⟩ is the SL(2, C) vacuum and x is the zero-mode of X(z, ¯z). That this identification
+is correct follows by applying the operator p:
+p |k⟩ = k |k⟩ .
+(7.79)
+The notation is consistent with the one of the SL(2, C) vacuum since p |0⟩ = 0.
+The vacuum is annihilated by the action of the positive-frequency modes:
+∀n > 0 :
+αn |k⟩ = 0,
+(7.80)
+which is equivalent to
+Nn |k⟩ = 0.
+(7.81)
+The different vacua are each ground state of a Fock space (they are all equivalent), but they
+are not ground states of the Hamiltonian since they have different energies:
+L+
+0 |k⟩ = 2ϵℓ2 k2 |k⟩ ,
+L−
+0 |k⟩ = 0,
+(7.82)
+using (7.70). The SL(2, C) vacuum is the lowest (highest) energy state if ϵ = 1 (ϵ = −1).
+The Fock space F(k) built from the vacuum at momentum k is found by acting repet-
+itively with the negative-frequency modes. A convenient basis, the oscillator basis, is given
+by the states:
+F(k) = Span
+�
+|k; {Nn}⟩
+�
+,
+(7.83a)
+|k; {Nn}⟩ :=
+�
+n≥1
+(α−n)Nn
+�
+nNnNn!
+|k⟩ ,
+Nn ∈ N∗
+(7.83b)
+(we don’t distinguish the notations between the number operators and their eigenvalues).
+The full Hilbert space is given by:
+H =
+�
+R
+dk F(k).
+(7.84)
+Computation – Equation (7.78)
+We provide a quick argument to justify the second form of (7.78). Take the limit of
+(7.57) with w = 0:
+lim
+z,¯z→0 eiϵkX(z,¯z) |0⟩ = lim
+z,¯z→0 exp iϵk
+�
+�x − i ℓ2
+2 p ln |z|2 + i
+�
+ℓ2
+2
+�
+n̸=0
+1
+n
+�
+αn z−n + ¯αn ¯z−n�
+�
+� |0⟩
+= lim
+z,¯z→0 exp
+�
+�iϵkx − ϵk
+�
+ℓ2
+2
+�
+n̸=0
+1
+n
+�
+αn z−n + ¯αn ¯z−n�
+�
+� |0⟩ .
+7To simplify the discussion, we do not consider winding but only vertex operators of the form (7.33).
+117
+
+The second term from the first line disappears because p |0⟩ = 0.
+For ϵk > 0, as
+z, ¯z → 0, the terms with αn and ¯αn for n < 0 disappear since they are accompanied
+with a positive power of zn and ¯zn. The modes with n > 0 diverge but the minus sign
+makes the exponential to vanish. A more rigorous argument requires to normal order
+the exponential and then to use (7.80).
+Computation – Equation (7.79)
+p |k⟩ = 1
+ℓ2
+1
+2πi
+� �
+dz i∂X(z) + d¯z i¯∂X(¯z)
+�
+Vk(0, 0) |0⟩
+= 1
+ℓ2
+1
+2πi
+� �dz
+z
+ℓ2k
+2
++ d¯z
+¯z
+ℓ2k
+2
+�
+Vk(0, 0) |0⟩
+= k Vk(0, 0) |0⟩
+using (7.44).
+Remark 7.4 (Fock space and Verma module isomorphism) Note that, in the absence
+of the so-called null states, there is a one-to-one map between states in the α−n oscillator
+basis and in the L−n Virasoro basis. This translates an isomorphism between the Fock space
+and the Verma module of Vk.
+One hint for this relation is that applying α−n and L−n
+changes the weight (eigenvalue of L0) by the same amount, and there are as many operators
+in both basis.
+7.1.7
+Euclidean and BPZ conjugates
+Since X is a real scalar field, it is self-adjoint (6.95) such that
+x† = x
+p† = p,
+α†
+n = α−n.
+(7.85)
+This implies that the Virasoro operators (7.65) are Hermitian:
+L†
+n = L−n,
+(7.86)
+as expected since T(z) is self-adjoint for a free scalar field.
+As a consequence of (7.85), the adjoint of the vacuum |k⟩ follows from (7.78):
+⟨k| = |k⟩‡ =⟨0| e−iϵkx,
+⟨k| p =⟨k| k.
+(7.87)
+The BPZ conjugate (6.111) of the mode αn is:
+αt
+n = −(±1)nα−n,
+(7.88)
+where the sign depends on the choice of I± in (6.111). Using (7.53), this implies that the
+momentum operator gets a minus sign:8
+pt = −p,
+⟨−k| = |k⟩t .
+(7.89)
+The inner product between two vacua |k⟩ and |k′⟩ is normalized as:
+⟨k|k′⟩ = 2π δ(k − k′)
+(7.90)
+8Be careful that |k⟩ is not the state associated to the operator p through the state–operator correspond-
+ence. Instead, they are associated to Vk, see (7.78). This explains why ⟨k| ̸= (|k⟩)t as in (6.136).
+118
+
+such that the conjugate state (6.145) of the vacuum reads
+⟨kc| = 1
+2π ⟨k| .
+(7.91)
+The Hermitian and BPZ conjugate states are related as:
+|k⟩‡ = − |k⟩t ,
+(7.92)
+which can be interpreted as a reality condition on |k⟩.
+7.2
+First-order bc ghost system
+First-order systems describe two free fields called ghosts which have a first-order action
+and whose conformal weights sum to 1. Commuting (resp. anti-commuting) fields are often
+denoted by β and γ (resp. b and c) and correspondingly first-order systems are also called
+βγ or bc systems. We will introduce a sign ϵ = ±1 to denote the Grassmann parity of the
+fields and always write them as b and c. In string theory, first-order systems describe the
+Faddeev–Popov ghosts associated to reparametrizations and supersymmetries (Sections 2.4
+and 17.1).
+7.2.1
+Covariant action
+A first-order system is defined by two symmetric and traceless fields bµ1···µλ and cµ1···µλ−1
+called ghosts. For fields of integer spins, the dynamics is governed by the first-order action
+S = 1
+4π
+�
+d2x√g gµν bµµ1···µλ−1∇νcµ1···µλ−1
+(7.93)
+after taking into account the symmetries of the field indices. Obviously, for λ = 2, one
+recovers the reparametrization ghost action (2.145). The action (7.93) is invariant under
+Weyl transformations (the fields and covariant derivatives are inert) such that it describes
+a CFT on flat space.
+When the fields have half-integer spins (and often denoted as β and γ in this case), they
+carry a spinor index. In this case, the action contains a Dirac matrix, and the covariant
+derivative a spin connection.
+The ghost action (7.93) is invariant under a global U(1) symmetry
+bµ1···µn −→ e−iθbµ1···µn,
+cµ1···µn−1 −→ eiθcµ1···µn−1.
+(7.94)
+7.2.2
+Action on the complex plane
+The simplest description of the system is on the complex plane. Due to the conditions im-
+posed on the fields, they have only two independent components for all n, and the equations
+of motion imply that one is holomorphic, and the other anti-holomorphic:
+b(z) := bz···z(z),
+¯b(¯z) := b¯z···¯z(¯z),
+c(z) := cz···z(¯z),
+¯c(¯z) := c¯z···¯z(z).
+(7.95)
+In this language, the action becomes
+S = 1
+2π
+�
+d2z
+�
+b¯∂c + ¯b∂¯c).
+(7.96)
+This action gives the correct equations of motion
+∂¯b = 0,
+¯∂b = 0,
+∂¯c = 0,
+¯∂c = 0.
+(7.97)
+119
+
+Since the fields split into holomorphic and anti-holomorphic sectors, it is convenient to study
+only the holomorphic sector as usual.
+This system is even simpler than the scalar field
+because the zero-modes don’t couple both sectors.9 All formulas for the anti-holomorphic
+sector are directly obtained from the holomorphic one by adding bars on quantities, except
+for conserved charges which have an index L or R and are both written explicitly.
+The action describes a CFT, and the weight of the fields are given by
+h(b) = λ,
+h(c) = 1 − λ,
+h(¯b) = λ,
+h(¯c) = 1 − λ,
+(7.98)
+where λ = n if the fields are in a tensor representation, and λ = n + 1/2 if they are in a
+spinor-tensor representation. The holomorphic energy–momentum reads
+T = −λ :b∂c: + (1 − λ) :∂b c:
+(7.99a)
+= −λ :∂(bc): + :∂b c:
+(7.99b)
+= (1 − λ) :∂(bc): − :b ∂c:.
+(7.99c)
+Normal ordering is taken with respect to the SL(2, C) vacuum (6.121).
+Finally, both fields can be classically commuting or anticommuting (see below for the
+quantum commutators):
+b(z)c(w) = −ϵ c(w)b(z),
+b(z)b(w) = −ϵ b(w)b(z),
+c(z)c(w) = −ϵ c(w)c(z), (7.100)
+where ϵ denotes the Grassmann parity
+ϵ =
+�
++1
+anticommuting,
+−1
+commuting.
+(7.101)
+Sometimes, if ϵ = +1, one denotes b and c respectively by β and γ. If b and c are ghosts
+arising from Faddeev–Popov gauge fixing, then ϵ = 1 if λ is integer; and ϵ = −1 if λ is
+half-integer (“wrong” spin–statistics assignment).
+The U(1) global symmetry (7.94) reads infinitesimally
+δb = −ib,
+δc = ic,
+δ¯b = −i¯b,
+δ¯c = i¯c.
+(7.102)
+It is generated by the conserved ghost current with components:
+j(z) = −:b(z)c(z):,
+¯ȷ(¯z) = −:¯b(¯z)¯c(¯z):
+(7.103)
+and the associated charge is called the ghost number
+Ngh = Ngh,L + Ngh,R,
+Ngh,L =
+�
+dz
+2πi j(z),
+Ngh,R = −
+�
+d¯z
+2πi ¯ȷ(¯z).
+(7.104)
+This charge counts the number of c ghosts minus the number of b ghosts, such that
+Ngh(c) = 1,
+Ngh(b) = −1,
+Ngh(¯c) = 1,
+Ngh(¯b) = −1.
+(7.105)
+The propagator can be derived from the path integral
+�
+d′b d′c
+δ
+δb(z)
+�
+b(w)e−S[b,c]�
+= 0
+(7.106)
+which gives the differential equation
+δ(2)(z − w) + 1
+2π ⟨b(w)¯∂c(z)⟩ = 0.
+(7.107)
+Using (B.2), the solution is easily found to be
+⟨c(z)b(w)⟩ =
+1
+z − w.
+(7.108)
+9For the scalar field, the coupling of both sectors happened because of the periodicity condition (7.62).
+120
+
+Remark 7.5 The propagator is constructed with the path integral. For convenience, the
+zero-modes are removed from the measure: reintroducing them, one finds that the propag-
+ator is computed not in the conformal vacuum (which has no operator insertion), but in
+a state with ghost insertions. This explains why the propagator (7.108) is not of the form
+(6.62b). However, this form is sufficient to extract the OPE as changing the vacuum does
+not introduce singular terms.
+7.2.3
+OPE
+The OPEs between the b and c fields are found from the propagator (7.108):
+c(z)b(w) ∼
+1
+z − w,
+b(z)c(w) ∼
+ϵ
+z − w,
+(7.109a)
+b(z)b(w) ∼ 0,
+c(z)c(w) ∼ 0.
+(7.109b)
+The OPE of each ghost with T confirms the conformal weights in (7.98):
+T(z)b(w) ∼ λ
+b(w)
+(z − w)2 + ∂b(w)
+z − w ,
+(7.110a)
+T(z)c(w) ∼ (1 − λ)
+c(w)
+(z − w)2 + ∂c(w)
+z − w .
+(7.110b)
+The OPE of T with itself is
+T(z)T(w) ∼
+cλ/2
+(z − w)4 +
+2T(w)
+(z − w)2 + ∂T(w)
+z − w ,
+(7.111)
+where the central charge is:
+cλ = 2ϵ(−1 + 6λ − 6λ2) = −2ϵ
+�
+1 + 6λ(λ − 1)
+�
+.
+(7.112)
+Introducing the ghost charge:
+qλ = ϵ(1 − 2λ),
+(7.113)
+the central charge can also be written as
+cλ = ϵ(1 − 3q2
+λ).
+(7.114)
+This parameter will appear many times in this section and its meaning will become clearer
+as we proceed.
+The OPE between the ghost current (7.103) and the b and c ghosts read
+j(z)b(w) ∼ − b(w)
+z − w,
+(7.115a)
+j(z)c(w) ∼ c(w)
+z − w.
+(7.115b)
+The coefficients of the (z − w)−1 terms correspond to the ghost number of the b and c fields
+(7.105). More generally, the ghost number Ngh(O) of any operator O(z) is defined by
+j(z)O(w) ∼ Ngh(O) O(w)
+z − w.
+(7.116)
+The OPE for j with itself is
+j(z)j(w) ∼
+ϵ
+(z − w)2 .
+(7.117)
+121
+
+Finally, the OPE of the current with T reads:
+T(z)j(w) ∼
+qλ
+(z − w)3 +
+j(w)
+(z − w)2 + ∂j(w)
+z − w .
+(7.118)
+Due to the presence of the z−3 term, the current j(z) is not a primary field if qλ ̸= 0, that is,
+if λ ̸= 1/2. In that case, its transformation under changes of coordinates gets an anomalous
+contribution:
+j(z) = dw
+dz j′(w) + qλ
+2
+d
+dz ln dw
+dz = dw
+dz j′(w) + qλ
+2
+∂2
+zw
+∂zw .
+(7.119)
+This implies in particular that the currents on the plane and on the cylinder (w = ln z) are
+related by:
+j(z) = dw
+dz
+�
+jcyl(w) − qλ
+2
+�
+,
+(7.120)
+which leads to the following relation between the ghost numbers on the plane and on the
+cylinder:
+Ngh = N cyl
+gh − qλ,
+Ngh,L = N cyl
+gh,L − qλ
+2 ,
+Ngh,R = N cyl
+gh,R − qλ
+2 .
+(7.121)
+For this reason, it is important to make clear the space with respect to which is given the
+ghost number: if not explicitly stated, ghost numbers in this book are given on the plane.10
+Due to this anomaly, one finds that the ghost number is not conserved on a curved space:
+N c − N b = −ϵ qλ
+2
+χg = (1 − 2λ)(g − 1),
+(7.122)
+where χg is the Euler characteristics (2.4), N b and N c are the numbers of b and c operators.
+In string theory, where the only ghost insertions are zero-modes, this translates into a
+statement on the number of zero-modes to be inserted. Hence, this can be interpreted as a
+generalization of (2.72). For a proof, see for example [24, p. 397].
+Computation – Equation (7.110a)
+T(z)b(w) =
+�
+− λ :b(z)∂c(z): + (1 − λ) :∂b(z) c(z):
+�
+b(w)
+∼ −λ :b(z)∂c(z): b(w) + (1 − λ) :∂b(z) c(z): b(w)
+∼ −λ b(z)∂z
+1
+z − w + (1 − λ) ∂b(z)
+1
+z − w
+∼ λ
+�
+b(w) +((((((
+(z − w)∂b(w)
+�
+1
+(z − w)2 + (1 − �λ) ∂b(w)
+z − w .
+10Other references, especially old ones, give it on the cylinder. This can be easily recognized if some ghost
+numbers in the holomorphic sector are half-integers: for the reparametrization ghosts, qλ is an integer such
+that the shift in (7.121) is a half-integer.
+122
+
+Computation – Equation (7.110b)
+T(z)c(w) =
+�
+− λ :b(z)∂c(z): + (1 − λ) :∂b(z)c(z):
+�
+c(w)
+∼ ϵλ :∂c(z)b(z): c(w) − ϵ(1 − λ) :c(z)∂b(z): c(w)
+∼ λ ∂c(z)
+z − w − (1 − λ) c(z) ∂z
+1
+z − w
+∼ λ ∂c(w)
+z − w + (1 − λ)
+�
+c(w) + (z − w)∂c(w)
+�
+1
+(z − w)2
+∼ (1 − λ)
+c(w)
+(z − w)2 +
+∂c(w)
+(z − w)2 .
+Computation – Equation (7.115a)
+j(z)b(w) = −:b(z)c(z): b(w) ∼ −:b(z)c(z): b(w) ∼ − b(z)
+z − w ∼ − b(w)
+z − w.
+Computation – Equation (7.115b)
+j(z)c(w) = −:b(z)c(z): c(w) ∼ ϵ :c(z)b(z): c(w) ∼ c(z)
+z − w ∼ c(w)
+z − w.
+Computation – Equation (7.117)
+j(z)j(w) = :b(z)c(z): :b(w)c(w):
+∼ :b(z)c(z): :b(w)c(w): + :b(z)c(z): :b(w)c(w): + :b(z)c(z): :b(w)c(w):
+∼
+ϵ
+(z − w)2 + ϵ :c(z)b(w):
+z − w
++ :b(z)c(w):
+z − w
+∼
+ϵ
+(z − w)2 .
+7.2.4
+Mode expansions
+The b and c ghosts are expanded as
+b(z) =
+�
+n∈Z+λ+ν
+bn
+zn+λ ,
+c(z) =
+�
+n∈Z+λ+ν
+cn
+zn+1−λ ,
+(7.123)
+where ν = 0, 1/2 depends on ϵ and on the periodicity of the fields, see (6.102). The modes
+are extracted with the contour formulas
+bn =
+�
+dz
+2πi zn+λ−1b(z),
+cn =
+�
+dz
+2πi zn−λc(z).
+(7.124)
+Ghosts with λ ∈ Z have integer indices and ν = 0 (we don’t consider ghosts with twisted
+boundary conditions). On the other hand, ghosts with λ ∈ Z + 1/2 have integer indices
+123
+
+and ν = 1/2 in the R sector, and half-integer indices and ν = 0 in the NS sector (see
+Section 6.4.4). The choices in the boundary conditions arise from the Z2 symmetry of the
+action:
+b −→ −b,
+c −→ −c.
+(7.125)
+The number operators N b
+n and N c
+n are defined to count the numbers of excitations above
+the SL(2, C) vacuum of b and c ghosts at level n:
+N b
+n = :b−ncn:,
+N c
+n = ϵ :c−nbn:.
+(7.126)
+The definitions follow from the commutators (7.134). Then, the level operators N b and N c
+are obtained by summing over n:
+N b =
+�
+n>0
+n N b
+n,
+N c =
+�
+n>0
+n N c
+n.
+(7.127)
+The Virasoro operators are
+Lm =
+�
+n
+�
+n − (1 − λ)m
+�
+:bm−ncn: =
+�
+n
+(λm − n) :bncm−n:.
+(7.128)
+Of particular importance is the zero-mode
+L0 = −
+�
+n
+n :bnc−n: =
+�
+n
+n :b−ncn:.
+(7.129)
+We will give the expression of L0 in terms of the level operators below, see (7.159). To
+do this, we will first need to change the normal ordering, which first requires to study the
+Hilbert space.
+The modes of the ghost current are
+jm = −
+�
+n
+:bm−ncn: = −
+�
+n
+:bncm−n:.
+(7.130)
+Note that the zero-mode of the current also equals the ghost number
+Ngh,L = j0 = −
+�
+n
+:b−ncn:.
+(7.131)
+When both the holomorphic and anti-holomorphic sectors enter, it is convenient to in-
+troduce the combinations
+b±
+n = bn ± ¯bn,
+c±
+n = 1
+2 (cn ± ¯cn).
+(7.132)
+The normalization of b±
+m is chosen to match the one of L±
+m (6.118), and the one of c±
+m such
+that (7.135) holds. Note the following useful identities:
+b−
+n b+
+n = 2bn¯bn,
+c−
+n c+
+n = 1
+2 cn¯cn.
+(7.133)
+124
+
+Computation – Equation (7.128)
+T = −λ :b∂c: + (1 − λ) :∂bc:
+=
+�
+m,n
+�
+λ :bmcn:
+n + 1 − λ
+zm+λzm+2−λ − (1 − λ) :bmcn:
+m + λ
+zm+λ+1zm+1−λ
+�
+=
+�
+m,n
+�
+λ (n + 1 − λ) − (1 − λ)(m + λ)
+� :bmcn:
+zm−n+2
+=
+�
+m,n
+�
+λ (n + 1 − λ) − (1 − λ)(m − n + λ)
+� :bm−ncn:
+zm+2
+=
+�
+m,n
+(n − m + λm) :bm−ncn:
+zm+2
+=
+�
+m
+Lm
+zm+2 .
+The fourth line follows from shifting m → m−n. The second equality in (7.128) follows
+by shifting n → m − n.
+Computation – Equation (7.130)
+j = −:bc: =
+�
+m,n
+:bmcn:
+zm+λzn+1−λ =
+�
+m,n
+:bm−ncn:
+zm+1
+=
+�
+m
+jm
+zm+1 .
+7.2.5
+Commutators
+The (anti)commutators between the modes bn and cn read:
+[bm, cn]ϵ = δm+n,0,
+[bm, bn]ϵ = 0,
+[cm, cn]ϵ = 0.
+(7.134)
+Therefore, the modes with n < 0 are creation operators and the modes with n > 0 are
+annihilation operators:
+• a b ghost excitation at level n > 0 is created by b−n and annihilated by cn;
+• a c ghost excitation at level n > 0 is created by c−n and annihilated by bn.
+In terms of b±
+m and c±
+m (7.132), we have:
+[b+
+m, c+
+n ]ϵ = δm+n,
+[b−
+m, c−
+n ]ϵ = δm+n.
+(7.135)
+The commutators of the number operators with the modes are:
+[N b
+m, b−n] = b−nδm,n,
+[N c
+m, c−n] = c−nδm,n,
+(7.136)
+while those between the Ln and the ghost modes are:
+[Lm, bn] =
+�
+m(λ − 1) − n
+�
+bm+n,
+[Lm, cn] = −(mλ + n)cm+n,
+(7.137)
+in agreement with (6.115).
+If n ∈ Z, each ghost field has zero-modes b0 and c0 which
+commutes with L0
+[L0, b0] = 0,
+[L0, c0] = 0.
+(7.138)
+125
+
+The commutator of the current modes reads
+[jm, jn] = m δm+n,0.
+(7.139)
+Then, the commutator with the Virasoro operators are
+[Lm, jn] = −njm+n + qλ
+2 m(m + 1)δm+n,0.
+(7.140)
+Finally, the commutators of the ghost number operator with the ghosts are:
+[Ngh, b(w)] = −b(w),
+[Ngh, c(w)] = c(w).
+(7.141)
+Computation – Equation (7.134)
+[bm, cn]ϵ = ϵ
+�
+C0
+dw
+2πi w−1
+�
+Cw
+dz
+2πi z−1 wn+λzm−λ+1b(z)c(w)
+∼ ϵ
+�
+C0
+dw
+2πi w−1
+�
+Cw
+dz
+2πi z−1 wn+λzm−λ+1
+ϵ
+z − w
+=
+�
+C0
+dw
+2πi wm+n−1 = δm+n,0.
+Computation – Equation (7.141)
+[Ngh, b(w)] =
+�
+dz
+2πi j(z)b(w) ∼ −
+�
+dz
+2πi
+b(w)
+z − w = −b(w).
+The computation for c is similar.
+7.2.6
+Hilbert space
+The SL(2, C) vacuum |0⟩ (6.121) is defined by:
+∀n > −λ :
+bn |0⟩ = 0,
+∀n > λ − 1 :
+cn |0⟩ = 0.
+(7.142)
+If λ > 1, there are positive modes which do not annihilate the vacuum.
+To simplify the notation, we consider the case λ ∈ Z, the half-integer case following by
+shifting the indices by 1/2. Since the modes {c1, . . . , cλ−1} do not annihilate |0⟩, one can
+create states
+|n1, . . . , nλ−1⟩ = cn1
+1 · · · cnλ−1
+λ−1 |0⟩
+(7.143)
+which have negative energies:
+L0 |n1, . . . , nλ−1⟩ = −
+�
+�
+λ−1
+�
+j=1
+j nj
+�
+� |n1, . . . , nλ−1⟩ ,
+(7.144)
+where (7.137) has been used. Moreover, this state is degenerate due to the existence of
+zero-modes since they commute with the Hamiltonian – see (7.138). As a consequence, it
+must be in a representation of the zero-mode algebra.
+If the ghosts are commuting (ϵ = −1), then it seems hard to make sense of the theory
+since one can find a state of arbitrarily negative energy since ni ∈ N. The zero-modes make
+the problem even worse. The appropriate interpretation of these states will be discussed in
+the context of the superstring theory for λ = 3/2 (superconformal ghosts).
+In the rest of this section, we focus on the Grassmann odd case ϵ = 1.
+126
+
+Energy vacuum (Grassmann odd)
+Since ni = 0 or ni = 1 for anticommuting ghosts (ϵ = 1), there is a state of lowest energy.
+This is the energy vacuum (6.129). Since the zero-modes b0 and c0 commute with L0, it is
+doubly degenerate. A convenient basis is
+�
+| ↓⟩ , | ↑⟩
+�
+,
+(7.145)
+where
+| ↓⟩ := c1 · · · cλ−1 |0⟩ ,
+| ↑⟩ := c0c1 · · · cλ−1 |0⟩ .
+(7.146)
+A general vacuum is a linear combination of the two basis vacua:
+|Ω⟩ = ω↓ | ↓⟩ + ω↑ | ↑⟩ ,
+ω↓, ω↑ ∈ C.
+(7.147)
+The algebra of these vacua is the one of a two-state system:
+b0 | ↑⟩ = | ↓⟩ ,
+c0 | ↓⟩ = | ↑⟩ ,
+b0 | ↓⟩ = 0,
+c0 | ↑⟩ = 0.
+(7.148)
+Hence, for the vacuum | ↓⟩ (resp. | ↑⟩), b0 (resp. c0) acts as an annihilation operator, and
+conversely c0 (resp. b0) acts as a creation operator. Finally, both states are annihilated by
+all positive modes:
+∀n > 0 :
+bn | ↓⟩ = bn | ↑⟩ = 0,
+cn | ↓⟩ = bn | ↓⟩ = 0.
+(7.149)
+Note that the SL(2, C) vacuum can be recovered by acting with b−n with n < λ:
+|0⟩ = b1−λ · · · b−1 | ↓⟩ = b1−λ · · · b−1b0 | ↑⟩ .
+(7.150)
+The zero-point energy (6.130) of these states is the conformal weight of the vacuum:
+L0 | ↓⟩ = aλ | ↓⟩ ,
+L0 | ↑⟩ = aλ | ↑⟩ ,
+(7.151)
+where aλ can be written in various forms:
+aλ = −
+λ−1
+�
+n=1
+n = −λ(λ − 1)
+2
+= cλ
+24 + 2
+24.
+(7.152)
+Taking into account the anti-holomorphic sector leads to a four-fold degeneracy. The
+basis
+�
+| ↓↓⟩ , | ↑↓⟩ , | ↓↑⟩ , | ↑↑⟩
+�
+,
+(7.153)
+is built as follows:
+| ↓↓⟩ := c1¯c1 · · · cλ−1¯cλ−1 |0⟩ ,
+| ↑↓⟩ := c0 | ↓↓⟩ ,
+| ↓↑⟩ := ¯c0 | ↓↓⟩ ,
+| ↑↑⟩ := c0¯c0 | ↓↓⟩ .
+(7.154)
+The modes b0 and ¯b0 can be used to flip the arrows downward, leading to the following
+algebra:
+c0 | ↓↓⟩ = | ↑↓⟩ ,
+¯c0 | ↓↓⟩ = | ↓↑⟩ ,
+c0 | ↓↑⟩ = −¯c0 | ↑↓⟩ = | ↑↑⟩ ,
+b0 | ↑↑⟩ = | ↓↑⟩ ,
+¯b0 | ↑↑⟩ = − | ↑↓⟩ ,
+b0 | ↑↓⟩ = ¯b0 | ↓↑⟩ = | ↓↓⟩ ,
+(7.155a)
+The vacua are annihilated by different combinations of the zero-modes:
+b0 | ↓↓⟩ = ¯b0 | ↓↓⟩ = 0,
+c0 | ↑↓⟩ = ¯b0 | ↑↓⟩ = 0,
+b0 | ↓↑⟩ = ¯c0 | ↓↑⟩ = 0,
+c0 | ↑↑⟩ = ¯c0 | ↑↑⟩ = 0.
+(7.155b)
+127
+
+In these manipulations, one has to be careful to correctly anti-commute the modes with the
+ones hidden in the definitions of the vacua.
+There is a second basis which is more natural when using the zero-modes c±
+0 and b±
+0
+(7.132):
+�
+| ↓↓⟩ , |+⟩ , |−⟩ , | ↑↑⟩
+�
+,
+(7.156)
+where the two vacua |±⟩ are combinations of the | ↓↑⟩ and | ↑↓⟩ vacua:
+|±⟩ = | ↑↓⟩ ± | ↓↑⟩ .
+(7.157)
+The different vacua are naturally related by acting with c±
+0 and b±
+0 which act as raising and
+lowering operators:
+c±
+0 | ↓↓⟩ = 1
+2 |±⟩ ,
+c∓
+0 |±⟩ = ± | ↑↑⟩ ,
+b±
+0 |±⟩ = ±2 | ↓↓⟩ ,
+b∓
+0 | ↑↑⟩ = ± |±⟩ .
+(7.158a)
+From the previous relations, it follows that the different vacua are annihilated by the zero-
+modes as follow:
+b+
+0 | ↓↓⟩ = b−
+0 | ↓↓⟩ = 0,
+c−
+0 |−⟩ = b+
+0 |−⟩ = 0,
+c+
+0 |+⟩ = b−
+0 |+⟩ = 0
+c+
+0 | ↑↑⟩ = c−
+0 | ↑↑⟩ = 0,
+(7.158b)
+This also means that we have
+c−
+0 c+
+0 | ↓↓⟩ = 1
+2 | ↑↑⟩ ,
+b+
+0 b−
+0 | ↑↑⟩ = 2 | ↓↓⟩ .
+(7.158c)
+Computation – Equation (7.158)
+2 c+
+0 |±⟩ = (c0 + ¯c0) | ↑↓⟩ ± (c0 + ¯c0) | ↓↑⟩ = ¯c0 | ↑↓⟩ ± c0 | ↓↑⟩ = (−1 ± 1) | ↑↑⟩
+b+
+0 |±⟩ = (b0 + ¯b0) | ↑↓⟩ ± (b0 + ¯b0) | ↓↑⟩ = b0 | ↑↓⟩ ± ¯b0 | ↓↑⟩ = (1 ± 1) | ↓↓⟩
+2 c±
+0 | ↓↓⟩ = (c0 ± ¯c0) | ↓↓⟩ = c0 | ↓↓⟩ ± ¯c0 | ↓↓⟩ = | ↑↓⟩ ± | ↓↑⟩ = |±⟩
+b±
+0 | ↑↑⟩ = (b0 ± ¯b0) | ↑↑⟩ = b0 | ↑↑⟩ ± ¯b0 | ↑↑⟩ = | ↓↑⟩ ∓ | ↑↓⟩ = ∓ |∓⟩
+Energy normal ordering (Grassmann odd)
+We now turn towards the definition of the energy normal ordering (6.150). Ultimately, it
+will be found that | ↓⟩ is the physical vacuum in string theory. For this reason, the energy
+normal ordering
+⋆
+⋆ · · ·
+⋆
+⋆ is associated to the vacuum | ↓⟩ in order to resolve the ambiguity
+of the zero-modes. In particular, b0 is an annihilation operator in this case, while c0 is a
+creation operator. In the rest of this section, we translate the normal ordering of expressions
+from the conformal vacuum to the energy vacuum.
+The Virasoro operators Ln for n ̸= 0 have no ordering problems since the modes which
+compose them commute. The expression of L0 (7.129) in the energy ordering becomes
+L0 =
+�
+n
+n
+⋆
+⋆b−ncn
+⋆
+⋆ + aλ = N b + N c + aλ
+(7.159)
+where aλ is the zero-point energy (7.152) and N b and N c are the ghost mode numbers
+(7.127). The contribution of the non-zero modes is denoted by:
+�L0 = N b + N c.
+(7.160)
+128
+
+The expression can be rewritten to encompass all modes:
+Lm =
+�
+n
+�
+n − (1 − λ)m
+� ⋆
+⋆bm−ncn
+⋆
+⋆ + aλ δm,0
+(7.161)
+Similarly, the expression of the ghost number is
+Ngh,L = j0 =
+�
+n
+⋆
+⋆b−ncn
+⋆
+⋆ −
+�qλ
+2 + 1
+2
+�
+(7.162a)
+=
+�
+n>0
+�
+N c
+n − N b
+n
+�
++ 1
+2
+�
+N c
+0 − N b
+0
+�
+− qλ
+2 ,
+(7.162b)
+and thus:
+jm =
+�
+n
+⋆
+⋆bm−ncn
+⋆
+⋆ −
+�qλ
+2 + 1
+2
+�
+δm,0.
+(7.163)
+It is useful to define the ghost number without ghost zero-modes:
+�
+Ngh,L :=
+�
+n>0
+�
+N c
+n − N b
+n
+�
+.
+(7.164)
+One can straightforwardly compute the ghost number of the vacua:
+j0 | ↓⟩ = (λ − 1) | ↓⟩ =
+�
+−qλ
+2 − 1
+2
+�
+| ↓⟩ ,
+(7.165a)
+j0 | ↑⟩ = λ | ↑⟩ =
+�
+−qλ
+2 + 1
+2
+�
+| ↑⟩ .
+(7.165b)
+This confirms that the SL(2, C) vacuum has vanishing ghost number since | ↓⟩ contains
+exactly λ − 1 ghosts:
+j0 |0⟩ = 0.
+(7.166)
+Using (7.121) allows to write the ghost numbers on the cylinder:
+jcyl
+0
+| ↓⟩ = −1
+2 | ↓⟩ ,
+jcyl
+0
+| ↑⟩ = 1
+2 | ↑⟩ .
+(7.167)
+That both ghost numbers have same magnitude but opposite signs could be expected: since
+the ghost number changes as Ngh → −Ngh when b ↔ c, the mean value of the ghost number
+should be zero.
+Remark 7.6 (Ghost number conventions) Since the ghost number is an additive quan-
+tum number, it is always possible to shift its definition by a constant. This can be used to
+set the ghost numbers of the vacua to some other values. For example, [24, p. 116] adds
+qλ/2 to the ghost number in order to get Ngh = ±1/2 on the plane (instead of the cylinder).
+We do not follow this convention in order to keep the symmetry between the vacuum ghost
+numbers on the cylinder.
+129
+
+Computation – Equation (7.159)
+Start with (7.129) and use (6.157):
+L0 = −
+�
+n
+n :bnc−n: = −
+�
+n≤−λ
+n bnc−n + ϵ
+�
+n>−λ
+n c−nbn
+=
+�
+n≥λ
+n b−ncn + ϵ
+�
+n>−λ
+n c−nbn
+=
+�
+n≥λ
+n b−ncn + ϵ
+�
+n>0
+n c−nbn + ϵ
+0
+�
+n=−λ+1
+n c−nbn
+=
+�
+n≥λ
+n b−ncn + ϵ
+�
+n>0
+n c−nbn + ϵ
+λ−1
+�
+n=0
+n b−ncn + aλ
+=
+�
+n>0
+n b−ncn + ϵ
+�
+n>0
+n c−nbn + aλ,
+=
+�
+n
+n
+⋆
+⋆b−ncn
+⋆
+⋆ + aλ,
+using that
+0
+�
+n=−λ+1
+c−nbn = −
+λ−1
+�
+n=0
+n cnb−n = −
+λ−1
+�
+n=0
+n (−ϵ b−ncn + 1) = ϵ
+λ−1
+�
+n=0
+n b−ncn + aλ.
+The result also follows from (6.163).
+Computation – Equation (7.162)
+j0 = −
+�
+n
+:b−ncn: = −
+�
+n≥λ
+b−ncn + ϵ
+�
+n>−λ
+c−nbn
+= −
+�
+n≥λ
+b−ncn + ϵ
+�
+n>0
+c−nbn + ϵ
+λ−1
+�
+n=1
+cnb−n + ϵ c0b0
+= −
+�
+n≥λ
+b−ncn + ϵ
+�
+n>0
+c−nbn −
+λ−1
+�
+n=1
+b−ncn + ϵ(λ − 1) + ϵ c0b0
+= −
+�
+n>0
+b−ncn + ϵ
+�
+n>0
+c−nbn + ϵ(λ − 1) + ϵ c0b0.
+Finally, one can write
+ϵ(λ − 1) = −qλ
+2 − ϵ
+2.
+(7.168)
+The result also follows from (6.163). The second expression is obtained by symmetrizing
+the last term such that
+ϵ c0b0 + ϵ(λ − 1) = ϵ
+2 c0b0 + 1
+2(−b0c0 + ϵ) + ϵ(λ − 1)
+= 1
+2 (ϵ c0b0 − b0c0) + ϵ
+�
+λ − 1
+2
+�
+.
+130
+
+Structure of the Hilbert space (Grassmann odd)
+Since the zero-modes commute with the Hamiltonian and with all other negative- and
+positive-frequency modes, the Hilbert space is decomposed in several subspaces, each as-
+sociated to a zero-mode.11
+Starting with the holomorphic sector only, the Hilbert space Hgh is:
+Hgh = Hgh,0 ⊕ c0Hgh,0,
+Hgh,0 := Hgh ∩ ker b0,
+(7.169)
+which follows from the 2-state algebra (7.148). Obviously, one has c0Hgh,0 = Hgh ∩ ker c0.
+The oscillator basis of the Hilbert space Hgh,0 is generated by applying the negative-
+frequency modes and has the structure of a fermionic Fock space without zero-modes:
+Hgh,0 = Span
+� ��↓; {N b
+n}; {N c
+n}
+� �
+,
+(7.170a)
+��↓; {N b
+n}; {N c
+n}
+�
+=
+�
+n≥1
+(b−n)Nb
+n(c−n)Nc
+n | ↓⟩ ,
+N b
+n, N c
+n ∈ N∗
+(7.170b)
+(again, number operators and their eigenvalues are not distinguished). This means that
+Hgh,0 can also be regarded as a Fock space built on the vacuum | ↓⟩, for which c0 and b0
+are respectively creation and annihilation operators. Conversely, c0 and b0 are respectively
+annihilation and creation operators for c0Hgh,0.
+In particular, this means that any state can be written as the sum of two states
+ψ = ψ↓ + ψ↑,
+ψ↓ ∈ Hgh,0,
+ψ↑ ∈ c0Hgh,0,
+(7.171)
+with ψ↓ and ψ↑ built respectively on top of the | ↓⟩ and | ↑⟩ vacua.
+This pattern generalizes when considering both the holomorphic and anti-holomorphic
+sectors. In that case, the Hilbert space is decomposed in four subspaces:12
+Hgh = Hgh,0 ⊕ c0Hgh,0 ⊕ ¯c0Hgh,0 ⊕ c0¯c0Hgh,0,
+Hgh,0 := Hgh ∩ ker b0 ∩ ker¯b0.
+(7.172)
+Basis states of the Hilbert space Hgh,0 are:
+��↓↓; {N b
+n}; {N c
+n}; { ¯N b
+n}; { ¯N c
+n}
+�
+=
+�
+n≥1
+(b−n)N b
+n(¯b−n)
+¯
+Nb
+n(c−n)Nc
+n(¯c−n)
+¯
+Nc
+n | ↓↓⟩ ,
+N b
+n, ¯N b
+n, N c
+n, ¯N c
+n ∈ N∗.
+(7.173)
+A general state of Hgh can be decomposed as
+ψ = ψ↓↓ + ψ↑↓ + ψ↓↑ + ψ↑↑,
+(7.174)
+where each state is built by acting with negative-frequency modes on the corresponding
+vacuum.
+In terms of the second basis (7.156), the Hilbert space admits a second decomposition:
+Hgh = Hgh,0 ⊕ c+
+0 Hgh,0 ⊕ c−
+0 Hgh,0 ⊕ c−
+0 c+
+0 Hgh,0,
+Hgh,0 := Hgh ∩ ker b−
+0 ∩ ker b+
+0 .
+(7.175)
+11Due to the specific structure of the inner product defined below, these subspaces are not orthonormal
+to each other.
+12The reader should not get confused by the same symbol Hgh,0 as in the case of the holomorphic sector.
+131
+
+In view of applications to string theory, it is useful to introduce two more subspaces:
+Hgh,± := Hgh ∩ ker b±
+0 = Hgh,0 ⊕ c∓
+0 Hgh,0,
+(7.176)
+and the associated decomposition
+Hgh = Hgh,± ⊕ c±
+0 Hgh,±.
+(7.177)
+In off-shell closed string theory, the principal Hilbert space will be H−
+gh due to the level-
+matching condition. In this case, H−
+gh has the same structure as Hgh in the pure holomorphic
+sector, and c+
+0 plays the same role as c0. A state in H−
+gh is built on top of the vacua | ↓↓⟩
+and |+⟩.
+7.2.7
+Euclidean and BPZ conjugates
+In order for the Virasoro operators to be Hermitian, the bn and cn must satisfy the following
+conditions:
+b†
+n = ϵb−n,
+c†
+n = c−n.
+(7.178)
+Hence, bn is anti-Hermitian if ϵ = −1. The BPZ conjugates of the modes are:
+bt
+n = (−1)λ b−n,
+ct
+n = (−1)1−λ c−n,
+(7.179)
+using I+(z) with (6.111).
+In the rest of this section, we consider only the case ϵ = 1 and λ ∈ N. The adjoints of
+the vacuum read:
+| ↓⟩‡ =⟨0| c1−λ · · · c−1,
+| ↑⟩‡ =⟨0| c1−λ · · · c−1c0.
+(7.180)
+The BPZ conjugates of the vacua are:
+⟨↓ | := | ↓⟩t = (−1)(1−λ)2⟨0| c−1 · · · c1−λ,
+⟨↑ | := | ↑⟩t = (−1)λ(1−λ)⟨0| c0c−1 · · · c1−λ.
+(7.181)
+The signs are inconvenient but will disappear when considering both the left and right vacua
+together as in (7.154). We have the following relations:
+⟨↓ | = (−1)aλ+(1−λ)(2−λ) | ↓⟩‡ ,
+⟨↑ | = (−1)aλ | ↑⟩‡ ,
+(7.182)
+where aλ is the zero-point energy (7.152).
+Computation – Equation (7.182)
+To prove the relation, we can start from the BPZ conjugate ⟨↓ | and reorder the modes
+to bring them in the same order as the adjoint:
+⟨↓ | = (−1)(1−λ)2+ 1
+2 (2−λ)(1−λ) | ↓⟩‡ = (−1)−aλ+(1−λ)(2−λ) | ↓⟩‡
+The reordering gives a factor (−1) to the power:
+λ−2
+�
+i=1
+i = 1
+2(2 − λ)(1 − λ) = −aλ + 1 − λ.
+Similarly, for the second vacuum:
+⟨↑ | = (−1)λ(1−λ)− 1
+2 λ(1−λ) | ↑⟩‡ = (−1)
+1
+2 λ(1−λ) | ↑⟩‡ .
+132
+
+We can identify the power with (7.152).
+Then, we have the following relations:
+⟨↑ | b0 =⟨↓ | ,
+⟨↓ | c0 =⟨↑ | ,
+⟨↓ | b0 = 0,
+⟨↑ | c0 = 0.
+(7.183)
+There is a subtlety in defining the inner product because the vacuum is degenerate. If
+we write the two vacua as vectors
+| ↓⟩ =
+�
+0
+1
+�
+,
+| ↑⟩ =
+�
+1
+0
+�
+,
+(7.184)
+then the zero-modes have the following matrix representation:
+b0 =
+�0
+0
+1
+0
+�
+,
+c0 =
+�0
+1
+0
+0
+�
+.
+(7.185)
+These matrices are not Hermitian as required by (7.178): since Hermiticity follows from the
+choice of an inner product, it means that the vacua cannot form an orthonormal basis. An
+appropriate choice for the inner products is:13
+⟨↓ | ↓⟩ = ⟨↑ | ↑⟩ = 0,
+⟨↑ | ↓⟩ =⟨↓ | c0 | ↓⟩ =⟨0| c1−λ · · · c−1c0c1 · · · cλ−1 |0⟩ = 1.
+(7.186)
+The effect of changing the definition of the inner product or to consider a non-orthonormal
+basis is represented by the insertion of c0. The last condition implies that the conjugate
+state (6.145) to the SL(2, C) vacuum is:
+⟨0c| =⟨0| c1−λ · · · c−1c0c1 · · · cλ−1,
+⟨↓c | =⟨↑ | .
+(7.187)
+7.2.8
+Summary
+In this section we summarize the values of the parameters for different theories of interest
+(Table 7.1). The (η, ξ) system will be introduced in Chapter 17 in the bosonization of the
+super-reparametrization (β, γ) ghosts.
+The ψ± system can be used to describe spin-1/2
+fermions.
+ϵ
+λ
+qλ
+cλ
+aλ
+b, c (diff.)
+1
+2
+−3
+−26
+−1
+β, γ (susy.)
+−1
+3/2
+2
+11
+3/8
+ψ±
+1
+1/2
+0
+1
+0
+η, ξ
+1
+1
+−1
+−2
+0
+Table 7.1: Summary of the first-order systems. Remember that h(b) = λ and h(c) = 1 − λ.
+7.3
+Suggested readings
+• Free scalar: general references [246, sec. 4.1.3, 4.3, 4.6.2, 54, sec. 5.3.1, 6.3, 24, sec. 4.2,
+193, 128], topological current and winding [109, 265, sec. 17.2–3].
+• First-order system: general references [24, chap. 5, sec. 13.1, 128, sec. 4.15, 193,
+sec. 2.5], ghost vacua [151, sec. 15.3].
+13To avoid confusions, let us note that the adjoint in (7.182) are defined only through the adjoint of
+the modes (6.110) but not with respect to the inner product given here, which would lead to exchanging
+| ↓⟩‡ ∼⟨↑ | and | ↑⟩‡ ∼⟨↓ |.
+133
+
+Chapter 8
+BRST quantization
+Abstract
+The BRST quantization can be introduced either by following the standard
+QFT treatment (outlined in Section 3.2), or by translating it in the CFT language. One
+can then use all the CFT techniques to extract information on the spectrum, which makes
+this approach more powerful. Moreover, this also provides an elegant description of states
+and string fields. In this chapter, we set the stage of the BRST quantization using the CFT
+language and we apply it to string theory. The main results of this chapter are a proof of
+the no-ghost theorem and a characterization of the BRST cohomology (physical states).
+8.1
+BRST for reparametrization invariance
+The BRST symmetry we are interested in results from gauge fixing the reparametrization
+invariance.
+In this chapter, we focus on the holomorphic sector: since both sectors are
+independent, most results follow directly, except those concerning the zero-modes.
+We
+consider a generic matter CFT coupled to reparametrization ghosts:
+1. matter: central charge cm, energy–momentum tensor Tm and Hilbert space Hm;
+2. reparametrization ghosts: bc ghost system (Sections 2.3 and 7.2) with ϵ = +1 and
+λ = 2, cgh = −26, energy–momentum tensor T gh and Hilbert space Hgh.
+The formulas for the reparametrization ghosts are summarized in Appendix B.3.5.
+For
+modes, the system (m, gh, b or c) is indicated as a superscript to not confuse it with the
+mode index.
+The total central charge, energy–momentum tensor and Hilbert space are
+denoted by:
+c = cm + cgh = cm − 26,
+T(z) = T m(z) + T gh(z),
+H = Hm ⊗ Hgh.
+(8.1)
+The goal is to find the physical states in the cohomology, that is, which are BRST closed
+QB |ψ⟩ = 0
+(8.2)
+but non exact (Section 3.2): the latter statement can be understood as an equivalence
+between closed states under shift by exact states:
+|ψ⟩ ∼ |ψ⟩ + QB |Λ⟩ .
+(8.3)
+We introduce the BRST current and study its CFT properties. Then, we give a compu-
+tation of the BRST cohomology when the matter CFT contains at least two scalar fields.
+134
+
+8.2
+BRST in the CFT formalism
+The BRST current can be found from (3.50) to be [193]:
+jB(z) = :c(z)
+�
+T m(z) + 1
+2 T gh(z)
+�
+: + κ ∂2c(z)
+(8.4a)
+= c(z)T m(z) + :b(z)c(z)∂c(z): + κ ∂2c(z),
+(8.4b)
+and similarly for the anti-holomorphic sector. This can be derived from (3.53): the generator
+of infinitesimal changes of coordinates (given by the Lie derivative) is the energy–momentum
+tensor. The factor of 1/2 comes from the expression (7.99) of the ghost energy–momentum
+tensor: the second term does not contribute while the first has a factor of 2. Since the
+transformation of c in (3.53) has no factor, the 1/2 is necessary to recover the correct
+normalization. Finally, one finds that the transformation of b is reproduced. The different
+computations can be checked using the OPEs given below. The last piece is a total derivative
+and does not contribute to the charge: for this reason, it cannot be derived from (3.53), its
+coefficient will be determined below. Note that it is the only total derivative of dimension
+1 and of ghost number 1.
+The BRST charge is then obtained by the contour integral:
+QB = QB,L + QB,R,
+QB,L =
+�
+dz
+2πi jB(z),
+QB,R =
+�
+d¯z
+2πi ¯ȷB(¯z).
+(8.5)
+As usual, QB ∼ QB,L when considering only the holomorphic sectors such that we generally
+omit the index.
+8.2.1
+OPE
+The OPE of the BRST current with T is
+T(z)jB(w) ∼
+�cm
+2 − 4 − 6κ
+�
+c(w)
+(z − w)4 + (3 − 2κ) ∂c(w)
+(z − w)3 +
+jB(w)
+(z − w)2 + ∂jB(w)
+z − w .
+(8.6)
+Hence, the BRST current is a primary operator only if
+cm = 26,
+κ = 3
+2.
+(8.7)
+The BRST current must be primary, otherwise, the BRST symmetry is anomalous, which
+means that the theory is not consistent. This provides another derivation of the critical
+dimension. In this case, the OPE becomes
+T(z)jB(w) ∼
+jB(w)
+(z − w)2 + ∂jB(w)
+z − w .
+(8.8)
+Remark 8.1 (Critical dimension in 2d gravity) The value cm = 26 (critical dimen-
+sion) was obtained in Section 2.3 by requiring that the Liouville field decouples from the
+path integral. In 2d gravity, where this condition is not necessary, (nor even desirable) the
+Liouville field is effectively part of the matter, such that cL + cm = 26. One can also study
+the BRST cohomology in this case.
+The OPE of jB(z) with the ghosts are
+jB(z)b(w) ∼
+2κ
+(z − w)3 +
+j(w)
+(z − w)2 + T(w)
+z − w,
+(8.9a)
+jB(z)c(w) ∼ :c(w)∂c(w):
+z − w
+.
+(8.9b)
+135
+
+Similarly, the OPE with any matter weight h primary field φ is
+jB(z)φ(w) ∼ h c(w)φ(w)
+(z − w)2 + :h ∂c(w)φ(w) + c(w)∂φ(w):
+z − w
+,
+(8.9c)
+using that c(w)2 = 0 to cancel one term.
+The OPE with the ghost current is
+jB(z)j(w) ∼
+2κ + 1
+(z − w)3 − 2∂c(w)
+(z − w)2 − jB(w)
+z − w ,
+(8.10)
+while the OPE with itself is (for κ = 3/2)
+jB(z)jB(w) ∼ −cm − 18
+2
+:c(w)∂c(w):
+(z − w)3
+− cm − 18
+4
+:c(w)∂2c(w):
+(z − w)2
+− cm − 26
+12
+:c(w)∂3c(w):
+z − w
+.
+(8.11)
+There is no first order pole if cm = 26: as we will see shortly, this implies that the BRST
+charge is nilpotent.
+8.2.2
+Mode expansions
+The mode expansion of the BRST charge can be written equivalently
+QB =
+�
+m
+:cm
+�
+Lm
+−m + 1
+2 Lgh
+−m
+�
+:
+(8.12a)
+=
+�
+m
+c−mLm
+m + 1
+2
+�
+m,n
+(n − m) :c−mc−nbm+n:
+(8.12b)
+In the energy ordering, this expression becomes
+QB =
+�
+m
+⋆
+⋆cm
+�
+Lm
+−m + 1
+2 Lgh
+−m
+�
+⋆
+⋆ − c0
+2
+(8.13a)
+=
+�
+n
+cmLm
+−m + 1
+2
+�
+m,n
+(n − m)
+⋆
+⋆c−mc−nbm+n
+⋆
+⋆ − c0,
+(8.13b)
+where the ordering constant is the same as in Lgh
+0
+(as can be checked by comparing both
+sides of the anticommutator). The simplest derivation of this term is to use the algebra
+and to ensure that it is consistent. The only ambiguity is in the second term, when one c
+does not commute with the b: this happens for −n + (m + n) = 0, such that the ordering
+ambiguity is proportional to c0. Then, one finds that it is equal to agh = −1.
+The BRST operator can be decomposed on the ghost zero-modes as
+QB = c0L0 − b0M + �QB
+(8.14a)
+where
+�QB =
+�
+m̸=0
+c−mLm
+m − 1
+2
+�
+m,n̸=0
+m+n̸=0
+(m − n)
+⋆
+⋆c−mc−nbm+n
+⋆
+⋆ ,
+(8.14b)
+M =
+�
+m̸=0
+m c−mcm
+(8.14c)
+136
+
+The interest of this decomposition is that L0, M and �Q do not contain b0 or c0, which
+make it very useful to act on states decomposed according to the zero-modes (7.169). The
+nilpotency of the BRST operator implies the relations
+[L0, M] = [ �QB, M] = [ �QB, L0] = 0,
+�Q2
+B = L0M.
+(8.15)
+Moreover, one has Ngh( �QB) = 1 and Ngh(M) = 2.
+8.2.3
+Commutators
+From the various OPEs, one can compute the (anti-)commutators of the BRST charge with
+the other operators. For the ghosts and a weight h primary field φ, one finds
+{QB, b(z)} = T(z),
+(8.16a)
+{QB, c(z)} = c(z)∂c(z),
+(8.16b)
+[QB, φ(z)] = h ∂c(z)φ(z) + c(z)∂φ(z).
+(8.16c)
+This reproduces correctly (3.53).
+Two facts will be useful in string theory. First, (8.16c) is a total derivative for h = 1:
+[QB, φ(z)] = ∂
+�
+c(z)φ(z)
+�
+.
+(8.17)
+Second, c(z)φ(z) is closed if h = 1
+{QB, c(z)φ(z)} = (1 − h)c(z)∂c(z)φ(z).
+(8.18)
+The commutator with the ghost current is
+[QB, j(z)] = −jB(z),
+(8.19)
+which confirms that the BRST charge increases the ghost number by 1
+[Ngh, QB] = QB.
+(8.20)
+One finds that the BRST charge is nilpotent
+{QB, QB} = 0
+(8.21)
+and commutes with the energy–momentum tensor
+[QB, T(z)] = 0
+(8.22)
+only if the matter central charge corresponds to the critical dimension:
+cm = 26.
+(8.23)
+The most important commutator for the modes is
+Ln = {QB, bn}.
+(8.24)
+Nilpotency of QB then implies that QB commutes with Ln:
+[QB, Ln] = 0.
+(8.25)
+137
+
+8.3
+BRST cohomology: two flat directions
+The simplest case for studying the BRST cohomology is when the target spacetime has at
+least two non-compact flat directions represented by two free scalar fields (X0, X1) (Sec-
+tion 7.1). The remaining matter fields are arbitrary as long as the critical dimension cm = 26
+is reached. The reason for introducing two flat directions is that the cohomology is easily
+worked out by introducing light-cone (or complex) coordinates in target spacetime.
+The field X0 can be spacelike or timelike ϵ0 = ±1, while we consider X1 to be always
+spacelike, ϵ1 = 1. The oscillators are denoted by α0
+m and α1
+m, and the momenta of the Fock
+vacua by k∥ = (k0, k1) such that
+k2
+∥ = ϵ0(k0)2 + (k1)2.
+(8.26)
+The rest of the matter sector, called the transverse sector ⊥, is an arbitrary CFT with
+energy–momentum tensor T ⊥, central charge c⊥ = 24 and Hilbert space H⊥. The ghost
+together with the two scalar fields form the longitudinal sector ∥. The motivation for the
+names longitudinal and transverse will become clear later: they will be identified with the
+light-cone and perpendicular directions in the target spacetime (and, correspondingly, with
+unphysical and physical states).
+The Hilbert space of the theory is decomposed as
+H := H∥ ⊗ H⊥,
+H∥ :=
+�
+dk0 F0(k0) ⊗
+�
+dk1 F1(k1) ⊗ Hgh,
+(8.27)
+where F0(k0) and F1(k1) are the Fock spaces (7.83a) of the scalar fields X0 and X1, and
+Hgh is the ghost Hilbert space (7.169). As a consequence, a generic state of H reads
+|ψ⟩ = |ψ∥⟩ ⊗ |ψ⊥⟩ ,
+(8.28)
+where ψ⊥ is a generic state of the transverse matter CFT H⊥ and ψ∥ is built by acting with
+oscillators on the Fock vacuum of H∥:
+|ψ∥⟩ = cNc
+0
+0
+�
+m>0
+(α0
+−m)N0
+m(α1
+−m)N1
+m (b−m)Nb
+m(c−m)Nc
+m |k0, k1, ↓⟩
+|k0, k1, ↓⟩ := |k0⟩ ⊗ |k1⟩ ⊗ | ↓⟩ ,
+N 0
+m, N 1
+m ∈ N,
+N b
+m, N c
+m = 0, 1.
+(8.29)
+Since the Virasoro modes commute with the ghost number, eigenstates of the Virasoro
+operators without zero-modes �L0, given by the sum of (7.68) and (7.160), can also be taken
+to be eigenstates of Ngh. It is also useful to define the Hilbert space of states lying in the
+kernel of b0:
+H0 = H ∩ ker b0
+(8.30)
+such that
+H = H0 ⊕ c0H0.
+(8.31)
+The full L0 operator reads
+L0 = Lm
+0 + Lgh
+0 = (Lm
+0 − 1) + N b + N c,
+(8.32)
+using (7.129) for Lgh
+0 . A more useful expression is obtained by separating the two sectors
+and by extracting the zero-modes using (7.67):
+L0 =
+�
+L⊥
+0 − m2
+∥,Lℓ2 − 1
+�
++ �L∥
+0,
+(8.33)
+where the longitudinal mass and total level operator are:
+m2
+∥,L = −p2
+∥,L,
+�L∥
+0 = N 0 + N 1 + N b + N c ∈ N.
+(8.34)
+138
+
+A state |ψ⟩ is said to be on-shell if it is annihilated by L0:
+on-shell:
+L0 |ψ⟩ = 0.
+(8.35)
+The absolute BRST cohomology Habs(QB) defines the physical states (Section 3.2) and
+is given by the states ψ ∈ H that are QB-closed but not exact:
+Habs(QB) :=
+�
+|ψ⟩ ∈ H
+�� QB |ψ⟩ = 0, ∄ |χ⟩ ∈ H
+�� |ψ⟩ = QB |χ⟩
+�
+.
+(8.36)
+Since QB commutes with L0, (8.25), the cohomology subspace is preserved under time
+evolution.
+Before continuing, it is useful to outline the general strategy for studying the cohomology
+of a BRST operator Q in the CFT language. The idea is to find an operator ∆ – called
+contracting homotopy operator – which, if it exists, trivializes the cohomology. Conversely,
+this implies that the cohomology is to be found within states which are annihilated by ∆
+or for which ∆ is not defined. Then, it is possible to restrict Q on these subspaces: this is
+advantageous when the restriction of the BRST charge on these subspaces is a simpler. In
+fact, we will find that the reduced operator is itself a BRST operator, for which one can
+search for another contracting homotopy operator.1
+Given a BRST operator Q, a contracting homotopy operator ∆ for Q is an operator such
+that
+{Q, ∆} = 1.
+(8.37)
+Interpreting Q as a derivative operator, ∆ corresponds to the Green function or propagator.
+The existence of a well-defined ∆ with empty kernel implies that the cohomology is empty
+because all closed states are exact. Indeed, consider a state |ψ⟩ ∈ H which is an eigenstate
+of ∆ and closed QB |ψ⟩ = 0. Inserting (8.37) in front of the state gives:
+|ψ⟩ = {QB, ∆} |ψ⟩ = QB
+�
+∆ |ψ⟩
+�
+.
+(8.38)
+If ∆ is well-defined on |ψ⟩ and |ψ⟩ /∈ ker ∆, then ∆ |ψ⟩ is another state in H, which implies
+that |ψ⟩ is exact. Hence, the BRST cohomology has to be found inside the subspaces ker ∆
+or on which ∆ is not defined.
+8.3.1
+Conditions on the states
+In this subsection, we apply explicitly the strategy just discussed to get conditions on the
+states. A candidate contracting homotopy operator for QB is
+∆ := b0
+L0
+(8.39)
+thanks to (8.24):
+L0 = {QB, b0}.
+(8.40)
+Indeed, suppose that |ψ⟩ is an eigenstate of L0, and that it is closed but not on-shell:
+QB |ψ⟩ = 0,
+L0 |ψ⟩ ̸= 0.
+(8.41)
+One can use (8.40) in order to write:
+|ψ⟩ = QB
+� b0
+L0
+|ψ⟩
+�
+.
+(8.42)
+1A similar strategy shows that there is no open string excitation for the open SFT in the tachyon vacuum.
+139
+
+The operator inside the parenthesis is ∆ defined above in (8.39). The formula (8.42) breaks
+down if ψ is in the kernel of L0 since the inverse is not defined. This implies that a necessary
+condition for a L0-eigenstate |ψ⟩ to be in the BRST cohomology is to be on-shell (8.35).
+Considering explicitly the subset of states annihilated by b0 is not needed at this stage since
+ker b0 ⊂ ker L0 for QB-closed states, according to (8.24). Hence, we conclude:
+Habs(QB) ⊂ ker L0.
+(8.43)
+Note that this statement holds only at the level of vector spaces, i.e. when considering
+equivalence classes of states |ψ⟩ ∼ |ψ⟩+Q |Λ⟩. This means that there exists a representative
+state of each equivalence class inside ker L0, but a generic state is not necessarily in ker L0.
+For example, consider a state |ψ⟩ ∈ ker L0 and closed. Then, |ψ′⟩ = |ψ⟩ + QB |Λ⟩ with
+|Λ⟩ /∈ ker L0 is still in Habs(QB) but |ψ′⟩ /∈ ker L0 since [L0, QB] = 0.
+Computation – Equation (8.42)
+For L0 |ψ⟩ ̸= 0, one has:
+|ψ⟩ = L0
+L0
+|ψ⟩ = 1
+L0
+{QB, b0} |ψ⟩ = 1
+L0
+QB
+�
+b0 |ψ⟩
+�
+where the fact that |ψ⟩ is closed has been used to cancel the second term of the anti-
+commutator. Note that L0 commutes with both QB and b0 such that it can be moved
+freely.
+This shows that ∆ = b0/L0 given by (8.39) is not a contracting homotopy operator. A
+proper definition involves the projector P0 on the kernel of L0:
+|ψ⟩ ∈ ker L0 :
+P0 |ψ⟩ = |ψ⟩ ,
+|ψ⟩ ∈ (ker L0)⊥ :
+P0 |ψ⟩ = 0.
+(8.44)
+Then, the appropriate contracting homotopy operator reads ∆(1−P0) and (8.37) is changed
+to:
+{QB, ∆(1 − P0)} = (1 − P0).
+(8.45)
+This parallels completely the definition of the Green function in presence of zero-modes, see
+(B.3). By abuse of language, we will also say that ∆ is a contracting homotopy operator,
+remembering that this statement is correct only when multiplying with (1 − P0).
+We will revisit these aspects later from the SFT perspective. In fact, we will find that QB
+is the kinetic operator of the gauge invariant theory, while ∆ is the gauge fixed propagator
+in the Siegel gauge. This is expected from experience with standard gauge theories: the
+inverse of the kinetic operator (Green function) is not defined when the gauge invariance is
+not fixed.
+The on-shell condition (8.35) is already a good starting point. In order to simplify the
+analysis further, one can restrict the question of computing the cohomology on the subspace:
+H0 := H ∩ ker b0 = Hm ⊗ Hgh,0,
+(8.46)
+where Hgh,0 = Hgh ∩ker b0 was defined in (7.2.6). This subspace contains all states |ψ⟩ such
+that:
+|ψ⟩ ∈ H0
+=⇒
+b0 |ψ⟩ = 0.
+(8.47)
+In this subspace, there is no exact state |ψ⟩ with L0 |ψ⟩ ̸= 0 such that b0 |ψ⟩ = QB |ψ⟩ = 0.
+Indeed, assuming these conditions, (8.42) leads to a contraction:
+b0 |ψ⟩ = QB |ψ⟩ = 0,
+L0 |ψ⟩ ̸= 0
+=⇒
+|ψ⟩ = 0.
+(8.48)
+140
+
+Note that the converse statement is not true: there are on-shell states such that b0 |ψ⟩ ̸= 0.
+This also makes sense because the ghost Hilbert space can be decomposed with respect to
+the ghost zero-modes. The cohomology of QB in the subspace H0 is called the relative
+cohomology:
+Hrel(QB) := H0(QB) =
+�
+|ψ⟩ ∈ H0
+�� QB |ψ⟩ = 0, ∄ |χ⟩ ∈ H
+�� |ψ⟩ = QB |χ⟩
+�
+.
+(8.49)
+The advantage of the subspace b0 = 0 is to precisely pick the representative of Habs which
+lies in ker L0. In particular, the operator L0 is simple and has a direct physical interpretation
+as the worldsheet Hamiltonian. This condition is also meaningful in string theory because
+these states are also mass eigenstates, which have a nice spacetime interpretation, and it
+will later be interpreted in SFT as fixing the Siegel gauge. Moreover, it is implied by the
+choice of ∆ in (8.39) as the contracting homotopy operator, which is particularly convenient
+to work with to derive the cohomology. However, there are other possible choices, which are
+interpreted as different gauge fixings.
+After having built this cohomology, we can look for the full cohomology by relaxing the
+condition b0 = 0. In view of the structure of the ghost Hilbert space (7.169), one can expect
+that Habs(QB) = Hrel(QB) ⊕ c0Hrel(QB), which is indeed the correct answer. But, we will
+see (building on Section 3.2.2) that, in fact, it is this cohomology which contains the physical
+states in string theory, instead of the absolute cohomology.
+As a summary, we are looking for QB-closed non-exact states annihilated by b0 and L0:
+QB |ψ⟩ = 0,
+L0 |ψ⟩ = 0,
+b0 |ψ⟩ = 0.
+(8.50)
+8.3.2
+Relative cohomology
+In (8.14a), the BRST operator was decomposed as:
+QB = c0L0 − b0M + �QB,
+�Q2
+B = L0M.
+(8.51)
+This shows that, on the subspace L0 = b0 = 0, �QB is nilpotent and equivalent to QB:
+|ψ⟩ ∈ H0 ∩ ker L0
+=⇒
+QB |ψ⟩ = �QB |ψ⟩ ,
+�Q2
+B |ψ⟩ = 0.
+(8.52)
+Hence, this implies that �QB is a proper BRST operator and the relative cohomology of QB
+is isomorphic to the cohomology of �QB:
+H0(QB) = H0( �QB).
+(8.53)
+Next, we introduce light-cone coordinates in the target spacetime. While it does not allow
+to write Lorentz covariant expressions, it is helpful mathematically because it introduces a
+grading of the Hilbert space, for which powerful theorems exist (even if we will need only
+basic facts for our purpose).
+Light-cone parametrization
+The two scalar fields X0 and X1 are combined in a light-cone (if ϵ0 = −1) or complex (if
+ϵ0 = 1) fashion:
+X±
+L =
+1
+√
+2
+�
+X0
+L ±
+i
+√ϵ0
+X1
+L
+�
+.
+(8.54)
+141
+
+The modes of X± are found by following (7.50):2
+α±
+n =
+1
+√
+2
+�
+α0
+n ±
+i
+√ϵ0
+α1
+n
+�
+,
+n ̸= 0,
+(8.55a)
+x±
+L =
+1
+√
+2
+�
+x0
+L ±
+i
+√ϵ0
+x1
+L
+�
+,
+p±
+L =
+1
+√
+2
+�
+p0
+L ±
+i
+√ϵ0
+p1
+L
+�
+,
+(8.55b)
+The non-zero commutation relations are:
+[α+
+m, α−
+n ] = ϵ0 m δm+n,0,
+[x±
+L, p∓
+L] = iϵ0.
+(8.56)
+This implies that negative-frequency (creation) modes α±
+−n are canonically conjugate to
+positive-frequency (annihilation) modes α∓
+n . Note the similarity with the first-order system
+(7.134).
+For later purposes, it is useful to note the following relations:
+2 p+
+Lp−
+L = (p0
+L)2 + ϵ0(p1
+L)2 = ϵ0 p2
+∥,L,
+(8.57a)
+x+p− + x−p+ = x0p0 + ϵ0 x1p1,
+(8.57b)
+�
+n
+α+
+n α−
+m−n = 1
+2
+�
+n
+�
+α0
+nα0
+m−n + ϵ0 α1
+nα1
+m−n
+�
+.
+(8.57c)
+In view of the commutators (8.56), the appropriate definitions of the light-cone number
+N ±
+n and level operators N ± are:
+N ±
+n = ϵ0
+n α±
+−nα∓
+n ,
+N ± =
+�
+n>0
+n N ±
+n .
+(8.58)
+The insertion of ϵ0 follows (7.63). Then, one finds the following relation:
+N + + N − = N 0 + N 1.
+(8.59)
+Using these definitions, the variables appearing in L0 (8.33)
+L0 =
+�
+L⊥
+0 − m2
+∥,Lℓ2 − 1
+�
++ �L∥
+0
+(8.60)
+can be rewritten as:
+m2
+∥,L = −2ϵ0 p+
+Lp−
+L,
+�L∥
+0 = N + + N − + N b + N c.
+(8.61)
+The expression for the sum of the Virasoro operators (7.65) easily follows from (8.57):
+L0
+m + L1
+m = ϵ0
+�
+n
+:α+
+n α−
+m−n: = ϵ0
+�
+n̸=0,m
+:α+
+n α−
+m−n: + ϵ0
+�
+α−
+0 α+
+m + α+
+mα−
+m
+�
+.
+(8.62)
+Computation – Equation (8.56)
+For the modes α±
+m, we have:
+[α+
+m, α±
+n ] = 1
+2
+��
+α0
+m +
+i
+√ϵ0
+α1
+m
+�
+,
+�
+α0
+n ±
+i
+√ϵ0
+α1
+n
+��
+= 1
+2
+�
+[α0
+m, α0
+n] ∓ 1
+ϵ0
+[α1
+m, α1
+n]
+�
+= ϵ0
+2 m δm+n,0(1 ∓ 1),
+2For ϵ0 = 1, this convention matches the ones from [29] for X0 = X and X1 = φ. For ϵ = −1, this
+convention matches [193].
+142
+
+where we used (7.72). The other commutators follow similarly from (7.73), for example:
+[x−
+L, p±
+L] = 1
+2
+��
+x0
+L −
+i
+√ϵ0
+x1
+L
+�
+,
+�
+p0
+L ±
+i
+√ϵ0
+p1
+L
+��
+= 1
+2
+�
+[x0
+L, p0
+L] ± ϵ0[x1
+L, p1
+L]
+�
+= ϵ0
+2 (1 ± 1).
+Computation – Equation (8.57)
+For the modes α±
+m, we have:
+�
+n
+α+
+n α−
+m−n = 1
+2
+�
+n
+�
+α0
+n +
+i
+√ϵ0
+α1
+n
+� �
+α0
+m−n −
+i
+√ϵ0
+α1
+m−n
+�
+= 1
+2
+�
+n
+�
+α0
+nα0
+m−n + ϵ0 α1
+nα1
+m−n +
+i
+√ϵ0
+(α0
+m−nα1
+n − α0
+nα1
+m−n)
+�
+.
+The last two terms in parenthesis cancel as can be seen by shifting the sum n → m − n
+in one of the term. Note that, for m ̸= 2n, there is no cross-term only after summing
+over n.
+The relations for the zero-modes follow simply by observing that expressions in both
+coordinates can be rewritten in terms of the 2-dimensional (spacetime) flat metric.
+Computation – Equation (8.59)
+Using (8.57), one finds:
+N 0 + N 1 =
+�
+n
+n
+�
+N 0
+n + N 1
+n
+�
+=
+�
+n
+n
+�
+N +
+n + N −
+n
+�
+= N + + N −.
+Reduced cohomology
+In terms of the light-cone variables, the reduced BRST operator �QB reads:
+�QB =
+�
+m̸=0
+c−m
+�
+L⊥
+m + ϵ0
+�
+n
+α+
+n α−
+m−n
+�
++ 1
+2
+�
+m,n
+(n − m) :c−mc−nbm+n:.
+(8.63)
+This operator can be further decomposed. Introducing the degree
+deg := N + − N − + �
+N c − �
+N b
+(8.64)
+such that
+∀m ̸= 0 :
+deg(α+
+m) = deg(cm) = 1,
+deg(α−
+m) = deg(bm) = −1,
+(8.65)
+and deg = 0 for the other variables, the operator �QB is decomposed as:3
+�QB = Q0 + Q1 + Q2,
+deg(Qj) = j,
+(8.66a)
+where
+Q1 =
+�
+m̸=0
+c−mL⊥
+m +
+�
+m,n̸=0
+m+n̸=0
+⋆
+⋆c−m
+�
+ϵ0 α+
+n α−
+m−n + 1
+2 (m − n) c−mbm+n
+�
+⋆
+⋆,
+Q0 =
+�
+n̸=0
+α+
+0 c−nα−
+n ,
+Q2 =
+�
+n̸=0
+α−
+0 c−nα+
+n .
+(8.66b)
+3The general idea behind this decomposition is the notion of filtration, nicely explained in [3, sec. 3, 36].
+143
+
+The nilpotency of �QB implies the following conditions on the Qj:
+Q2
+0 = Q2
+2 = 0,
+{Q0, Q1} = {Q1, Q2} = 0,
+Q2
+1 + {Q0, Q2} = 0.
+(8.67)
+Hence, Q0 and Q2 are both nilpotent and define a cohomology.
+One can show that the cohomologies of �QB and Q0 are isomorphic4
+H0( �QB) ≃ H0(Q0)
+(8.68)
+under general conditions [29], in particular, if the cohomology is ghost-free (i.e. all states
+have Ngh = 1).
+The contracting homotopy operator for Q0 is
+�∆ := B
+�L∥
+0
+,
+B := ϵ0
+�
+n̸=0
+1
+α+
+0
+α+
+−nbn.
+(8.69)
+Indeed, it is straightforward to check that
+�L∥
+0 = {Q0, B}
+=⇒
+{Q0, �∆} = 1.
+(8.70)
+As a consequence, a necessary condition for a closed �L∥
+0-eigenstate |ψ⟩ to be in the
+cohomology of Q0 is to be annihilated by �L∥
+0:
+�L∥
+0 |ψ⟩ = 0,
+=⇒
+N ± |ψ⟩ = N c |ψ⟩ = N b |ψ⟩ = 0,
+(8.71)
+since �L∥
+0 is a sum of positive integers. This means that the state ψ contains no ghost or
+light-cone excitations α±
+−n, b−n and c−n, and lies in the ground state of the Fock space H∥,0.
+Then, we need to prove that this condition is sufficient: states with �L∥
+0 = 0 are closed.
+First, note that a state |ψ⟩ ∈ H0 with �L∥
+0 has ghost number 1 since there are no ghost
+excitations on top of the vacuum | ↓⟩, which has Ngh = 1. Second, �L0 and Q0 commute,
+such that:
+0 = Q0�L∥
+0 |ψ⟩ = �L∥
+0Q0 |ψ⟩ .
+(8.72)
+Since Q0 increases the ghost number by 1, one can invert �L∥
+0 = N b + N c + · · · in the last
+term since �L∥
+0 ̸= 0 in this subspace. This gives:
+Q0 |ψ⟩ = 0.
+(8.73)
+Hence, the condition �L∥
+0 |ψ⟩ = 0 is sufficient for |ψ⟩ to be in the cohomology. This has to be
+contrasted with Section 8.3.1 where the condition L∥
+0 = 0 is necessary but not sufficient.
+In this case, the on-shell condition (8.33) reduces to
+L0 = L⊥
+0 − m2
+∥,Lℓ2 − 1 = 0.
+(8.74)
+But, additional states can be found in ker B or in a subspace of H on which B is singular.
+We have ker B = ker �L∥
+0 such that nothing new can be found there. However, the operator B
+is not defined for states with vanishing momentum α+
+0 ∝ p+
+L = 0. In fact, one must also have
+α−
+0 ∝ p−
+L = 0 (otherwise, the contracting operator for Q2 is well-defined and can be used
+instead). But, these states do not satisfy the on-shell condition (except for massless states
+with L⊥
+0 = 1), as it will be clear later (see [245, sec. 2.2] for more details). For this reason,
+we assume that states have a generic non-zero momentum and that there is no pathology.
+4The role of Q0 and Q2 can be reversed by changing the sign in the definition of the degree and the role
+of P ±
+n .
+144
+
+Full relative cohomology
+This section aims to construct states in H0( �QB) from states in H(Q0).
+We follow the
+construction from [29].
+Given a state |ψ0⟩ ∈ H0(Q0), the state Q1 |ψ0⟩ is Q0-closed since Q0 and Q1 anticommute
+(8.67):
+{Q0, Q1} |ψ0⟩ = 0
+=⇒
+Q0
+�
+Q1 |ψ0⟩
+�
+= 0.
+(8.75)
+Since Q1 |ψ0⟩ is not in ker �L∥
+0 (because Q1 increases the ghost number by 1), the state Q1 |ψ0⟩
+is Q0-exact and can be written as Q0 of another state |ψ1⟩:
+Q1 |ψ0⟩ =: −Q0 |ψ1⟩
+=⇒
+|ψ1⟩ = − B
+�L∥
+0
+Q1 |ψ0⟩ .
+(8.76)
+Computation – Equation (8.76)
+Start from the definition and insert (8.70) since �L0 is invertible:
+Q1 |ψ0⟩ =
+�
+Q0, B
+�L∥
+0
+�
+Q1 |ψ0⟩ = Q0
+�
+B
+�L∥
+0
+Q1 |ψ0⟩
+�
+.
+The state |ψ1⟩ is identified with minus the state inside the parenthesis (up to a BRST
+exact state).
+As for |ψ0⟩, apply {Q0, Q1} on ψ1:
+{Q0, Q1} |ψ1⟩ = Q0
+�
+Q1 |ψ1⟩ + Q2 |ψ0⟩
+�
+.
+(8.77)
+This implies that the combination in parenthesis is Q0-closed and, for the same reason as
+above, it is exact:
+Q1 |ψ1⟩ + Q2 |ψ0⟩ = Q0 |ψ2⟩ ,
+|ψ2⟩ = − B
+�L∥
+0
+�
+Q1 |ψ1⟩ + Q2 |ψ0⟩
+�
+.
+(8.78)
+Computation – Equation (8.77)
+{Q0, Q1} |ψ1⟩ = Q0Q1 |ψ1⟩ − Q2
+1 |ψ0⟩ = Q0Q1 |ψ1⟩ + {Q0, Q2} |ψ0⟩ .
+The first equality follows from (8.76), the second by using (8.67). The final result is
+obtained after using that |ψ0⟩ is Q0-closed.
+Iterating this procedure leads to a series of states:
+|ψk+1⟩ = − B
+�L∥
+0
+�
+Q1 |ψk⟩ + Q2 |ψk−1⟩
+�
+.
+(8.79)
+We claim that a state in the relative cohomology |ψ⟩ ∈ H0( �QB) is built by summing all
+these states:
+|ψ⟩ =
+�
+k∈N
+|ψk⟩ .
+(8.80)
+Indeed, it is easy to check that |ψ⟩ is �QB-closed:
+�QB |ψ⟩ = 0.
+(8.81)
+145
+
+We leave aside the proof that ψ is not exact (see [29]). Note that ψ and ψ0 have the same
+ghost numbers
+Ngh(ψ) = Ngh(ψ0) = 1
+(8.82)
+since Ngh(BQj) = 0.
+In fact, since ψ0 does not contain longitudinal modes, it is annihilated by Q1 and Q2
+(these operators contain either a ghost creation operator together with a light-cone annihil-
+ation operator, or the reverse):
+Q1 |ψ0⟩ = Q2 |ψ0⟩ = 0.
+(8.83)
+As a consequence, one has ψk = 0 for k ≥ 1 and ψ = ψ0.
+Computation – Equation (8.81)
+�QB |ψ⟩ =
+�
+k∈N
+�QB |ψk⟩
+= Q0 |ψ0⟩ + Q1 |ψ0⟩ + Q0 |ψ1⟩
+�
+��
+�
+=0
++ Q2 |ψ0⟩ + Q1 |ψ1⟩ + Q0 |ψ2⟩
+�
+��
+�
+=0
++ · · ·
+= 0.
+8.3.3
+Absolute cohomology, states and no-ghost theorem
+The absolute cohomology is constructed from the relative cohomology:
+Habs(QB) = Hrel(QB) ⊕ c0 Hrel(QB).
+(8.84)
+The interested reader is refereed to [29] for the proof. A simple motivation is that the Hilbert
+space is decomposed in terms of the ghost zero-modes as in (7.169). Since the zero-modes
+commute with �Q0, linear combination of states in Hrel(QB) and c0Hrel(QB) are expected
+to be in the cohomology. Obviously, one has to work out the other terms of QB and prove
+that there are no other states.
+It looks like there is a doubling of the physical states, one built on | ↓⟩ and one on | ↑⟩.
+The remedy is to impose the condition b0 = 0 on the states (see also Section 3.2.2 and [245,
+sec. 2.2] for more details). As already pointed out, states in Habs form equivalence class
+under |ψ⟩ ∼ |ψ⟩ + QB |Λ⟩, and it is necessary to select a single representative. This is what
+the condition b0 = 0 achieves. Obviously, it is always possible to add BRST exact states to
+write another representative (for example, to restore the Lorentz covariance).
+The last step is to discuss the no-ghost theorem: the latter states that there is no
+negative-norm states in the BRST cohomology of string theory. This follows straightfor-
+wardly from the condition �L0 = 0: it implies that there are no ghost and no light-cone
+excitations. The ghosts and the time direction (if X0 is timelike) are responsible for negative-
+norm states. Hence, the cohomology has no negative-norm states if the transverse CFT is
+unitary (which implies that all states in H⊥ have a positive-definite inner-product).
+Physical states |ψ⟩ ∈ Hrel(QB) are thus of the form:
+|ψ⟩ = |k0, k1, ↓⟩ ⊗ |ψ⊥⟩ ,
+|ψ⊥⟩ ∈ H⊥,
+(8.85a)
+�
+L⊥
+0 − m2
+∥,Lℓ2 − 1
+�
+|ψ⟩ = 0,
+p2
+L,∥ = −m2
+∥,Lℓ2.
+(8.85b)
+This form can be made covariant: taking a state of the form |ψ⟩⊗| ↓⟩ with |ψ⟩ ∈ Hm, acting
+with QB implies the equivalence with the old covariant quantization:
+(Lm
+0 − 1) |ψ⟩ = 0,
+∀n > 0 :
+Lm
+n |ψ⟩ = 0.
+(8.86)
+146
+
+This means that ψ must be a weight 1 primary field of the matter CFT.
+Remark 8.2 (Open string) The results of this section provide, in fact, the cohomology
+for the open string after taking pL = p (instead of pL = p/2 for the closed string).
+8.3.4
+Cohomology for holomorphic and anti-holomorphic sectors
+It remains to generalize the computation of the cohomology when considering both the
+holomorphic and anti-holomorphic sectors.
+In this case, the BRST operator is
+QB = c0L0 − b0M + �QB + ¯c0 ¯L0 − ¯b0 ¯
+M + �QB.
+(8.87)
+It is useful to rewrite this expression in terms of L±
+0 , b±
+0 and c±
+0 :
+QB = c+
+0 L+
+0 − b+
+0 M + + c−
+0 L−
+0 − b−
+0 M − + �Q+
+B,
+(8.88)
+where
+L+
+0 =
+�
+L⊥+
+0
+−
+m2
+∥ℓ2
+2
+− 2
+�
++ �L∥+
+0 ,
+L−
+0 = L⊥−
+0
++ �L∥−
+0
+(8.89)
+and
+M ± := 1
+2(M ± ¯
+M).
+(8.90)
+Because of the relations L±
+0 = {QB, b±
+0 }, we find that states in the cohomology must be
+on-shell L+
+0 = 0 and must satisfy the level-matching condition L−
+0 = 0:5
+L+
+0 |ψ⟩ = L−
+0 |ψ⟩ = 0.
+(8.91)
+Again, it is possible to reduce the cohomology by imposing conditions on the zero-modes
+such that the above conditions are automatically satisfied (see also Section 3.2.2). Imposing
+first the condition b−
+0 = 0 defines the semi-relative cohomology. The relative cohomology is
+found by imposing b±
+0 = 0 and in fact corresponds to the physical space (see [245, sec. 2.3] for
+more details). The rest of the derivation follows straightforwardly because the two sectors
+commute: we find that the cohomology is ghost-free and has no light-cone excitations:
+�L∥±
+0
+= N 0 ± ¯N 0 + N 1 ± ¯N 1 + N b ± ¯N b + N c ± ¯N c = 0.
+(8.92)
+In general, it is simpler to work with a covariant expression and to impose the necessary
+conditions. Taking a state |ψ⟩ ⊗ | ↓↓⟩ with |ψ⟩ ∈ Hm, we find that ψ is a weight (1, 1)
+primary field of the matter CFT:
+(Lm
+0 + ¯Lm
+0 − 2) |ψ⟩ = 0,
+(Lm
+0 − ¯Lm
+0 ) |ψ⟩ = 0,
+∀n > 0 :
+Lm
+n |ψ⟩ = ¯Lm
+n |ψ⟩ = 0.
+(8.93)
+An important point is that the usual mass-shell condition k2 = −m2 is provided by the first
+condition only. This also shows that states in the cohomology naturally appears with c¯c
+insertion since
+| ↓↓⟩ = c(0)¯c(0) |0⟩ = c1¯c1 |0⟩ .
+(8.94)
+This hints at rewriting of scattering amplitudes in terms of unintegrated states (3.29) only.
+A state is said to be of level (ℓ, ¯ℓ) and denoted as ψℓ,¯ℓ if it satisfies:
+�L0 |ψℓ,¯ℓ⟩ = ℓ |ψℓ,¯ℓ⟩ ,
+�¯L0 |ψℓ,¯ℓ⟩ = ¯ℓ |ψℓ,¯ℓ⟩ .
+(8.95)
+5In the current case, the propagator is less easily identified. We will come back on its definition later.
+147
+
+Example 8.1 – Closed string tachyon
+As an example, let’s construct the state ψ0,0 with level zero for a spacetime with D
+non-compact dimensions. In this case, the transverse CFT contains D − 2 free scalars
+which combine with X0 and X1 into D scalars Xµ. The Fock space is built on the
+vacuum |k⟩ and we define the mass such that on-shell condition reduces to the standard
+QFT expression:
+k2 = −m2,
+m2 := 2
+ℓ2 (N + ¯N − 2),
+(8.96)
+where N and ¯N are the matter level operators. The state in the remaining transverse
+CFT (without the D − 2 scalars) is the SL(2, C) vacuum with L⊥
+0 = ¯L⊥
+0 = 0 (this is
+the state with the lowest energy for a unitary CFT). In this case, the on-shell condition
+reads
+m2ℓ2 = −4 < 0.
+(8.97)
+Since the mass is negative, this state is a tachyon. The vertex operator associated to
+the state reads:
+V (k, z, ¯z) = c(z)¯c(¯z)eik·X(z,¯z).
+(8.98)
+8.4
+Summary
+In this chapter, we have described the BRST quantization from the CFT point of view. We
+have first considered only the holomorphic sector (equivalently, the open string). We proved
+that the cohomology does not contain negative-norm states and we provided an explicit way
+to construct the states. Finally, we glued together both sectors and characterized the BRST
+cohomology of the closed string.
+What is the next step? We could move to computations of on-shell string amplitudes,
+but this falls outside the scope of this book. We can also start to consider string field theory.
+Indeed, the BRST equation QB |ψ⟩ = 0 and the equivalence |ψ⟩ ∼ |ψ⟩ + QB |Λ⟩ completely
+characterize the states. In QFT, states are solutions of the linearized equations of motion:
+hence, the BRST equation can provide a starting point for building the action. This is the
+topic of Chapter 10.
+8.5
+Suggested readings
+• The general method to construct the absolute cohomology follows [29, 193]. Other
+works and reviews include [20, 28, 57, 118, 119, 170, 177].
+• String states are discussed in [24, sec. 3.3, 193, sec. 4.1].
+148
+
+Part II
+String field theory
+149
+
+Chapter 9
+String field
+In this chapter, we introduce general concepts about the string field. The goal is to give an
+idea of which type of object it is and of the different possibilities for describing it. We will
+see that the string field is a functional and, for this reason, it is more convenient to work
+with the associated ket field, which can itself be represented in momentum space. We focus
+on what to expect from a free field, taking inspiration from the worldsheet theory. The
+interpretation becomes more difficult when taking into account the interactions.
+9.1
+Field functional
+A string field, after quantization, is an operator which creates or destroys a string at a given
+time. Since a string is a 1-dimension extended object, the string field Ψ must depend on the
+spatial positions of each point of the string denoted collectively as Xµ. Hence, the string
+field is a functional Ψ[Xµ]. The fact that it is a functional rather than a function makes the
+construction of a field theory much more challenging: it asks for revisiting all concepts we
+know in point-particle QFT without any prior experience with a simple model.1
+It is important that the dependence is only on the shape and not on the parametrization.
+However, it is simpler to first work with a specific parametrization X(σ) and make sure that
+nothing depends on it at the end (equivalent to imposing the invariance under reparamet-
+rization of the worldsheet). This leads to work with a functional Ψ[X(σ)] of fields on the
+worldsheet (at fixed time). To proceed, one should first determine the degrees of freedom of
+the string, and then to find the interactions. The simplest way to achieve the first step is to
+perform a second-quantization of the string wave-functional: the string field is written as a
+linear combination of first-quantized states with spacetime wave functionals as coefficients.2
+This provides a free Hamiltonian; trying to add interactions perturbatively does not work
+well.
+It is not possible to go very far with this approach and one is lead to choose a specific
+gauge, breaking the manifest invariance under reparametrizations. The simplest is the light-
+cone gauge since one works only with the physical degrees of freedom of the string. While
+this approach is interesting to gain some intuitions and to show that, in principle, it is
+possible to build a string field theory, it requires making various assumptions and ends up
+1The problem is not in working with the wordline formalism and writing a BRST field theory, but really
+to take into account the spatial extension of the objects.
+In fact, generalizing further to functionals of
+extended (p > 1)-dimensional objects – branes – shows that SFT is the simplest of such field theories.
+2The description of the first-quantized states depends on the CFT used to describe the theory. This
+explains the lack of manifest background independence of SFT. Unfortunately, no better approach has been
+found until now.
+150
+
+with problems (especially for superstrings).3
+Since worldsheet reparametrization invariance is just a kind of gauge symmetry – maybe
+less familiar than the non-Abelian gauge symmetries in Yang–Mills, but still a gauge sym-
+metry –, one may surmise that it should be possible to gauge fix this symmetry and to
+introduce a BRST symmetry in its place. This is the program of the BRST (or covariant)
+string field theory in which the string field depends not only of the worldsheet (at fixed
+time), but also on the ghosts: Ψ[X(σ), c(σ)]. There is no dependence on the b ghost because
+the latter is the conjugate momentum of the c ghost: in the operator language, b(σ) ∼
+δ
+δc(σ).
+The BRST formalism has the major advantage to allow to move easily from D = 26
+dimensions – described by Xµ scalars (µ = 0, . . . , 25) – to a (possibly curved) D-dimensional
+spacetime and a string with some internal structure – described by a more general CFT,
+in which D scalars Xµ represent the non-compact dimensions and the remaining system
+with central charge 26 − D describes the compactification and structure. It is sufficient to
+consider the string field as a general functional of all the worldsheet fields. For simplicity,
+we will continue to write X in the functional dependence, keeping the other matter fields
+implicit.
+It is complicated to find an explicit expression for the string field as a functional of X(σ)
+and c(σ). In fact, the field written in this way is in the position representation and, as usual
+in quantum mechanics, one can choose to work with the representation independent ket |Ψ⟩:
+Ψ[X(σ), c(σ)] := ⟨X(σ), c(σ)|Ψ⟩ .
+(9.1)
+It is often more convenient to work with |Ψ⟩ (which we will also denote simply as Ψ, not
+distinguishing between states and operators). The latter will be the basic object of SFT in
+most of this book.
+Writing a field theory in terms of |Ψ⟩ may not be intuitive since in point-particle QFT,
+one is used to work with the position or momentum representation. In fact, there is a very
+simple way to recover a formulation in terms of spacetime point-particle fields, which can be
+used almost whenever there is a doubt about what is going on. Indeed, as is well-known from
+standard worldsheet string theory, the string states behave like a collection of particles. This
+is because the modes of the CFT fields (like αµ
+n) carry spacetime indices (Lorentz, group
+representation. . . ) such that the states themselves carries indices. Indeed, these quantum
+numbers classify eigenstates of the operators L0 and ¯L0. On the other hand, positions and
+shapes are not eigenstates of any simple CFT operator.
+9.2
+Field expansion
+It follows that the second-quantized string field can be written as a linear combination
+of first-quantized off-shell states |φα(k)⟩ = Vα(k; 0, 0) |0⟩ (which form a basis of the CFT
+Hilbert space H):
+|Ψ⟩ =
+�
+α
+�
+dDk
+(2π)D ψα(k) |φα(k)⟩ ,
+(9.2)
+where k is the D-dimensional momentum of the string (conjugated to the position of the
+centre-of-mass) and α is a collection of discrete quantum numbers (Lorentz indices, group
+representation. . . ). When inserting this expansion inside the action, we find that it reduces
+to a standard field theory with an infinite number of particles described by the spacetime
+3While this approach has been mostly abandoned, recent results show that it can still be used when
+defined with a proper regularization [4, 114–117].
+151
+
+fields ψα(k) (momentum representation).
+The fields can also be written in the position
+representation by Fourier transforming only the momentum k to the centre-of-mass x:
+ψα(x) =
+�
+dDk
+(2π)D eik·xψα(k).
+(9.3)
+However, we will see that it is often not convenient because the action is non-local in position
+space (including for example exponentials of derivatives).
+The physical intuition is that the string is a non-local object in spacetime. It can be
+expressed in momentum space through a Fourier transformation: variables dual to non-
+compact (resp. compact) dimensions are continuous (discrete). As a consequence, the mo-
+mentum is continuous since the centre-of-mass move in the non-compact spacetime, while
+the string itself has a finite extension and the associated modes are discrete but still not
+bounded (and similarly for compact dimensions). This indicates that the spectrum is the
+collection of a set of continuous and discrete modes. Hence, the non-locality of the string
+(due to the spatial extension) is traded for an infinite number of modes which behave like
+standard particles. In this description, the non-locality arises: 1) in the infinite number of
+fields, 2) in the coupling between the modes, 3) as a complicated momentum-dependence of
+the action.
+When we are not interested in the spacetime properties, we will write a generic basis of
+the Hilbert space H as {φr}:
+|Ψ⟩ =
+�
+r
+ψr |φr⟩ .
+(9.4)
+The sum over r includes discrete and continuous labels.
+Example 9.1 – Scalar field
+In order to illustrate the notations for a point-particle, consider a scalar field φ(x). It
+can be expanded in Fourier modes as:
+φ(x) =
+�
+dDk
+(2π)D φ(k)eik·x.
+(9.5)
+The corresponding ket |φ⟩ is found by expanding on a basis {|k⟩}:
+|φ⟩ =
+�
+dDk
+(2π)D φ(k) |k⟩ ,
+φ(k) = ⟨k|φ⟩ .
+(9.6)
+Similarly, the position space field is defined from the basis {|x⟩} such that:
+φ(x) = ⟨x|φ⟩ =
+�
+dDk
+(2π)D ⟨x|k⟩ ⟨k|φ⟩ ,
+⟨x|k⟩ = eik·x.
+(9.7)
+9.3
+Summary
+In this chapter, we introduced general ideas about what a string field is. We now need to
+write an action. In general, one proceeds in two steps:
+1. build the kinetic term (free theory):
+(a) equations of motion → physical states
+(b) equivalence relation → gauge symmetry
+2. add interactions and deform the gauge transformation
+152
+
+We consider the first point in the next chapter, but we will have to introduce more machinery
+in order to discuss interactions.
+9.4
+Suggested readings
+• General discussions of the string field and of the ideas of string field theory can be
+found in [192, sec. 4, 261].
+• Light-cone SFT is reviewed in [245, 124, chap. 6, 125, chap. 9].
+153
+
+Chapter 10
+Free BRST string field theory
+Abstract
+In this chapter, we construct the BRST (or covariant) free bosonic string field
+theories. It is useful to first ignore the interactions in order to introduce some general tools
+and structures in a simpler setting. Moreover, the free SFT is easily constructed and does
+not require as much input as the interactions. In this chapter, we discuss mostly the open
+string, keeping the closed string for the last section. We start by describing the classical
+theory: equations of motion, action, gauge invariance and gauge fixing. Then, we perform
+the path integral quantization and compute the action in terms of spacetime fields for the
+first two levels (tachyon and gauge field).
+10.1
+Classical action for the open string
+Contrary to most of this book, we will exemplify the discussion with the open string. The
+reason is that most computations are the same in both the open and closed string theories,
+but the latter requires twice more writing. There are also a few subtleties which can be more
+easily explained once the general structure is understood. Everything needed for the open
+string for this chapter can be found in Chapter 8: in fact, describing the open string (at this
+level) is equivalent to consider only the holomorphic sector of the CFT and to set pL = p
+(instead of p/2). We consider a generic matter CFT in addition to the ghost system and we
+denote as H the space of states. The open and closed string fields are denoted respectively
+by Φ and Ψ, such that it is clear which theory is studied.
+An action can be either constructed from first principles, or it can be derived from the
+equations of motion. Since the fundamental structure of string field theory is not (really)
+known, one needs to rely on the second approach.
+But do we already know the (free)
+equations of motion for the string field? The answer is yes. But, before showing how these
+can be found from the worldsheet formalism, we will study the case of the point-particle to
+fix ideas and notations.
+10.1.1
+Warm-up: point-particle
+The free (or linearized) equation of motion for a scalar particle reads:
+(−∆ + m2)φ(x) = 0.
+(10.1)
+Solutions to this equation provides one-particle state of the free theory: a convenient basis
+is {eikx}, where each state satisfies the on-shell condition
+k2 = −m2.
+(10.2)
+154
+
+The field φ(x) is decomposed on the basis as
+φ(x) =
+�
+dk φ(k)eikx,
+(10.3)
+where φ(k) are the coefficients of the expansion. Since the field is off-shell, the condition
+k2 = −m2 is not imposed. Following Chapter 9, the field can also be represented as a ket:
+φ(x) = ⟨x|φ⟩ ,
+φ(k) = ⟨k|φ⟩ ,
+(10.4)
+or, conversely:
+|φ⟩ =
+�
+dx φ(x) |x⟩ =
+�
+dk φ(k) |k⟩ .
+(10.5)
+Writing the kinetic operator as a kernel:
+K(x, x′) :=⟨x| K |x′⟩ = δ(x − x′) (−∆x + m2),
+(10.6)
+the equations of motion reads
+�
+dx′ K(x, x′)φ(x′) = 0
+⇐⇒
+K |φ⟩ = 0.
+(10.7)
+An action can easily be found from the equation of motion by multiplying with φ(x) and
+integrating:
+S = 1
+2
+�
+dx φ(x)(−∆ + m2)φ(x) = 1
+2
+�
+dxdx′ φ(x)K(x, x′)φ(x′).
+(10.8)
+It is straightforward to write the action in terms of the ket:
+S = 1
+2 ⟨φ| K |φ⟩ .
+(10.9)
+There is one hidden assumption in the previous lines: the definition of a scalar product.
+A natural inner product is provided in the usual quantum mechanics by associating a bra to
+a ket. Similarly, integration provides another definition of the inner product when working
+with functions. We will find that the definition of the inner product requires more care in
+closed SFT. To summarize, to write the kinetic term of the action, one needs the linearized
+equation of motion and an appropriate inner product on the space of states.
+10.1.2
+Open string action
+The worldsheet equation which yields precisely all the string physical states |ψ⟩ is the BRST
+condition:
+QB |ψ⟩ = 0.
+(10.10)
+Considering the open string field Φ to be a linear combination of all possible one-string
+states |ψ⟩
+Φ ∈ H,
+(10.11)
+the equation of motion is:
+QB |Φ⟩ = 0.
+(10.12)
+Moving away from the physical state condition, the string field Φ is off-shell and is expanded
+on a general basis {φr} of H. This presents a first difficulty because the worldsheet approach
+– and the description of amplitudes – looks ill-defined for off-shell states: extending the usual
+155
+
+formalism will be the topic of Chapter 11. However, this is not necessary for the free theory
+and we can directly proceed.
+Next, we need to find an inner product ⟨·, ·⟩ on the Hilbert space H. A natural candidate
+is the BPZ inner product since it is not degenerate
+⟨A, B⟩ := ⟨A|B⟩ ,
+(10.13)
+where ⟨A| = |A⟩t is the BPZ conjugate (6.98) of |A⟩, using I−. This leads to the action:
+S = 1
+2 ⟨Φ, QBΦ⟩ = 1
+2 ⟨Φ| QB |Φ⟩ .
+(10.14)
+Due to the definition of the BPZ product, the action is equivalent to a 2-point correlation
+function on the disk.
+The inner product satisfies the following identities:
+⟨A, B⟩ = (−1)|A||B|⟨B, A⟩,
+⟨QBA, B⟩ = −(−1)|A|⟨A, QBB⟩,
+(10.15)
+where |A| denotes the Grassmann parity of the operator A.
+A first consistency check is to verify that the ghost number of the string can be defined
+such that the action is not vanishing. Indeed, the ghost number anomaly on the disk implies
+that the total ghost number must be Ngh = 3. Since physical states have Ngh = 1, it is
+reasonable to take the string field to satisfy the same condition, even off-shell:
+Ngh(Φ) = 1.
+(10.16)
+This condition means that there is no ghost at the classical level beyond the one of the
+energy vacuum | ↓⟩, which has Ngh = 1. Moreover, the BRST charge has Ngh(QB) = 1,
+such that the action has ghost number 3.
+One needs to find the Grassmann parity of the string field. Using the properties of the
+BPZ inner product, the string field should be Grassmann odd
+|Φ| = 1
+(10.17)
+for the action to be even. This is in agreement with the fact that the string field has ghost
+number 1 and that the ghosts are Grassmann odd. One must impose a reality condition on
+the string field (a complex field would behave like two real fields and have too many states).
+The appropriate reality condition identifies the Euclidean and BPZ conjugates:
+|Φ⟩‡ = |Φ⟩t .
+(10.18)
+That this relation is correct will be checked a posteriori for the tachyon field in Section 10.4.
+Computation – Equation (10.17)
+⟨Φ, QBΦ⟩ = (−1)|Φ|(|QBΦ|)⟨QBΦ, Φ⟩ = (−1)|Φ|(1+|Φ|)⟨QBΦ, Φ⟩
+= ⟨QBΦ, Φ⟩ = −(−1)|Φ|⟨Φ, QBΦ⟩,
+where both properties (10.15), together with the fact that |Φ|(1 + |Φ|) is necessarily
+even. In order for the bracket to be non-zero, one must have |Φ| = 1.
+Since the Hilbert space splits as H = H0 ⊕ c0H0 with H0 = H ∩ ker b0, see (8.31), it is
+natural to split the field as (this is discussed further in Section 10.2):
+|Φ⟩ = |Φ↓⟩ + c0 |�Φ↓⟩ ,
+(10.19)
+156
+
+where
+Φ↓, �Φ↓ ∈ H0
+=⇒
+b0 |Φ↓⟩ = b0 |�Φ↓⟩ = 0.
+(10.20)
+The ghost number of each component is
+Ngh(Φ↓) = 1,
+Ngh(�Φ↓) = 0.
+(10.21)
+Remembering the decomposition (8.14a) of the BRST operator
+QB = c0L0 − b0M + �QB,
+(10.22)
+inserting the decomposition (10.19) in the action (10.14) gives:
+S = 1
+2⟨Φ↓| c0L0 |Φ↓⟩ + 1
+2⟨�Φ↓| c0M |�Φ↓⟩ +⟨�Φ↓| c0 �QB |Φ↓⟩ .
+(10.23)
+The equations of motion are obtained by varying the different fields:
+0 = −M |�Φ↓⟩ + �QB |Φ↓⟩ ,
+0 = c0L0 |Φ↓⟩ + c0 �QB |�Φ↓⟩ .
+(10.24)
+Computation – Equation (10.23)
+Let’s introduce the projector Πs = b0c0 on the space H0 = H∩ker b0 and the orthogonal
+projector ¯Πs = c0b0 such that
+|Φ⟩ = |Φ↓⟩ + |Φ↑⟩ ,
+|Φ↓⟩ = Πs |Φ⟩ ,
+|Φ↑⟩ = ¯Πs |Φ⟩ .
+(10.25)
+We then have:
+ΠsQB |Φ⟩ = −b0M |Φ↑⟩ + �QB |Φ↓⟩ ,
+¯ΠsQB |Φ⟩ = c0L0 |Φ↓⟩ + �QB |Φ↑⟩ ,
+(10.26)
+using
+[Πs, �QB] = [Πs, M] = [Πs, L0] = 0.
+(10.27)
+Then, we need the fact that Π†
+s = ¯Πs. to compute the action:
+S = 1
+2 ⟨Φ, QBΦ⟩
+= 1
+2 ⟨ΠsΦ + ¯ΠsΦ, QBΦ⟩
+= 1
+2 ⟨ΠsΦ, ¯ΠsQBΦ⟩ + 1
+2 ⟨¯ΠsΦ, ΠsQBΦ⟩
+= 1
+2 ⟨Φ↓, c0L0Φ↓ + �QBΦ↑⟩ + 1
+2 ⟨Φ↑, −b0MΦ↑ + �QBΦ↓⟩
+= 1
+2 ⟨Φ↓, c0L0Φ↓⟩ + 1
+2 ⟨Φ↓, �QBΦ↑⟩ − 1
+2 ⟨Φ↑, b0MΦ↑⟩ + 1
+2 ⟨Φ↑, �QBΦ↓⟩.
+The result follows by setting |Φ↑⟩ = c0 |�Φ⟩, using (10.15) and that the BPZ conjugate
+of c0 is −c0.
+10.1.3
+Gauge invariance
+In writing the action, only the condition that the states are BRST closed has been used. One
+needs to interpret the condition that the state are not BRST-exact, or phrased differently
+that two states differing by a BRST exact state are equivalent:
+|φ⟩ ∼ |ψ⟩ + QB |λ⟩ .
+(10.28)
+157
+
+Uplifting this condition to the string field, the most direct interpretation is that it corres-
+ponds to a gauge invariance:
+|Φ⟩ −→ |Φ′⟩ = |Φ⟩ + δΛ |Φ⟩ ,
+δΛ |Φ⟩ = QB |Λ⟩
+Ngh(Λ) = 0.
+(10.29)
+In order for the ghost numbers to match, the gauge parameter has vanishing ghost number.
+The action (10.14) is obviously invariant since the BRST charge is nilpotent.
+10.1.4
+Siegel gauge
+In writing the action (10.14), the condition b0 |ψ⟩ = 0 has not been imposed on the string
+field. In Section 3.2.2, this condition was found by restricting the BRST cohomology, pro-
+jecting out states built on the ghost vacuum | ↑⟩, as required by the behaviour of the on-shell
+scattering amplitudes. In Chapter 8, we obtained it by finding that the absolute cohomology
+contains twice more states as necessary. This was also understood as a way to work with
+a specific representative of the BRST cohomology. Since the field is off-shell and since the
+action computes off-shell Green functions, these arguments cannot be used, which explains
+why we did not use this condition earlier.
+On the other hand, the condition
+b0 |Φ⟩ = 0
+(10.30)
+can be interpreted as a gauge fixing condition, called Siegel gauge. It can be reached from
+any field through a gauge transformation (10.29) with
+|Λ⟩ = −∆ |Φ⟩ ,
+∆ = b0
+L0
+,
+(10.31)
+where ∆ was defined in (8.39) and will be identified with the propagator. Note that b0 = 0
+does not imply L0 = 0 since the string field is not BRST closed.
+This gauge choice is well-defined and completely fixes the gauge symmetry off-shell,
+meaning that no solution of the equation of motion is pure gauge after the gauge fixing.
+This is shown as follows: assume that |ψ⟩ = QB |χ⟩ is an off-shell pure-gauge state with
+L0 ̸= 0, then, because it is also annihilated by b0, one finds:
+0 = {QB, b0} |ψ⟩ = L0 |ψ⟩
+(10.32)
+which yields a contradiction.
+The gauge fixing condition breaks down for L0 = 0, but this does not pose any problem
+when working with Feynman diagrams since they are not physical by themselves (nor are
+the off-shell and on-shell Green functions). Only the sum giving the scattering amplitudes
+(truncated on-shell Green functions) is physical: in this case, the singularity L0 = 0 cor-
+responds to the on-shell condition and it is well-known how such infrared divergences for
+intermediate states are removed (through the LSZ prescription, mass renormalization and
+tadpole cancellation).
+Computation – Equation (10.31)
+Performing a gauge transformation gives
+b0 |Φ′⟩ = b0 |Φ⟩ + b0QB |Λ⟩ = 0.
+(10.33)
+Then, one writes
+b0 |Φ⟩ = b0{QB, ∆} |Φ⟩ = b0QB∆ |Φ⟩ ,
+(10.34)
+using the relation (8.37), the expression (8.39) and the fact that b2
+0 = 0. Plugging this
+158
+
+back in the first equation gives:
+b0QB (∆ |Φ⟩ + |Λ⟩) = 0.
+(10.35)
+The factor of b0 can be removed by multiplying with c0, and the parenthesis should
+vanish (since it is not identically closed), which means that (10.31) holds up to a BRST
+exact state.
+Example 10.1 – Gauge fixing and singularity
+In Maxwell theory, the gauge transformation
+A′
+µ = Aµ + ∂µλ,
+(10.36)
+is used to impose the Lorentz condition
+∂µA′
+µ = 0
+=⇒
+∆λ = −∂µAµ.
+(10.37)
+In momentum space, the parameter reads
+λ = −kµ
+k2 Aµ.
+(10.38)
+It is singular when k is on-shell, k2 = 0. However, this does not prevent from computing
+Feynman diagrams.
+To understand the effect of the gauge fixing on the string field components, decompose
+the field as (10.19) |Φ⟩ = |Φ↓⟩ + c0 |�Φ↓⟩. Then, imposing the condition (10.30) yields
+|�Φ↓⟩ = 0
+=⇒
+|Φ⟩ = |Φ↓⟩ .
+(10.39)
+This has the expected effect of dividing by two the number of states and show that they are
+not physical.
+Plugging this condition in the action (10.23) leads to gauge fixed action
+S = 1
+2 ⟨Φ| c0L0 |Φ⟩ ,
+(10.40)
+for which the equation of motion is
+L0 |Φ⟩ = 0.
+(10.41)
+But, note that this equation contains much less information than the original (10.12): as |�Φ↓⟩
+is truncated from (10.40), a part of the equations of motion is lost. The missing equation
+can be found by setting |�Φ⟩ = 0 in (10.24) and must be imposed on top of the action:
+�QB |Φ⟩ = 0.
+(10.42)
+It is called out-of-Siegel gauge constraint and is equivalent to the Gauss constraint in elec-
+tromagnetism: the equations of motion for pure gauge states contain also the physical fields,
+thus, when one fixes a gauge, these relations are lost and must be imposed on the side of the
+action. This procedure mimics what happens in the old covariant theory, where the Virasoro
+constraints are imposed after choosing the flat gauge (if Φ contains no ghost on top of | ↓⟩,
+then �QB = 0 implies Ln = 0, see Section 8.3.3). Moreover, the states which do not satisfy
+the condition b0 = 0 do not propagate: this restricts the external states to be considered in
+amplitudes.
+159
+
+Remark 10.1 Another way to derive (10.40) is to insert {b0, c0} = 1 in the action:
+S = 1
+2 ⟨Φ| QB{c0, b0} |Φ⟩ = 1
+2 ⟨Φ| QBb0c0 |Φ⟩
+= 1
+2 ⟨Φ| {b0, QB}c0 |Φ⟩ − 1
+2 ⟨Φ| b0QBc0 |Φ⟩
+= 1
+2 ⟨Φ| c0L0 |Φ⟩ .
+The drawback of this computation is that it does not show directly how the constraints (10.42)
+arise.
+Remark 10.2 (Generalized gauge fixing) It is possible to generalize the Siegel gauge,
+in the same way that the Feynman gauge generalizes the Lorentz gauge. This has been studied
+in [1, 2].
+In this section, we have motivated different properties and adopted some normalizations.
+The simplest way to check that they are consistent is to derive the action in terms of the
+spacetime fields and to check that it has the expected properties from standard QFT. This
+will be the topic of Section 10.4.
+10.2
+Open string field expansion, parity and ghost num-
+ber
+A basis for the off-shell Hilbert space H is denoted by {φr}, where the ghost numbers and
+parity of the states are written as:
+nr := Ngh(φr),
+|φr| = nr
+mod 2.
+(10.43)
+The corresponding basis of dual (or conjugate) states {φc
+r} is defined by (6.145):
+⟨φc
+r|φs⟩ = δrs.
+(10.44)
+The basis states can be decomposed according to the ghost zero-modes
+|φr⟩ = |φ↓,r⟩ + |φ↑,r⟩ ,
+b0 |φ↓,r⟩ = c0 |φ↑,r⟩ = 0.
+(10.45)
+Finally, each state ψ↑ ∈ c0H can be associated to a state �ψ:
+|ψ↑⟩ = c0 | �ψ↓⟩ ,
+b0 | �ψ↓⟩ = 0,
+Ngh(ψ↑) = Ngh( �ψ↓) + 1.
+(10.46)
+More details can be found in Section 11.2.
+Any field Φ can be expanded as
+|Φ⟩ =
+�
+r
+ψr |φr⟩ ,
+(10.47)
+where the ψr are spacetime fields (remembering that r denotes collectively the continuous
+and discrete quantum numbers).1
+1The notation is slightly ambiguous: from (10.45), it looks like both components of φr have the same
+coefficient ψr. But, in fact, one sums over all linearly independent states: in terms of the components of φr,
+different basis can be considered; for example {φ↓,r, φ↑,r}, or {φ↓,r ± φ↑,r}. A more precise expression can
+be found in (10.54) and (10.56).
+160
+
+Obviously, the coefficients do not carry a ghost number since they are not worldsheet
+operators. However, they can be Grassmann even or odd such that each term of the sum
+has the same parity, so that the field has a definite parity:
+∀r :
+|Φ| = |ψr| |φr|.
+(10.48)
+If the field is Grassmann odd (resp. even) then the coefficients ψr and the basis states must
+have opposite (resp. identical) parities, such that |Φ| = 1.
+Since the parity results from worldsheet ghosts and since there would be Grassmann odd
+states even in a purely bosonic theory, it suggests that the parity of the coefficients ψr is
+also related to a spacetime ghost number G defined as:
+G(ψr) = 1 − nr.
+(10.49)
+The normalization is chosen such that the component of a classical string field (Ngh = 1)
+are classical spacetime fields with G = 0 (no ghost). We will see later that this definition
+makes sense.
+A quantum string field Φ generally contains components Φn of all worldsheet ghost
+numbers n:
+Φ =
+�
+n∈Z
+Φn,
+Ngh(Φn) = n.
+(10.50)
+The projections on the positive and negative (cylinder) ghost numbers are denoted by Φ±:
+Φ = Φ+ + Φ−,
+Φ+ =
+�
+n>1
+Φn,
+Φ− =
+�
+n≤1
+Φn.
+(10.51)
+The shift in the indices is explained by the relation (B.56) between the cylinder and plane
+ghost numbers.
+For a field Φn of fixed ghost number, coefficients of the expansion vanish whenever the
+ghost number of the basis state does not match the one of the field:
+∀nr ̸= n :
+ψr = 0.
+(10.52)
+Another possibility to define the field Φn is to insert a delta function:
+|Φn⟩ = δ(Ngh − n) |Ψ⟩ =
+�
+r
+δ(nr − n) ψr |φr⟩ .
+(10.53)
+According to (10.45), a string field Φ can also be separated in terms of the ghost zero-
+modes:
+|Φ⟩ = |Φ↓⟩ + |Φ↑⟩ = |Φ↓⟩ + c0 |�Φ↓⟩ ,
+(10.54a)
+|Φ↑⟩ = c0 |�Φ↓⟩ ,
+|�Φ↓⟩ = b0 |Φ↑⟩ ,
+(10.54b)
+where the components satisfy the constraints
+b0 |Φ↓⟩ = 0,
+c0 |Φ↑⟩ = 0,
+b0 |�Φ↓⟩ = 0.
+(10.55)
+The fields |Φ↓⟩ and |Φ↑⟩ (or |�Φ↓⟩) are called the down and top components and they can be
+expanded as:
+|Φ↓⟩ =
+�
+r
+ψ↓,r |φ↓,r⟩ ,
+|Φ↑⟩ =
+�
+r
+ψ↑,r |φ↑,r⟩ .
+(10.56)
+161
+
+10.3
+Path integral quantization
+The string field theory can be quantized with a path integral:
+Z =
+�
+dΦcl e−S[Φcl] =
+�
+dΦcl e− 1
+2⟨Φcl|QB|Φcl⟩.
+(10.57)
+An index has been added to the field to emphasize that it is the classical field (no spacetime
+ghosts). The simplest way to define the measure is to use the expansion (9.4) such that
+Z =
+� �
+s
+dψs e−S[{ψr}].
+(10.58)
+10.3.1
+Tentative Faddeev–Popov gauge fixing
+The action can be gauge fixed using the Faddeev–Popov formalism. The gauge fixing con-
+dition is
+F(Φcl) := b0 |Φcl⟩ = 0.
+(10.59)
+Its variation under a gauge transformation (10.29) reads
+δF = b0QB |Λcl⟩ ,
+(10.60)
+which implies that the Faddeev–Popov determinant is
+det δF
+δΛcl
+= det b0QB.
+(10.61)
+This determinant is rewritten as a path integral by introducing a ghost C and an antighost
+B′ string fields (the prime on B′ will become clear below):
+det b0QB =
+�
+dB′dC e−SFP,
+SFP = −⟨B′| b0QB |C⟩ .
+(10.62)
+The ghost numbers are attributed by selecting the same ghost number for the C ghost and for
+the gauge parameter, and then requiring that the Faddeev–Popov action is non-vanishing:
+Ngh(B′) = 3,
+Ngh(C) = 0.
+(10.63)
+The ghosts can be expanded as
+|B′⟩ = δ(Ngh − 3)
+�
+r
+b′
+r |φr⟩ ,
+|C⟩ = δ(Ngh)
+�
+r
+cr |φr⟩ ,
+(10.64)
+where the coefficients br and cr are Grassmann odd in order for the determinant formula to
+make sense:
+|br| = |cr| = 1.
+(10.65)
+Then, since the basis states appearing in B′ and C are respectively odd and even, this
+implies
+|B′| = 0,
+|C| = 1.
+(10.66)
+However, there is a redundancy in the gauge fixing because the Faddeev–Popov action
+is itself invariant under two independent transformations:
+δ |C⟩ = QB |Λ−1⟩ ,
+Ngh(Λ−1) = −1,
+(10.67a)
+δ |B′⟩ = b0 |Λ′⟩ ,
+Ngh(Λ′) = 4.
+(10.67b)
+162
+
+This residual invariance arises because not all |Λcl⟩ generate a gauge transformation. Indeed,
+if
+|Λ⟩ = |Λ0⟩ + QB |Λ−1⟩ ,
+(10.68)
+the field transforms as
+|Φ′
+cl⟩ −→ |Φcl⟩ + QB |Λ0⟩
+(10.69)
+and there is no trace left of |Λ−1⟩, so it should not be counted.
+The second invariance (10.67b) is not problematic because b0 is an algebraic operator (the
+Faddeev–Popov action associated to the determinant has no dynamics). The decompositions
+of the gauge parameter Λ′ and the B′ field into components (10.54) read:
+|B′⟩ = |B′
+↓⟩ + c0 |B⟩ ,
+|B⟩ := | �B′
+↓⟩ ,
+(10.70a)
+|Λ′⟩ = |Λ′
+↓⟩ + c0 |�Λ′
+↓⟩ .
+(10.70b)
+The gauge transformations act on the components as:
+δ |B′
+↓⟩ = |�Λ′
+↓⟩ ,
+δ |B⟩ = 0.
+(10.71)
+This shows that B is gauge invariant and B′
+↓ can be completely removed by the gauge
+transformation. This makes sense because B′
+↓ does not appear in the action (10.62). The
+gauge transformation (10.67b) can be used to fix the gauge:
+|F ′⟩ = c0 |B′⟩ = 0
+=⇒
+|B′
+↓⟩ = 0.
+(10.72)
+This fixes completely the gauge invariance since the field B is restricted to satisfy b0 |B⟩ = 0,
+and the component form (10.71) of the gauge transformation shows that no transformation
+is allowed. Moreover, there is no need to introduce a Faddeev–Popov determinant for this
+gauge fixing because the corresponding ghosts would not couple to the other fields (and this
+would continue to hold even in the presence of interactions, see Remark 10.4). Indeed, from
+the absence of derivatives in the gauge transformation, one finds that the determinant is
+constant and thus a ghost-representation is not necessary:
+det δF ′
+δΛ′ = det c0b0 = det c0 det b0 = 1
+2 det{b0, c0} = 1
+2.
+(10.73)
+Then, redefining the measure, the partition function and action reduce to
+∆FP =
+�
+dB dC e−SFP[B,C],
+SFP =⟨B| QB |C⟩ .
+(10.74)
+Note that the field B satisfies
+b0 |B⟩ = 0,
+Ngh(B) = 2,
+|B| = 1.
+(10.75)
+Since both fields are Grassmann odd, the action can be rewritten in a symmetric way:
+SFP = 1
+2
+�
+⟨B| QB |C⟩ +⟨C| QB |B⟩
+�
+.
+(10.76)
+Remark 10.3 (Ghost and anti-ghost definitions) The definition of the anti-ghost B
+and ghost C is appropriate because the worldsheet and spacetime ghost numbers are related
+by a minus sign (and a shift of one unit). In the BV formalism, we will see that the fields
+contain the matter and ghost fields, while the antifields contain the anti-ghosts. These two
+sets are respectively defined with Ngh ≤ 1 and Ngh > 1.
+163
+
+The constraint b0 |B⟩ = 0 can be lifted by adding a top component:
+|B⟩ = |B↓⟩ + c0 | �B↓⟩
+(10.77)
+together with the gauge invariance
+δ |B⟩ = QB |Λ1⟩ .
+(10.78)
+Note the difference with (10.70): while B = �B′
+↓ was the top component of the B′ field, here,
+it is defined to be the down component, such that |B↓⟩ = | �B′
+↓⟩. However, for the moment,
+we keep B to satisfy b0 |B⟩ = 0.
+Remark 10.4 (Decoupling of the ghosts) Since the theory is free the Faddeev–Popov
+action (10.74) could be ignored and absorbed in the normalization because it does not couple
+to the field. On the other hand, when interactions are included, the gauge transformation
+is modified and the ghosts couple to the matter fields.
+But this is true only for the C
+transformation (10.67a), not for (10.67b). Then it means that ghosts introduced for gauge
+fixing (10.67b) will never couple to the matter and other ghosts.
+The invariance (10.67a) is a gauge invariance for C and must be treated in the same
+way as (10.29). Then, following the Faddeev–Popov procedure, one is lead to introduce new
+ghosts for the ghosts. But, the same structure appears again. This leads to a residual gauge
+invariance, which has the same form. This process continues recursively and one finds an
+infinite tower of ghosts.
+10.3.2
+Tower of ghosts
+In order to simplify the notations, all the fields are denoted by Φn where n gives the ghost
+number:
+• Φ1 := Φcl is the original physical field
+• Φ0 := C and, more generally, Φn with n < 1 are ghosts
+• Φ2 := B and, more generally, Φn with n > 1 are anti-ghosts
+The recipe is that each pair of ghost fields (Φn+2, Φ−n) is associated to a gauge parameter
+Λ−n−1 with n ≥ 0. It is then natural to gather all the fields in a single field
+|Φ⟩ =
+�
+n
+|Φn⟩
+(10.79)
+satisfying the gauge fixing constraint:
+b0 |Φ⟩ = 0
+=⇒
+b0 |Φn⟩ = 0.
+(10.80)
+For n ≤ 1, these constraints are gauge fixing conditions for the invariance δ |Φn⟩ = QBΛn.
+For n > 1, they arise by considering only the top component of the B field.
+Finally, the gauge fixing condition can be incorporated inside the action by using a
+Lagrange multiplier β, which is an auxiliary string field containing also components of all
+ghost numbers:
+|β⟩ =
+�
+n∈Z
+|βn⟩ .
+(10.81)
+The path integral then reads
+Z =
+�
+dΦdβ e−S[Φ,β],
+(10.82)
+164
+
+where
+S[Φ, β] = 1
+2⟨Φ| QB |Φ⟩ +⟨β| b0 |Φ⟩
+(10.83a)
+=
+�
+n∈Z
+�1
+2⟨Φ2−n| QB |Φn⟩ +⟨β4−n| b0 |Φn⟩
+�
+.
+(10.83b)
+The first term of the action has the same form as the classical action (10.14), but now includes
+fields at every ghost number. The complete BV analysis is relegated to the interacting theory.
+Removing the auxiliary field β = 0, one finds that the action is invariant under the
+extended gauge transformation
+δ |Φ⟩ = QB |Λ⟩ ,
+(10.84)
+where the gauge parameter has also components of all ghost numbers:
+|Λ⟩ =
+�
+n∈Z
+|Λn⟩ .
+(10.85)
+10.4
+Spacetime action
+In order to make the string field action more concrete, and as emphasized in Chapter 9,
+it is useful to expand the string field in spacetime fields and to write the action for the
+lowest modes. This also helps to check that the normalization chosen until here correctly
+reproduces the standard QFT normalizations. For simplicity we focus on the open bosonic
+string in D = 26.
+We build the string from the vacuum |k, ↓⟩ (Chapter 8) by acting with the ghost positive-
+frequency modes b−n and c−n, the zero-mode c0, and from the scalar oscillators iαµ
+−n.
+Up to level ℓ = 1, the classical open string field can be expanded as
+|Φ⟩ =
+1
+√
+α′
+�
+dDk
+(2π)D
+�
+T(k) + Aµ(k)αµ
+−1 + i
+�
+α′
+2 B(k)b−1c0 + · · ·
+�
+|k, ↓⟩
+(10.86)
+before gauge fixing. The spacetime fields are T(k), Aµ(k) and B(k): their roles will be
+interpreted below. The first two terms are part of the |Φ↓⟩ component, while the last term
+is part of the |Φ↑⟩ component.
+All terms are correctly Grassmann even and they have
+vanishing spacetime ghost numbers. The normalizations are chosen in order to retrieve the
+canonical normalization in QFT. The factor of i in front of B is needed for the field B to
+be real (as can be seen below, this leads to the expected factor ikµ which maps to ∂µ in
+position space).
+The equation (10.12) leads to the following equations of motion of the spacetime fields:
+(α′k2 − 1)T(k) = 0,
+k2Aµ(k) + ikµB(k) = 0,
+kµAµ(k) + iB(k) = 0.
+(10.87)
+Moreover, plugging the last equation into the second one gives
+k2Aµ(k) − kµk · A(k) = 0.
+(10.88)
+After Fourier transformation, the equations in position space read:
+(α′∆ + 1) T = 0,
+B = ∂µAµ,
+∆Aµ = ∂µB.
+(10.89)
+This shows that T(k) is a tachyon with mass m2 = −1/α′ and Aµ(k) is a massless gauge
+field. The field B(k) is the Nakanishi–Lautrup auxiliary field: it is completely fixed once
+Aµ is known since its equation has no derivative. Siegel gauge imposes B = 0 which shows
+that it generalizes the Feynman gauge to the string field.
+165
+
+Computation – Equation (10.87)
+Keeping only the levels 0 and 1 terms in the string field, it is sufficient to truncate the
+BRST operator as
+QB = c0L0 − b0M + �QB,
+M ∼ 2c−1c1,
+�QB ∼ c1Lm
+−1 + c−1Lm
+1 ,
+Lm
+1 ∼ α0 · α1,
+Lm
+−1 ∼ α0 · α−1.
+(10.90)
+Acting on the string field gives
+QB |Φ⟩ =
+1
+√
+α′
+�
+dDk
+(2π)D
+�
+T(k)c0L0 |k, ↓⟩ + Aµ(k)
+�
+c0L0 + ηνρc−1αν
+1αρ
+0
+�
+αµ
+−1 |k, ↓⟩
++ i
+�
+α′
+2 B(k)
+�
+− 2b0c−1c1 + ηνρc1αν
+−1αρ
+0
+�
+b−1c0 |k, ↓⟩
+�
+=
+1
+√
+α′
+�
+dDk
+(2π)D
+�
+T(k)(α′k2 − 1)c0 |k, ↓⟩
++ Aµ(k)
+�
+α′k2c0αµ
+−1 +
+√
+2α′ηνρηµν kρ c−1
+�
+|k, ↓⟩
++ i
+�
+α′
+2 B(k)
+�
+2c−1 +
+√
+2α′ηνρkραν
+−1c0
+�
+|k, ↓⟩
+�
+=
+1
+√
+α′
+�
+dDk
+(2π)D
+�
+T(k)(α′k2 − 1)c0 |k, ↓⟩
++ α′�
+Aµ(k)k2 + ikµB(k
+�
+c0αµ
+−1 |k, ↓⟩
++
+√
+2α′
+�
+kµAµ(k) + iB(k)
+�
+c−1 |k, ↓⟩
+�
+.
+One needs to be careful when anticommuting the ghosts and we used that pL = k and
+α0 =
+√
+2α′k for the open string. It remains to require that the coefficient of each state
+vanishes.
+In order to confirm that Aµ is indeed a gauge field, we must study the gauge transform-
+ation. The gauge parameter is expanded at the first level:
+|Λ⟩ =
+i
+√
+2α′
+�
+dDk
+(2π)D
+�
+λ(k) b−1 |k, ↓⟩ + · · ·
+�
+.
+(10.91)
+Note that b−1 | ↓⟩ is the SL(2, C) ghost vacuum. Since
+QB |Λ⟩ =
+i
+√
+α′
+�
+dDk
+(2π)D λ(k)
+�
+−
+�
+α′
+2 k2 b−1c0 + kµαµ
+−1
+�
+|k, ↓⟩ ,
+(10.92)
+matching the coefficients in (10.29) gives
+δAµ = −ikµλ,
+δB = k2λ.
+(10.93)
+This is the appropriate transformation for a U(1) gauge field.
+Finally, one can derive the action; for simplicity, we work in the Siegel gauge. We consider
+only the tachyon component:
+|T⟩ =
+�
+dDk
+(2π)D T(k)c1 |k, 0⟩ ,
+(10.94)
+166
+
+with c1 |0⟩ = | ↓⟩. The BPZ conjugate and Hermitian conjugates are respectively:
+⟨T| =
+�
+dDk
+(2π)D T(k)⟨−k, 0| c−1,
+(10.95a)
+⟨T ‡| =
+�
+dDk
+(2π)D T(k)∗⟨k, 0| c−1.
+(10.95b)
+since ct
+1 = c−1 when using the operator I− in (6.111). Imposing equality of both leads to
+the reality condition
+T(k)∗ = T(−k),
+(10.96)
+which agrees with the fact that the tachyon is real (the integration measure changes as
+dDk → −dDk, but the contour is reversed).
+Then, the action reads:
+S[T] = 1
+2
+�
+dDk
+(2π)D T(−k)
+�
+k2 − 1
+α′
+�
+T(k).
+(10.97)
+This shows that the action is canonically normalized as it should for a real scalar field.
+Similarly, one can compute the action for the gauge field:
+S[A] = 1
+2
+�
+dDk
+(2π)D Aµ(−k)k2Aµ(k).
+(10.98)
+The correct normalization of the tachyon (real scalar field of negative mass) gives a jus-
+tification a posteriori for the normalization of the action (10.14).
+Typically, string field
+actions are normalized in this way, by requiring that the first physical spacetime fields has
+the correct normalization. Note how this implies the correct normalization for all the others
+physical fields. Generalizing this computation for higher-levels, one always find the kinetic
+term to be:
+L+
+0
+2
+= 1
+2(k2 + m2),
+(10.99)
+which is the canonical normalization.
+Computation – Equation (10.97)
+⟨T| c0L0 |T⟩ = 1
+α′
+�
+dDk
+(2π)D
+dDk′
+(2π)D T(k)T(k′)⟨−k′, 0| c−1c0L0c1 |k, 0⟩
+= 1
+α′
+�
+dDk
+(2π)D
+dDk′
+(2π)D T(k)T(k′)(α′k2 − 1)⟨−k′, 0| c−1c0c1 |k, 0⟩
+= 1
+α′
+�
+dDk
+(2π)D dDk′ T(k)T(k′)(α′k2 − 1) δ(D)(k + k′),
+where we used ⟨0| c−1c0c1 |0⟩ = 1 and ⟨k′|k⟩ = (2π)Dδ(D)(k + k′).
+10.5
+Closed string
+The derivation of the BRST free action for the closed string is very similar. The starting
+point is the equation of motion
+QB |Ψ⟩ = 0
+(10.100)
+167
+
+for the closed string field |Ψ⟩. The difference with (10.12) is that the BRST charge QB now
+includes both the left- and right-moving sectors. In the case of the open string, the field Φ
+was free of any constraint: we will see shortly that this is not the case for the closed string.
+The next step is to find an inner product ⟨·, ·⟩ to write the action:
+S = 1
+2 ⟨Ψ, QBΨ⟩.
+(10.101)
+Following the open string, it seems logical to give the string field Ψ the same ghost number
+as the states in the cohomology:
+Ngh(Ψ) = 2.
+(10.102)
+In this case, the ghost number of the arguments of ⟨·, ·⟩ in (10.101) is Ngh = 5. The ghost
+number anomaly requires the total ghost number to be 6, that is:
+Ngh(⟨·, ·⟩) = 1.
+(10.103)
+There is no other choice because Ngh(Ψ) must be integer.
+The simplest solution is to
+insert one c zero-mode c0 or ¯c0, or a linear combination. The BRST operator QB contains
+both L±
+0 (see the decomposition (8.88)): the natural expectation (and by analogy with the
+open string) is that the gauge fixed equation of motion (to be discussed below) should be
+equivalent to the on-shell equation L+
+0 = 0 (see also Section 8.3.4). This is possible only if
+the insertion is c−
+0 . With this insertion, ⟨·, ·⟩ can be formed from the BPZ product:
+⟨A, B⟩ =⟨A| c−
+0 |B⟩ .
+(10.104)
+Then, the action reads:
+S = 1
+2 ⟨Ψ| c−
+0 QB |Ψ⟩ .
+(10.105)
+However, the presence of c−
+0 has a drastic effect because it annihilates part of the string
+field. Decomposing the Hilbert space as in (7.175)
+H = H− ⊕ c−
+0 H−,
+H− := H ∩ ker b−
+0 ,
+(10.106)
+the string field reads:
+|Ψ⟩ = |Ψ−⟩ + c−
+0 |�Ψ−⟩ ,
+Ψ−, �Ψ− ∈ H−,
+(10.107)
+such that
+c−
+0 |Ψ⟩ = c−
+0 |Ψ−⟩ .
+(10.108)
+The problem in such cases is that the kinetic term may become non-invertible. This motiv-
+ates to project out the component �Ψ− by imposing the following constraint on the string
+field:
+b−
+0 |Ψ⟩ = 0.
+(10.109)
+The constraint (10.109) is stronger than the constraint L−
+0 = 0 for states in the cohomo-
+logy (Section 8.3.1), so there is no information lost on-shell by imposing it. For this reason,
+we will also impose the level-matching condition:
+L−
+0 |Ψ⟩ = 0,
+(10.110)
+such that
+Ψ ∈ H− ∩ ker L−
+0 .
+(10.111)
+This will later be motivated by studying the propagator and the off-shell scattering amp-
+litudes. To avoid introducing more notations, we will not use a new symbol for this space
+and keep implicit that Ψ ∈ ker L−
+0 .
+168
+
+The necessity of this condition can be understood differently. We had found that it is
+necessary to ensure that the closed string parametrization is invariant under translations
+along the string (Section 3.2.2). Since there is no BRST symmetry associated to this sym-
+metry, one needs to keep the constraint.2 This suggests that one may enlarge further the
+gauge symmetry and interpret (10.109) as a gauge fixing condition. This would be quite
+desirable: one could argue that a fundamental field should be completely described by the
+Lagrangian (if such a description exists) and that it should not be necessary to supplement
+it with constraints imposed by hand. While this can be achieved at the free level, this idea
+runs into problems in the presence of interactions (Section 13.3.1) and the interpretation is
+not clear.3
+The action (10.105) is gauge invariant under:
+|Ψ⟩ −→ |Ψ′⟩ = |Ψ⟩ + δΛ |Ψ⟩ ,
+δΛ |Ψ⟩ = QB |Λ⟩ ,
+(10.112)
+where the gauge parameter has ghost number 1 and also lives in H− ∩ ker L−
+0 :
+Ngh(Λ) = 1,
+L−
+0 |Λ⟩ = 0,
+b−
+0 |Λ⟩ = 0.
+(10.113)
+As for the open string, the gauge invariance (10.112) can be gauge fixed in the Siegel
+gauge:
+b+
+0 |Ψ⟩ = 0.
+(10.114)
+Then, the action reduces to:
+S = 1
+2 ⟨Ψ| c−
+0 c+
+0 L+
+0 |Ψ⟩ = 1
+4 ⟨Ψ| c0¯c0L+
+0 |Ψ⟩ .
+(10.115)
+The equation of motion is equivalent to the on-shell condition as expected:
+L+
+0 |Ψ⟩ = 0.
+(10.116)
+Additional constraints must be imposed to ensure that only the physical degrees of freedom
+propagate.
+Computation – Equation (10.115)
+c−
+0 QB = (c0 − ¯c0)(c0L0 + ¯c0 ¯L0) = c0¯c0(L0 + ¯L0).
+10.6
+Summary
+In this chapter, we have shown how the BRST conditions defining the cohomology can be
+interpreted as an equation of motion for a string field together with a gauge invariance. We
+found a subtlety for the closed string due to the ghost number anomaly and because of the
+level-matching condition. Then, we studied several basic properties in order to prove that
+the free action has the expected properties.
+The next step is to add the interactions to the action, but we don’t know first principles
+to write them. For this reason, we need to take a detour and to consider off-shell amplitudes.
+By introducing a factorization of the amplitudes, it is possible to rewrite them as Feynman
+diagrams, where fundamental interactions are connected by propagators (which we will find
+to match the one in the Siegel gauge). This can be used to extract the interacting terms of
+the action.
+2Yet another reason can be found in Section 3.2.2 (see also Section 8.3.4): to motivate the need of the
+b+
+0 condition, we could take the on-shell limit from off-shell states because L+
+0 is continuous. However, the
+L−
+0 operator is discrete and there is no such limit we can consider [245]. So we must always impose this
+condition, both off- and on-shell.
+3A recent proposal can be found in [179].
+169
+
+10.7
+Suggested readings
+• The free BRST string field theory is discussed in details in [245] (see also [124, chap. 7,
+125, chap. 9, 235, chap. 11]). Shorter discussions can be found in [261, 192, sec. 4,
+253, 1, 244].
+• Spacetime fields and actions are discussed in [242, 192, sec. 4].
+• Gauge fixing [147, 242, sec. 6.5, 7.2, 7.4, 1, 134, sec. 2.1, 2, 26].
+• General properties of string field (reality, parity, etc.) [1, 262].
+170
+
+Chapter 11
+Introduction to off-shell string
+theory
+Abstract
+In this chapter, we introduce a framework to describe off-shell amplitudes in
+string theory. We first start by motivating various concepts – in particular, local coordinates
+and factorization – by focusing on the 3- and 4-point amplitudes. We then prepare the stage
+for a general description of off-shell amplitudes. We focus again on the closed bosonic string
+only.
+11.1
+Motivations
+11.1.1
+3-point function
+The tree-level 3-point amplitude of 3 weight hi vertex operators1 Vi is given by
+A0,3 =
+� 3
+�
+i=1
+Vi(zi)
+�
+S2
+∝ (z1 − z2)h3−h1−h2 × perms × c.c.
+(11.1)
+There is no integration since dim M0,3 = 0.
+The amplitude is independent of the zi only if the matter state is on-shell, hi = 0, for
+example if Vi = c¯cVi with h(Vi) = 1. Indeed, if hi ̸= 0, then A0,3 is not invariant under
+conformal transformations (6.38):
+z −→ fg(z) = az + b
+cz + d ∈ SL(2, C)
+(11.2)
+(it transforms covariantly). This is a consequence of the punctures: the presence of the latter
+modifies locally the metric, since they act as sources of negative curvature. When performing
+a conformal transformation, the metric around the punctures changes in a different way as
+away from them. This implies that the final result depends on the metric chosen around
+the punctures. This looks puzzling because the original path integral derivation (Chapter 3)
+indicates that the 3-point amplitude should not depend on the locations of the operators
+because its moduli space is empty (hence, all choices of zi should be equivalent).
+1The quantum number (k, j) of the vertex operator is mostly irrelevant for the discussion of the current
+and next chapters, and they are omitted.
+We will distinguish them by a number and reintroduce the
+momentum k when necessary. We also omit the overall normalization of the amplitudes.
+171
+
+The solution is to introduce local coordinates wi with a flat metric |dwi|2 around each
+puncture conventionally located at wi = 0. The local coordinates are defined by the maps:
+z = fi(wi),
+zi = fi(0).
+(11.3)
+This is also useful to characterize in a simpler way the dependence of off-shell amplitudes
+rather than using the metric around the punctures (computations may be more difficult with
+a general metric).
+The expression of a local operator in the local coordinate system is found by applying
+the corresponding change of coordinates (6.48):
+f ◦ V (w) = f ′(w)hf ′(w)
+¯h V
+�
+f(w)
+�
+.
+(11.4)
+The amplitude reads then
+A0,3 =
+� 3
+�
+i=1
+fi ◦ Vi(0)
+�
+S2
+=
+� 3
+�
+i=1
+f ′
+i(0)hif ′
+i(0)
+¯hi
+� � 3
+�
+i=1
+Vi
+�
+fi(0)
+�
+�
+S2
+(11.5a)
+∝
+� 3
+�
+i=1
+f ′
+i(0)hif ′
+i(0)
+¯hi
+�
+�
+f1(0) − f2(0)
+�h3−h1−h2 × perms × c.c.
+(11.5b)
+The amplitude depends on the local coordinate choice fi, but not on the metric around the
+punctures. It is also invariant under SL(2, C): the transformation (11.2) written in terms of
+the local coordinates is
+fi −→ afi + b
+cfi + d
+(11.6)
+from which we get:
+f ′
+i −→
+f ′
+i
+(cfi + d)2 ,
+fi − fj −→
+fi − fj
+(cfi + d)(cfj + d).
+(11.7)
+All together, this implies the invariance of the 3-point amplitude since the factors in the
+denominator cancel.
+When the states are on-shell hi = 0, the dependence in the local
+coordinate cancels, showing that the latter is non-physical.
+One can ask how Feynman graphs can be constructed in string theory. By definition, an
+amplitude is the sum of Feynman graphs contributing at that order in the loop expansion
+and for the given number of external legs. The Feynman graphs are themselves built from a
+set of Feynman rules. These correspond to the data of the fundamental interactions together
+with the definition of a propagator. Since a tree-level cubic interaction is the interaction of
+the lowest order, it makes sense to promote it to a fundamental cubic vertex2 V0,3:
+V0,3(V1, V2, V3) :=
+= A0,3(V1, V2, V3).
+(11.8)
+The index 0 reminds that it is a tree-level interaction.
+2The notation will become clear later, and should not be confused with the vertex operators.
+172
+
+11.1.2
+4-point function
+The tree-level 4-point amplitude is expressed as
+A0,4 =
+�
+d2z4
+� 3
+�
+i=1
+c¯cVi(zi) V4(z4)
+�
+S2
+.
+(11.9)
+The conformal weights are denoted by h(Vi) = hi. For on-shell states, hi = 1: while there
+is no dependence on the positions z1, z2 and z3, there are divergences for
+z4 −→ z1, z2, z3,
+(11.10)
+corresponding to collisions of punctures in the integration process. Moreover, the expression
+does not look symmetric: it would me more satisfactory if all the insertions were accompanied
+by ghost insertions and if all the puncture locations were treated on an equal footing.
+Example 11.1 – Tachyons
+Given tachyon states Vi = eiki·X, the amplitude reads:
+A0,4 ∝
+3
+�
+i,j=1
+i<j
+|zi − zj|2+ki·kj
+�
+d2z4
+3
+�
+i=1
+|z4 − zi|ki·k4.
+(11.11)
+The integral diverges for z4 → zi if ki · k4 ≤ 0. This can happen for physical values of
+the momenta ki.
+The idea is to cut out regions around z1, z2 and z3 in the z4-plane and to change the
+interpretation of these contributions. First, we consider the case z4 → z3, which corresponds
+to cutting a region around z3. Writing z4 = qy4 with y4 ∈ C fixed, the contribution of this
+region to the amplitude is denoted by F(s)
+0,4. For simplicity, we take z3 = 0. The contribution
+reads:
+F(s)
+0,4 =
+� d2q
+|q|2 ⟨c¯cV1(z1)c¯cV2(z2)c¯cV3(0)|qy4|2V4(qy4)⟩.
+(11.12)
+The implicit radial ordering pushes V3 to the left of V4 and using the OPE between the b
+and c ghosts gives:
+F(s)
+0,4 = −
+� d2q
+|q|2
+�
+c¯cV1(z1)c¯cV2(z2)
+�
+|w|=|q|1/2
+dw w b(w)
+�
+|w|=|q|1/2
+d ¯w ¯w b(w) c¯cV4(qy4)c¯cV3(0)
+�
+.
+(11.13)
+The sign arises by anti-commuting c and ¯b. The integration variable q can be removed from
+the argument of V4 using the L0 and ¯L0 operators:
+F(s)
+0,4 = −
+� d2q
+|q|2
+�
+c¯cV1(z1)c¯cV2(z2)
+�
+dw w b(w)
+�
+d ¯w ¯w b(w) qL0 ¯q
+¯L0c¯cV4(y4)c¯cV3(0)
+�
+.
+(11.14)
+This expression is more satisfactory because all vertex operators are accompanied with c-
+ghost insertions and none of the arguments is integrated over. But, in fact, even better can
+be achieved.
+Inserting two complete sets of states {φr} (see Section 11.2) inside this expression gives
+(restoring a generic z3-dependence):
+F(s)
+0,4 = ⟨c¯cV1(z1)c¯cV2(z2)φr(0)⟩ ⟨c¯cV3(z3)c¯cV4(y4)φs(0)⟩
+� d2q
+|q|2
+�
+φc
+rqL0 ¯q
+¯L0b0¯b0φc
+s
+�
+, (11.15)
+173
+
+where the sum over r and s is implicit. The conjugate states φc
+r are defined by ⟨φc
+r|φs⟩ = δrs.
+The first two terms are cubic interactions (11.8), and the last term connects both. It is then
+tempting to identify the latter with a propagator ∆
+∆
+�
+φc
+r, φc
+s
+�
+:=⟨φc
+r| ∆ |φc
+s⟩ := −
+� d2q
+|q|2
+�
+φc
+rqL0 ¯q
+¯L0b0¯b0φc
+s
+�
+,
+(11.16)
+such that:
+F(s)
+0,4 = V0,3
+�
+c¯cV1(z1), c¯cV2(z2), φr(0)
+�
+× ∆
+�
+φc
+r, φc
+s
+�
+× V0,3
+�
+c¯cV3(z3), c¯cV4(y4), φs(0)
+�
+=
+(11.17)
+To make this more precise, change the coordinates as:
+q = e−s+iθ,
+s ∈ R+,
+θ ∈ [0, 2π),
+(11.18)
+such that the integral becomes:
+� d2q
+|q|2 qL0 ¯q
+¯L0 = 2
+� ∞
+0
+ds
+� 2π
+0
+dθ e−s(L0+¯L0)eiθ(L0−¯L0) =
+2
+L0 + ¯L0
+δL0,¯L0.
+(11.19)
+This shows that the propagator can be rewritten as
+∆ = − 2b0¯b0
+L0 + ¯L0
+δL0,¯L0 = b+
+0
+L+
+0
+b−
+0 δL−
+0 ,0,
+(11.20)
+where L±
+0 = L0 ± ¯L0 and b±
+0 = b0 ±¯b0. The sign is added by anticipating the normalization
+to be derived later. Its properties will be studied in details in Section 14.2.2.
+Taking the basis states φr := φα(k) to be eigenstates of L0 and ¯L0
+L0 |φα(k)⟩ = ¯L0 |φα(k)⟩ = α′
+4 (k2 + m2
+α) |φα(k)⟩
+(11.21)
+allows to rewrite the last term of F(s)
+0,4 as
+∆αβ(k) =
+� d2q
+|q|2
+�
+φc
+α(k)qL0 ¯q
+¯L0b0¯b0φc
+β(−k)
+�
+= Mαβ(k)
+k2 + m2α
+.
+(11.22)
+The finite-dimensional matrix Mαβ gives the overlap of states of identical masses:
+Mαβ(k) := 2
+α′ ⟨φc
+α(k)| b+
+0 b−
+0 |φc
+β(−k)⟩ .
+(11.23)
+The propagator depends only on one momentum because ⟨k|k′⟩ ∼ δ(D)(k − k′). This is
+exactly the standard propagator one finds in QFT and this justifies the above claim. The
+contribution F(s)
+0,4 to the amplitude can be seen as a s-channel Feynman graph obtained by
+gluing two cubic fundamental vertices with a propagator. We will see later the interpretation
+in terms of Riemann surfaces.
+174
+
+The same procedure can be followed by considering z4 ∼ z2 and z4 ∼ z1. This leads to
+contributions F(t)
+0,4 and F(u)
+0,4 corresponding to t- and u-channel Feynman graphs:
+F(t)
+0,4 =
+F(u)
+0,4 =
+(11.24)
+In general, the sum of the three contributions F(s,t,u)
+0,4
+does not reproduce the full amp-
+litude A0,4. Said differently, the regions cut in the z4-plane does not cover it completely. It is
+then natural to interpret the remaining part as a fundamental tree-level quartic interaction
+denoted by
+V0,4 =
+(11.25)
+such that
+A0,4 = F(s)
+0,4 + F(t)
+0,4 + F(u)
+0,4 + V0,4.
+(11.26)
+Up to (11.14) it was sufficient to consider on-shell states, but the insertion of the complete
+basis requires to consider also off-shell states since on-shell states do not form a basis of the
+Hilbert space.
+As discussed for the 3-point functions, it is necessary to introduce local
+coordinates to describe off-shell states properly.
+With the 3- and 4-point functions, we motivated the use of off-shell states and introduced
+the two important ideas of local coordinates and amplitude factorization. We also indicated
+that amplitudes can be written in a more symmetric way (see also the discussion at the
+end of Section 3.1.2). In the rest of this chapter, we give additional ideas on off-shell string
+theory.
+Remark 11.1 (Riemann surface interpretation) The interpretation of the insertion of
+a propagator in terms of Riemann surface consists in gluing two of them thanks to the
+plumbing fixture procedure (Section 12.3).
+11.2
+Off-shell states
+An off-shell state is a generic state of the CFT Hilbert space
+H = Hm ⊗ Hgh.
+(11.27)
+without any constraint. A basis for the off-shell states is denoted by
+H = Span{|φr⟩}.
+(11.28)
+175
+
+Since there is no constraint on the states, the ghost number of φr are arbitrary and denoted
+as:
+nr := Ngh(φr) ∈ Z
+(11.29)
+(the ghost number is restricted for states in the cohomology of QB). The Grassmann parity
+of a state φr is denoted as |φr|. When there is no fermions in the matter sector (usually the
+case for the bosonic string), only ghosts are odd. Then, the Grassmann parity of a state is
+odd (resp. even) if its ghost number is odd (resp. even):
+|φr| := Ngh(φr)
+mod 2 =
+�
+0
+Ngh(φr) even,
+1
+Ngh(φr) odd.
+(11.30)
+The dual basis {|φc
+r⟩} is defined from the BPZ inner product:
+⟨φc
+r|φs⟩ = δrs.
+(11.31)
+Denoting the ghost numbers of the dual states by
+nc
+r := Ngh(φc
+r),
+(11.32)
+the product is non-vanishing if
+nc
+r + nr = 6,
+(11.33)
+due to the ghost number anomaly on the sphere. This condition cannot be satisfied if the
+dual state φc
+r is simply taken to be the BPZ conjugate φt
+r since the BPZ conjugation does
+not change the ghost number. This implies that
+⟨φr|φs⟩ = 0
+(11.34)
+from the ghost anomaly for every state, except for the closed string states with Ngh = 3 (in
+fact, the inner product of these states is also zero as can be seen after investigation). One
+can show that
+⟨φr|φc
+s⟩ = (−1)|φr| δrs.
+(11.35)
+Hence, the resolution of the identity can be written in the two equivalent ways
+1 =
+�
+r
+|φr⟩⟨φc
+r| =
+�
+r
+(−1)|φr| |φc
+r⟩⟨φr| .
+(11.36)
+Following (7.176), the Hilbert space can be decomposed as:
+H = H± ⊕ c±
+0 H±,
+(11.37)
+where
+H± := H ∩ ker b±
+0 = H0 ⊕ c∓
+0 H0,
+H0 := H ∩ ker b−
+0 ∩ ker b+
+0 .
+(11.38)
+In fact, we will find that a consistent description of the off-shell amplitudes for the closed
+string requires to impose some conditions on the states even at the off-shell level. The off-
+shell states will have to satisfy the level-matching condition and to be annihilated by b−
+0 :
+L−
+0 |φ⟩ = 0,
+b−
+0 |φ⟩ = 0.
+(11.39)
+This implies that the off-shell states will be elements of H− ∩ ker L−
+0 . This will appear as
+consistency conditions on the geometry of the moduli space and by studying the propagator.
+In general, we shall work with H and indicate when necessary the restriction to H− (keeping
+the condition ker L−
+0 implicit to avoid new notations).
+176
+
+The Hilbert space H can be separated according to the ghost zero-modes
+H ∼ H↓↓ ⊕ H↓↑ ⊕ H↑↓ ⊕ H↑↑,
+(11.40)
+with the following definitions:
+H↓↑ ∼ H0,
+H↓↑ ∼ ¯c0H↓↓
+H↑↓ ∼ c0H↓↓
+H↑↑ ∼ c0¯c0H↓↓.
+(11.41)
+Accordingly, every basis state can be split as
+φr = φ↓↓,r + φ↓↑,r + φ↑↓,r + φ↑↑,r
+(11.42)
+such that
+b0 |φ↓↓,r⟩ = ¯b0 |φ↓↓,r⟩ = 0,
+b0 |φ↓↑,r⟩ = ¯c0 |φ↓↑,r⟩ = 0,
+c0 |φ↑↓,r⟩ = ¯b0 |φ↑↓,r⟩ = 0,
+c0 |φ↑↑,r⟩ = ¯c0 |φ↑↑,r⟩ = 0.
+(11.43)
+Moreover, the basis can be indexed such that
+|φ↓↑,r⟩ = ¯c0 |φ↓↓,r⟩
+|φ↑↓,r⟩ = c0 |φ↓↓,r⟩
+|φ↑↑,r⟩ = c0¯c0 |φ↓↓,r⟩ .
+(11.44)
+A dual state φc
+r is also expanded:
+φc
+r = φc
+↓↓,r + φc
+↓↑,r + φc
+↑↓,r + φc
+↑↑,r
+(11.45)
+and the components satisfy
+⟨φc
+↓↓,r| c0 =⟨φc
+↓↓,r| ¯c0 = 0,
+⟨φc
+↓↑,r| c0 =⟨φc
+↓↑,r|¯b0 = 0,
+⟨φc
+↑↓,r| b0 =⟨φc
+↑↓,r| ¯c0 = 0,
+⟨φc
+↑↑,r| b0 =⟨φc
+↑↑,r|¯b0 = 0.
+(11.46)
+The indexing of the basis is chosen such that:
+⟨φc
+↓↑,r| =⟨φc
+↓↓,r|¯b0
+⟨φc
+↑↓,r| =⟨φc
+↓↓,r| b0
+⟨φc
+↑↑,r| =⟨φc
+↓↓,r|¯b0b0.
+(11.47)
+such that
+⟨φc
+x,r|φy,s⟩ = δxyδrs,
+(11.48)
+where x, y =↓↓, ↑↓, ↓↑, ↑↑.
+Consider the Hilbert space H−, then a basis state must satisfy
+b−
+0 |φr⟩ = 0
+=⇒
+b0 |φr⟩ = ¯b0 |φr⟩ .
+(11.49)
+The expansion (11.42) gives the relation
+φ↑↓,r + φ↑↑,r = φ↓↓,r + φ↓↑,r
+(11.50)
+such that
+φr = 2(φ↓↓,r + φ↓↑,r).
+(11.51)
+For convenience, the factor of 2 can be omitted (which amounts to rescaling the basis states).
+11.3
+Off-shell amplitudes
+In this section, we provide a guideline of what we need to look for in order to write an
+off-shell amplitude. The geometrical tools will be described in the next chapter, and the
+construction of off-shell amplitudes in the following one.
+177
+
+11.3.1
+Amplitudes from the marked moduli space
+In Chapter 3, the scattering amplitudes were written as an integral over the moduli space
+Mg of the Riemann surface Σg. As a consequence, the moduli of Mg and the positions
+of the vertex operators are not treated on an equal footing. Moreover, the insertions of
+operators is not symmetric since some are integrated, and others have factors of c. These
+problems can be solved by reinterpreting the scattering amplitudes in a more geometrical
+way.
+The key is to consider the punctures where vertex operators are inserted as part of the
+geometry and not as external data added on top of the Riemann surface Σg.
+Then, the worldsheet with the external states is described as a punctured (or marked)
+Riemann surface Σg,n, which is a Riemann surface Σg with n punctures (marked points) zi.
+The Euler number of such a surface was given in (3.4):
+χg,n := χ(Σg,n) = 2 − 2g − n.
+(11.52)
+This makes sense since punctures can be interpreted as disks (boundaries). Note that the
+punctures are labeled and are thus distinguishable.
+Since the marked points are distinguished, marked Riemann surfaces with identical g
+and n but with punctures located at different points are seen as different (this statement
+requires some care for g = 0 and g = 1 due to the presence of CKV). The corresponding
+moduli space is denoted by Mg,n, and it can be viewed as a fibre bundle with Mg as the
+base and the puncture positions as the fibre. The dimension of Mg,n is
+Mg,n := dimR Mg,n = 6g − 6 + 2n,
+for
+�
+�
+�
+�
+�
+g ≥ 2,
+g = 1, n ≥ 1,
+g = 0, n ≥ 3.
+(11.53)
+These cases are equivalent to χg,n < 0, that is, when the surfaces have a negative curvature.
+The corresponding coordinates are denoted by tλ, λ = 1, . . . , Mg,n. Comparing with (2.51)
+and (2.93), this corresponds to the situation where Σg,n has no CKV left unfixed.
+Example 11.2 – 4-punctured sphere Σ0,4
+The positions of the punctures are denoted by zi with i = 1, . . . , 4. Since there are three
+CKV, the positions of three punctures (say z1, z2 and z3) can be fixed, leaving only
+one position which characterizes Σ0,4. Hence, the moduli space has dimension M0,4 = 2
+and M0,4 is parametrized by {z4}.
+Example 11.3 – 2-punctured torus Σ1,2
+The positions of the punctures are denoted by zi with i = 1, 2. One puncture can be
+fixed using the single CKV of the surface, which leaves one position. Together with the
+moduli parameter τ of the torus, this gives M1,2 = 4 and the coordinates of M1,2 are
+{z2, τ}.
+The g-loop n-point scattering amplitude with external states {Vi} can be written as an
+integral over Mg,n of some Mg,n-form ω(g,n)
+Mg,n :
+Ag,n(V1, . . . , Vn) =
+�
+Mg,n
+ωg,n
+Mg,n(V1, . . . , Vn).
+(11.54)
+The integration over Mg,n has the correct dimension to reproduce the formulas from Sec-
+tion 3.1.
+178
+
+While it is possible to derive this amplitude from the path integral (see the comments
+at the end of Section 3.1.2), we will make only use of the properties of CFT on Riemann
+surfaces in the next chapter. This provides an alternative point of view on the computation
+of scattering amplitudes and how to derive the formulas, which can be helpful when the
+manipulation of the path integral is more complicated (for example, with the superstring).
+The expression of the form ωg,n
+Mg,n must 1) provide a measure on the moduli space and 2)
+extract a function of the moduli from the states Vi. It is natural to achieve the second point
+by computing a correlation function on the Riemann surfaces Σg,n. Moreover, Chapters 2
+and 3 indicate that the ghosts are part of the definition of the measure. Hence, one can
+expect the ωg,n
+Mg,n to have the form:
+ωg,n
+Mg,n(V1, . . . , Vn) =
+�
+ghosts ×
+n
+�
+i=1
+Vi
+�
+Σg,n
+×
+Mg,n
+�
+λ=1
+dtλ.
+(11.55)
+We will motivate an expression in Chapter 14 before checking that it has the correct prop-
+erties. The ghost insertions are necessary to saturate the number of zero-modes to obtain
+a non-vanishing result. By convention, the ghosts are inserted on the left: while this does
+not make difference for on-shell closed states, this will for off-shell states and for the other
+types of strings (open and supersymmetric) since the operators can be Grassmann odd.
+11.3.2
+Local coordinates
+The next step is to consider off-shell states Vi ∈ H. As motivated previously, one needs to
+introduce local coordinates defined by the maps:
+z = fi(wi),
+zi = fi(0).
+(11.56)
+There is one local coordinate for each operator, which is inserted at the origin.
+Local
+coordinates on the surfaces can be seen in two different fashions (Figure 11.1).
+Either
+as describing patches on the surface, in which case the maps fi correspond to transition
+functions. Or, one can interpret them by cutting disks centred at the punctures and whose
+interiors are mapped to complex planes, and the maps fi tell how to insert the plane inside
+the disk.
+When the amplitude Ag,n is defined in terms of local coordinates, it will depend on
+the maps fi and one needs to ensure that this cancels when the Ai are on-shell. But, the
+choice of the maps fi is arbitrary: selecting a specific set hides that all choices are physically
+equivalent and should lead to the same results on-shell. For this reason, the geometry can
+be enriched with the local coordinates, in the same way that the puncture locations were
+added as a fibre to the moduli space Mg to get the marked moduli space Mg,n. Hence,
+the fundamental geometrical object is the fibre bundle Pg,n with Mg,n being the base and
+the local coordinates the fibre. Since there is an infinite number of functions, the fibre is
+infinite-dimensional, and so is the space Pg,n.
+Every point of Pg,n corresponds to a genus-g Riemann surface with n punctures together
+with a choice of local coordinates around the punctures. The form ωg,n
+Mg,n is defined in this
+bigger space and the integration giving the off-shell amplitude (11.54) is performed over a
+Mg,n-dimensional section Sg,n ⊂ Pg,n (Figure 11.2):
+Ag,n(V1, . . . , Vn)Sg,n =
+�
+Sg,n
+ωg,n
+Mg,n(V1, . . . , Vn)
+��
+Sg,n.
+(11.57)
+The subscript in the LHS indicates that the amplitudes depend on Sg,n through the choice
+of local coordinates. The on-shell independence of Ag,n on the local coordinates translate
+179
+
+(a) Original surface with three punctures.
+(b) Disks delimiting the local coordinate patches.
+(c) Complex plane mapped to disks centred around the previous
+puncture location.
+Figure 11.1: Usage of local coordinates for a Riemann surface.
+180
+
+into the independence on the choice of the section:
+∀Sg,n :
+Ag,n(V1, . . . , Vn)Sg,n = Ag,n(V1, . . . , Vn)
+(on-shell).
+(11.58)
+The section is taken to be continuous, which means that two neighbouring surfaces of the
+moduli space must have close local coordinates.
+Figure 11.2: Section Sg,n of the fibre bundle Pg,n, the latter having Mg,n as a base the local
+coordinates as a fibre.
+In order to define the amplitude, one needs to find the expression of the Mg,n-form ωg,n
+Mg,n
+on Pg,n. It is in fact simpler to define general p-forms ωg,n
+p
+on Pg,n, in particular, for proving
+general properties about the forms and the amplitudes. Given a manifold, a p-form is an
+element of the cotangent space and it can be defined through its contraction with vectors
+(tangent space). Since vectors correspond to small variation of the manifold coordinates, it
+is necessary to find a parametrization of Pg,n. The geometry of Pg,n – and of its relevant
+subspaces and tangent space – is studied in the next chapter. Then, we will come back on
+the construction of the amplitudes in Chapter 13.
+11.4
+Suggested readings
+• General references on off-shell string theory include [262, sec. 7, 215, sec. 2, 42, 193].
+• Interpretation of local coordinates [193, sec. 5.2].
+181
+
+Chapter 12
+Geometry of moduli spaces and
+Riemann surfaces
+Abstract
+In this chapter, we describe how to parametrize the moduli space Mg,n and
+the local coordinates which together form the fibre bundle Pg,n introduced in the previous
+chapter. Then, we can characterize the tangent space which we will need in the next chapter
+to write the p-forms on Pg,n necessary to write the amplitudes. Finally, we introduce the
+notion of plumbing fixture, an operation which glue together punctures located on the same
+or different surfaces.
+12.1
+Parametrization of Pg,n
+The first step is to find a parametrization of the Riemann surfaces.
+As we have seen
+(Chapter 11), the dependence of the surface on the punctures can be described by local
+coordinates, that is, transition functions. The patch is defined by cutting a disk around
+each puncture and the transition functions are defined on the circle given by the intersection
+of the disk with the rest of the surface. The number of disks is simply:
+#disks = n.
+(12.1)
+It makes sense to look for a similar description of the other moduli (associated to the
+genus) by introducing additional coordinate patches. One can imagine that all the depend-
+ence of the moduli and punctures will reside in the transition functions between patches
+if the different patches are isomorphic to a surface without any moduli: the 3-punctured
+sphere Σ0,3. Hence, one can look for a decomposition of the surface by cutting disks such
+that one is left with 3-punctured spheres only, and transition functions are defined on the
+circles at the intersections of the spheres.
+Next, we need to find the number of spheres with 3 holes (or punctures). We start first
+with Σ0,n: in this case, it is straightforward to find that there will be n − 2 spheres. Indeed,
+for each additional puncture beyond n = 3, an additional sphere is created by cutting a
+circle. For g ≥ 1, natural places to split the surface are handles: two circles can be cut for
+each of them. By inspection, one finds that it leads to 2 spheres for each handle (one on the
+right and one on the left).1 This shows that the number of spheres is:
+#spheres = 2g − 2 + n.
+(12.2)
+1The simplest way to find this result is to consider Σg,2 and to write one puncture at each side of the
+surface (as in Figure 12.2). To generalize further, one can consider a generic n and put all punctures but
+one on one side of the surfaces.
+182
+
+The number of circles corresponds to the number of boundaries divided by two since the
+boundaries are glued pairwise: each disk has one boundary and each sphere has 3, which
+leads to:
+#circles = n + 3(2g − 2 + n)
+2
+= 3g − 3 + 2n.
+(12.3)
+The idea of the construction is to split the surface into elementary objects (spheres and
+disks) such that the full surface is seen as the union of all of them (gluing along circles),
+and no information is left in the individual geometries. This parametrization is particularly
+useful because there are simple coordinate systems on spheres and disks and these surfaces
+are easy to visualize and to work with. For example, they can be easily mapped to the
+complex plane.
+To conclude, a genus-g Riemann surface Σg,n with n punctures can be seen as the
+collection of:
+• 2g − 2 + n three-punctured spheres {Sa} with coordinates za
+• n disks {Di} with coordinates wi around each puncture
+• 3g − 3 + 2n circles {Cα} at the intersections of the spheres and disks
+Examples for Σ0,4 and Σ2,2 are given in Figures 12.1 and 12.2.
+Figure 12.1: Parametrization of Σ0,4.
+Figure 12.2: Parametrization of Σ2,2.
+183
+
+There are two types of circles: respectively, the ones at the overlap between two spheres,
+and between a disk and a 3-sphere:
+CΛ(ab) := Sa ∩ Sb,
+Ci(a) := Sa ∩ Di,
+{Cα} = {CΛ, Ci},
+(12.4)
+where Λ counts in fact the number of moduli:
+Λ = 1, . . . , Mc
+g,n,
+Mc
+g,n = 3g − 3 + n.
+(12.5)
+On the overlap circles, the coordinate systems are related by transitions functions:
+on CΛ(ab):
+za = Fab(zb),
+on Ci(a):
+za = fai(wi).
+(12.6)
+Then, the set of functions {Fab, fai} completely specifies the Riemann surface Σg,n together
+with the choice of the local coordinate systems around the punctures. The transition func-
+tions can thus be used to parametrize the moduli space Mg,n and the fibre bundle Pg,n, but
+it is highly redundant because many different functions lead to the same Riemann surface.
+A unique characterization of the different spaces is obtained by making identifications up to
+symmetries.
+In the previous chapter, we have seen that the metric in the local coordinate system is
+flat, ds2 = |dw|2. This means that two systems differing by a global phase rotation
+wi −→ ˜wi = eiαiwi
+(12.7)
+lead to surfaces with local coordinates which cannot be distinguished. Correspondingly, the
+two maps fi and ˜fi which relate the local coordinates wi and ˜wi to the coordinate z
+z = fi(wi),
+z = ˜fi( ˜wi)
+(12.8)
+are related as:
+fi(wi) = ˜fi(eiαiwi).
+(12.9)
+Hence, this motivates to consider the smaller space
+ˆPg,n = Pg,n/U(1)n,
+(12.10)
+where the action of each U(1) is defined by the equivalence (12.9). The necessity to consider
+this subspace will be strengthen further later, and will correspond to the level-matching
+condition.
+Below, global phase rotations are also interpreted in terms of the plumbing
+fixture, see (12.57).
+The different spaces which we need are parametrized by the transition functions up to
+the following identifications:
+• Pg,n = {Fab, fai} modulo reparametrizations of za
+• ˆPg,n = {Fab, fai} modulo reparametrizations of za and phase rotations of wi
+• Mg,n = {Fab, fai} modulo reparametrizations of za and of wi keeping the points wi = 0
+fixed
+• Mg = {Fab, fai} modulo reparametrizations of za and wi
+At each step, the dimension of the space is reduced because one divides by bigger and bigger
+groups. The highest reduction occurs when dividing by the reparametrizations of wi which
+form an infinite-dimensional group (phase rotations form a finite-dimensional subgroup of
+them).
+184
+
+For concreteness, it is useful to introduce explicit coordinates xs on Pg,n (s ∈ N since the
+space is infinite-dimensional). The transition functions on the Riemann surface depend on
+the xs which explains why they can be used to parametrize the moduli spaces. Describing
+the spheres Sa by complex planes with punctures located at za,1, za,2 and za,3, the transition
+functions on the Mg,n circles CΛ(ab) = Sa ∩ Sb for all a < b can be taken to be:
+on CΛ(ab):
+za − za,m =
+qΛ
+zb − zb,n
+,
+(12.11)
+where za,m and zb,n denote the punctures of Sa and Sb lying in CΛ. Then, the complex
+parameters qΛ with Λ = 1, . . . , Mc
+g,n are coordinates on the moduli space Mg,n. On the
+remaining n circles Ci(a) = Sa ∩ Di, the transition functions can be expanded in series
+on Ci(a):
+za − za,m = wi +
+∞
+�
+N=1
+pi,NwN
+i ,
+(12.12)
+where za,m is the puncture of Sa lying in Ci. There is no negative index in the series because
+the RHS must vanish for wi = 0 which maps to za = za,m (puncture location). The complex
+coefficients of the series pi,N (i = 1, . . . , n and N ∈ N∗) provide coordinates for the fibre.
+Thus, coordinates for Pg,n are:
+{xs} = {qΛ, pi,N}.
+(12.13)
+As usual, derivatives with respect to xs are abbreviated by ∂s.
+When the dependence in the xs must be stressed, the transition functions (12.6) are
+denoted by:
+za = Fab(zb; xs),
+za = fai(wi; xs).
+(12.14)
+In this coordinate system, each parameter appears in only one transition function and it looks
+like one can separate the fibre from the basis. But, this is not an invariant statement as this
+would not hold in other coordinate systems. For example, one can rescale the coordinates
+to lump all dependence on qΛ in a single circle.
+Since both cases are formally identical, it is convenient to fix the orientation of each Cα
+and to denote by σα (resp. τα) the coordinate on the left (resp. right) of the contour, such
+that the transition functions reads:
+on Cα:
+σα = Fα(τα; xs).
+(12.15)
+Now that we have coordinates on Pg,n, it is possible to construct tangent vectors.
+12.2
+Tangent space
+A tangent vector Vs ∈ TPg,n corresponds to an infinitesimal variation of the coordinates on
+the manifold
+δxs = ϵ Vs,
+(12.16)
+where ϵ is a small parameter, such that functions of xs vary as:
+ϵ Vs ∂sf = f(xs + ϵ Vs) − f(xs).
+(12.17)
+The transition functions Fα provide an equivalent (but redundant) set of coordinates for
+Pg,n. Hence, vectors in TPg,n can also be obtained by considering small variations of the
+transition functions Fα:
+Fα −→ Fα + ϵ δFα.
+(12.18)
+185
+
+Considering an overlap circle Cα, a deformation of the transition function
+σα = Fα(τα)
+(12.19)
+for fixed τα can be interpreted as a change of the coordinate σα:
+σ′
+α = Fα(τα) + ϵ δFα(τα) = σα + ϵ δFα(τα) = σα + ϵ δFα
+�
+F −1
+α (σα)
+�
+.
+(12.20)
+This transformation is generated by a vector field v(α) on the Riemann surface Σg,n:
+σ′
+α = σα + ϵv(α)(σα),
+v(α) = δFα ◦ F −1
+α .
+(12.21)
+The situation is symmetrical and one can obviously fix σα and vary τα. The vector field is
+regular around the circle Cα (to have a well-defined changes of coordinates) but it can have
+singularities away from the circle Cα. Hence, the vector field v(α) together with the circle
+Cα define a vector of Pg,n:
+V (α) ∼
+�
+v(α), C(α)
+�
+.
+(12.22)
+This provides a basis of TPg,n.
+This is sufficient when using the coordinate system
+(12.13), but, in more general situations, one needs to consider linear combinations. For
+example, if a modulus appears in several transitions functions, then the associated vector
+field will be defined on the corresponding circles. A general vector V is described by a vector
+field v with support on a subset C of the circles Cα:
+V ∼
+�
+v, C
+�
+,
+C ⊆
+�
+α
+Cα,
+(12.23)
+and the restriction of v on the various circles is written as:
+v|Cα = v(α).
+(12.24)
+Note that the vector field v(α) and its complex conjugate ¯v(α) are independent and are
+associated to different tangent vectors. This construction is called the Schiffer variation.
+The simplest tangent vectors ∂s are given by varying one coordinate of Pg,n while keeping
+the other fixed:
+xs −→ xs + ϵ δxs.
+(12.25)
+On each circle Cα, this gives a deformation of the transition functions
+Cα :
+Fα −→ Fα + ϵ δFα,
+δFα = ∂Fα
+∂xs
+δxs
+(12.26)
+(no sum over s), such that the change of coordinates reads
+σ′
+α = σα + ϵ v(α)
+s
+(σα) δxs,
+v(α)
+s
+(σα) = ∂Fα
+∂xs
+�
+F −1
+α (σα)
+�
+.
+(12.27)
+If the xs are given by (12.13), the vectors have support in only one circle.
+There is, however, a redundancy in these vectors. Not all of them leads to a motion in
+Pg,n because some modifications can be absorbed with a reparametrization of the za. For
+example, if a given v(α) can be extended holomorphically outside the circle Cα in the neigh-
+bour sphere, then its effect can be undone by reparametrizing the corresponding coordinate.
+A similar discussion holds for the other spaces and relations can be found by restricting the
+vector on subspaces. A non-trivial vector (v(i), Ci) of Pg,n becomes trivial on Mg,n if it can
+be cancelled with a reparametrization of wi which leaves the origin fixed.
+186
+
+12.3
+Plumbing fixture
+The plumbing fixture is a way to glue together two Riemann surfaces (separating case) or
+two parts of the same surface (non-separating case) together, in order to build a surface with
+a higher number of holes and punctures. This geometric operation will correspond precisely
+to the concept of gluing two Feynman graphs with a propagator in Siegel gauge.
+The plumbing fixture depends on a (complex) one-parameter, which leads to a family of
+surfaces. This provides the correct number of moduli for the surface obtained after gluing.
+This brings to the question of describing the moduli spaces Mg,n in terms of the moduli
+spaces with lower genus and number of punctures.
+12.3.1
+Separating case
+Consider two Riemann surfaces Σg1,n1 and Σg2,n2 with local coordinates w(1)
+1 , . . . , w(1)
+n1 and
+w(2)
+1 , . . . , w(2)
+n2 .
+The first step is to cut two disks D(1)
+q
+and D(2)
+q
+of radius |q|1/2 around a puncture on
+each surface, taken to be the n1-th and n2-th for definiteness:
+D(1)
+q
+=
+�
+|w(1)
+n1 | ≤ |q|1/2�
+,
+D(2)
+q
+=
+�
+|w(2)
+n2 | ≤ |q|1/2�
+,
+(12.28)
+where q ∈ C is fixed (Figure 12.4).2 Then, both surfaces can be glued (indicated by the
+binary operation #) together into a new surface
+Σg,n = Σg1,n1#Σg2,n2,
+�
+g = g1 + g2,
+n = n1 + n2 − 2
+(12.29)
+by removing the disks D(1)
+q
+and D(2)
+q
+and by identifying the circles ∂D(1)
+q
+and ∂D(2)
+q . At the
+level of the coordinates, this is achieved by the plumbing fixture operation:
+w(1)
+n1 w(2)
+n2 = q,
+|q| ≤ 1,
+(12.30)
+The restriction on q arises because we have |w(1)
+n1 |, |w(2)
+n2 | ≤ 1 (for a discussion, see [71]). This
+case is called separating because cutting the new tube splits the surface in two components.
+Locally, the new surface looks like Figure 12.3. It is also convenient to parametrize q as
+q = e−s+iθ,
+s ∈ R+,
+θ ∈ [0, 2π).
+(12.31)
+The parameters s and θ are interpreted below as moduli of the Riemann surface.
+Figure 12.3: Integration contour on the circle between the two local coordinates which are
+glued together.
+The geometry of the new surface can be viewed in three different ways:
+2The disks D(i)
+q
+should be equal or smaller than the disks D(1)
+n1 and D(2)
+n2 .
+187
+
+Figure 12.4: Disks around one puncture of the surfaces Σ1,1 and Σ0,3. The disks appear as
+a cap because it is on top of the surface, which is curved.
+1. both surfaces Σg1,n1 and Σg2,n2 (with the disks removed) are connected directly at
+their boundaries (Figure 12.5a);
+2. both surfaces Σg1,n1 and Σg2,n2 (with the disks removed) are connected by a cylinder
+of finite size (Figure 12.5b);
+3. the surface Σg2,n2 is inserted inside the disk D(1)
+q , or conversely Σg1,n1 inside D(2)
+q
+(Figures 12.5c and 12.5d).
+The first interpretation is the most direct one: the disks are simply removed and the
+boundaries are glued together by a small tube of radius |w(1)
+1 | < |q|1/2. The connection
+between both surfaces can be smoothed (and figures are often drawn in this way – for
+example Figure 12.6), but this is not necessary (the smoothing is achieved by cutting disks
+of size |q|1/2 − ϵ and gluing the boundaries to the disks of radius |q|1/2).
+In the second interpretation, one rescales the local coordinates in order to bring the
+radius of the disk to 1 instead of |q|1/2. In terms of these new coordinates, the surfaces are
+connected by a tube of length s = − ln |q| after a Weyl transformation (see [193, sec. 9.3]
+for a longer discussion).
+The last interpretation is obtained by performing a conformal mapping of the second
+case: the region |w(1)
+n1 | < |q|1/2 is mapped to the region
+|w(2)
+n2 | =
+|q|
+|w(1)
+n1 |
+> |q|1/2,
+(12.32)
+and conversely. The idea is that the disk D(1)
+q
+of Σg1,n1 is removed and replaced by the
+complement of D(2)
+q
+in Σg2,n2, i.e. the full surface Σg2,n2 −D(2)
+q
+is glued inside D(1)
+q . While it
+is clear geometrically, this statement may look confusing from the coordinate point of view
+because the local coordinates w(1)
+n1 and w(2)
+n2 do not cover completely the Riemann surfaces,
+but their relation still encodes information about the complete surface. The reason is that
+one can always use transition functions to relate the coordinates on the two surfaces.
+Example 12.1
+Denote by S(1)
+a
+and S(2)
+b
+the spheres sharing a boundary with D(1)
+n1 and D(2)
+n2 , and write
+the corresponding coordinates by z(1)
+a
+and z(2)
+b
+such that the transition functions are
+z(1)
+a
+= f (1)
+an1(w(1)
+n1 ),
+z(2)
+b
+= f (2)
+bn2(w(2)
+n2 ).
+(12.33)
+188
+
+(a) Direct gluing of circles.
+(b) Connection by a long tube.
+(c) Insertion of the second surface into the first one.
+(d) Insertion of the first surface
+into the second one.
+Figure 12.5: Different representations of the surface Σ1,2 obtained after gluing Σ1,1 and Σ0,3
+through the plumbing fixture.
+Figure 12.6: Smoothed connection between both surfaces.
+189
+
+Then the coordinates za and zb are related by
+z(1)
+a
+= f (1)
+an1
+�
+w(1)
+n1
+�
+= f (1)
+an1
+�
+q
+w(2)
+n2
+�
+= f (1)
+an1
+�
+q
+f (2)−1
+bn2
+�
+z(2)
+b
+�
+�
+(12.34)
+such that the new transition function reads
+za = Fab(zb),
+Fab = f (1)
+an1 ◦ (q · I) ◦ f (2)−1
+bn2
+,
+(12.35)
+where I is the inversion (the superscript on the coordinates za and zb has been removed
+to indicate that they are now seen as coordinates on the same surface Σg,n).
+The Riemann surface Σg,n is a point of Mg,n. By varying the moduli parameters of
+Σg1,n1 and Σg2,n2, one obtains other surfaces in Mg,n.
+But the number of parameters
+furnished by Σg1,n1 and Σg2,n2 does not match the dimension (11.53) of Mg,n:
+Mg1,n1 + Mg2,n2 = 6g1 − 6 + 2n1 + 6g2 − 6 + 2n2 = Mg,n − 2.
+(12.36)
+This means that the subspace of Mg,n obtained by gluing all the possible surfaces in Mg1,n1
+and Mg2,n2 is of codimension 2.
+The missing complex parameter is q: in writing the
+plumbing fixture, it was taken to be fixed, but it can be varied to generate a 2-parameter
+family of Riemann surfaces in Mg,n, with the moduli of the original surfaces held fixed.
+The surface Σg,n is equipped with local coordinates inherited from the original surfaces
+Σg1,n1 and Σg2,n2. Hence, the plumbing fixture of points in Pg1,n1 and Pg2,n2 automatically
+leads to a point of Pg,n. The fact that the local coordinates are inherited from lower-order
+surfaces is called gluing compatibility. It is also not necessary to add parameters to describe
+the fibre direction.
+12.3.2
+Non-separating case
+In the previous section, the plumbing fixture was used to glue punctures on two different
+surfaces. In fact, one can also glue two punctures on the same surface to get a new surface
+with an additional handle:
+Σg,n = #Σg1,n1,
+�
+g = g1 + 1,
+n = n1 − 2,
+(12.37)
+defining # as a unary operator. This gluing is called non-separating because there is a single
+surface before the identification of the disks.
+In terms of the local coordinates, the gluing relation reads
+w(1)
+n1−1w(1)
+n1 = q,
+(12.38)
+where we consider the last two punctures for definiteness.
+The dimensions of both moduli spaces are related by
+Mg1,n1 = Mg,n − 2.
+(12.39)
+Again, the two missing parameters are provided by varying q and we obtain a Mg,n-
+dimensional subspace of Mg,n.
+Example 12.2
+Here are some examples of surfaces obtained by gluing:
+190
+
+• Σ0,4 = Σ0,3#Σ0,3
+• Σ0,5 = Σ0,3#Σ0,3#Σ0,3, Σ0,3#Σ0,4
+• Σ1,1 = #Σ0,3
+• Σ1,2 = #Σ0,4, Σ1,1#Σ0,3
+Note that the moduli on the LHS and RHS are fixed (we will see later that not all
+surfaces can be obtained by gluing).
+12.3.3
+Decomposition of moduli spaces and degeneration limit
+We have seen that the separating and non-separating plumbing fixtures yield a family of
+surfaces in Mg,n described in terms of lower-dimensional moduli spaces. The question is
+whether all points in Mg,n can be obtained in this way by looking at all the possible gluing
+(varying g1, n1, g2 and n2). It turns out that this is not possible, which is at the core of the
+difficulties to construct a string field theory.
+Which surfaces are obtained from this construction? In order to interpret the regions
+of Mg,n covered by the plumbing fixture, the parametrization (12.31) is the most useful.
+Previously, we explained that s gives the size of the tube connecting the two surfaces. Since
+the latter is like a sphere with two punctures, it corresponds to a cylinder (interpreted as an
+intermediate closed string propagating). The angle θ in (12.31) is the twist of the cylinder
+connecting both components. This amounts to start with θ = 0, then to cut the cylinder,
+to twist it by an angle θ and to glue again.
+The limit s → ∞ (|q| → 0) is called the degeneration limit: the degenerate surface
+Σg,n reduces to Σg1,n1 and Σg2,n2 connected by a very long tube attached to two punctures
+(separating case), or to Σg−1,n+2 with a very long handle (non-separating case). So it means
+that the family of surfaces described by the plumbing fixture are “close” to degeneration.
+Another characterization (for the separating case) is that the punctures on Σg1,n1 are closer
+(according to some distance, possibly after a conformal transformation) to each other than
+to the punctures on Σg2,n2.
+Conversely, there are surfaces which cannot be described in this way: the plumbing
+fixture does not cover all the possible values of the moduli. For a given Mg,n, we denote
+the surfaces which cannot be obtained by the plumbing fixture by Vg,n. This space does
+not contain any surface arbitrarily close to degeneration (i.e. with long handles or tubes).
+In terms of punctures, it also means that there is no conformal frame where the punctures
+split in two sets.
+In the previous subsection, we considered two specific punctures, but any other punctures
+could be chosen. Hence, there are many ways to split Σg,n in two surfaces Σg1,n1 and Σg2,n2
+(with fixed g1, g2, n1 and n2): every partition of the punctures and holes in two sets lead to
+different degeneration limits (because they are associated to different moduli – Figure 12.7).
+Since each puncture is described by a modulus, choosing different punctures for gluing give
+different set of moduli for Σg,n, such that each possibility covers a different subspace of
+Mg,n. The part of the moduli space Mg,n covered by the plumbing fixture of all surfaces
+Σg1,n1 and Σg2,n2 (with fixed g1, g2, n1, n2) is denoted by Mg1,n1#Mg2,n2:
+Mg1,n1#Mg2,n2 ⊂ Mg,n,
+(12.40)
+where the operation # includes the plumbing fixture for all values of q and all pairs of
+punctures. Similarly, the part covered by the non-separating plumbing fixture is written as
+#Mg1,n1:
+#Mg1,n1 ⊂ Mg,n.
+(12.41)
+Importantly, the regions covered by the plumbing fixture depend on the choice of the
+local coordinates because (12.30) is written in terms of local coordinates. The subspaces
+Mg1,n1#Mg2,n2 and #Mg1,n1 are not necessarily connected (in the topological sense).
+191
+
+(a) Degeneration 12 → 34
+(b) Degeneration 13 → 24
+(c) Degeneration 14 → 23
+Figure 12.7: Permutations of punctures while gluing two spheres: they correspond to differ-
+ent (disconnected) parts of M0,4.
+The moduli space Mg,n cannot be completely covered by the plumbing fixture of lower-
+dimensional surfaces. We define the propagator and fundamental vertex regions Fg,n and
+Vg,n as the subspaces which can and cannot be described by the plumbing fixture:
+Fg,n := #Mg−1,n+2
+�
+�
+�
+n1+n2=n+2
+g1+g2=g
+Mg1,n1#Mg2,n2
+�
+,
+(12.42a)
+Vg,n := Mg,n − Fg,n,
+(12.42b)
+In the RHS, it is not necessary to consider multiple non-separating plumbing fixtures for
+the first term because #Mg−2,n+4 ⊂ Mg−1,n+2, etc. For the same reason, it is sufficient to
+consider a single separating plumbing fixture. Note that Vg,n and Fg,n are in general not
+connected subspaces. A simple illustration is given in Figure 12.9. The actual decomposition
+of M0,4 is given in Figure 12.8. Importantly, Fg,n and Vg,n depend on the choice of the
+local coordinates for all Vg′,n′ appearing in the RHS.
+It is also useful to define the subspaces F1PR
+g,n
+and V1PI
+g,n of Mg,n which can and cannot
+be described with the separating plumbing fixture only:
+F1PR
+g,n :=
+�
+n1+n2=n+2
+g1+g2=g
+Mg1,n1#Mg2,n2,
+(12.43a)
+V1PI
+g,n := Mg,n − F1PR
+g,n .
+(12.43b)
+1PR (1PI) stands for 1-particle (ir)reducible, a terminology which will become clear later.
+Note the relation:
+V1PI
+g,n = Vg,n
+�
+� �
+g′
+#Mg−g′,n+g′
+�
+.
+(12.44)
+The two plumbing fixtures behave as follow:
+• separating: increases both n and g (if both surfaces have a non-vanishing g);
+192
+
+Figure 12.8: In white are the subspaces of the moduli space M0,4 covered by the plumb-
+ing fixture. The three different regions correspond to the three different ways to pair the
+punctures (see Figure 12.7). In grey is the fundamental vertex region V0,4.
+Figure 12.9: Schematic illustration of the covering of Mg,n from the plumbing fixture of
+lower-dimensional spaces. The fundamental region Vg,n (usually disconnected) is not covered
+by the plumbing fixture.
+193
+
+• non-separating plumbing: increases g but decreases n.
+The construction is obviously recursive: starting from the lowest-dimensional moduli space,
+which is M0,3 (no moduli), one has:
+V0,3 = M0,3,
+F0,3 = ∅.
+(12.45)
+Next, the subspace of M0,4 obtained from the plumbing fixture is:
+F0,4 = V0,3#V0,3,
+(12.46)
+and V0,4 is characterized as the remaining region. Then, one has:
+F0,5 = M0,4#M0,3
+= F0,4#V0,3 + V0,4#V0,3 = V0,3#V0,3#V0,3 + V0,4#V0,3,
+(12.47)
+and V0,5 is what remains of M0,5. The pattern continues for g = 0. The same story holds
+for g ≥ 1: the first such space is
+F1,1 = #V0,3,
+(12.48)
+and V1,1 = M1,1 − F1,1. The gluing of a 3-punctured sphere and the addition of a handle
+are the two most elementary operations.
+To keep track of which moduli spaces can contribute, it is useful to find a function of
+Σg,n, called the index, which increases by 1 for each of the two elementary operations:
+r(Σg1,n1#Σ0,3) = r(Σg1,n1) + 1,
+r(#Σg1,n1) = r(Σg1,n1) + 1.
+(12.49)
+An appropriate function is
+r(Σg,n) = 3g + n − 2 ∈ N∗.
+(12.50)
+which is normalized such that:
+r(Σ0,3) = 1.
+(12.51)
+For a generic separating plumbing fixture, we find:
+r(Σg1,n1#Σg2,n2) = r(Σg1,n1) + r(Σg2,n2).
+(12.52)
+Since the index increases, surfaces with a given r can be obtained by considering all the
+gluings of surfaces with r′ < r.
+12.3.4
+Stubs
+To conclude this chapter, we introduce the concept of stubs. Previously in (12.31), the range
+of the parameter s was the complete line of positive numbers, s ∈ R+. This means that
+tubes of all lengths were considered to glue surfaces. But, we could also introduce a minimal
+length s0 > 0, called the stub parameter, for the tube. In this case, the plumbing fixture
+parameter is generalized to:
+q = e−s+iθ,
+s ∈ [s0, ∞),
+θ ∈ [0, 2π),
+s0 ≥ 0.
+(12.53)
+What is the effect on the subspaces Fg,n(s0) and Vg,n(s0)? Obviously, less surfaces can be
+described by the plumbing fixture if s0 > 0 than if s0 = 0, since the plumbing fixture cannot
+describe anymore surfaces which contain a tube of length less than s0. Equivalently, the
+values of the moduli described by the plumbing fixture is more restricted when s0 > 0. More
+generally, one has:
+s0 < s′
+0 :
+Fg,n(s′
+0) ⊂ Fg,n(s0)
+Vg,n(s0) ⊂ Vg,n(s′
+0).
+(12.54)
+194
+
+Figure 12.10: In light grey is the subspace covered by the V0,4(s0) as in Figure 12.8. In dark
+grey is the difference δV0,4 = V0,4(s0 + δs0) − V0,4(s0) with δs0 > 0.
+This is illustrated on Figure 12.10. Even if s0 is very large, Vg,n still does not include surfaces
+arbitrarily close to degeneracy. In general, we omit the dependence in s0 except when it is
+necessary.
+To interpret the stub parameter, consider two local coordinates w1 and w2 and rescale
+them by λ ∈ C with Re λ > 0:
+w1 = λ ˜w1,
+w2 = λ ˜w2.
+(12.55)
+Then, the plumbing fixture (12.30) becomes
+˜w1 ˜w2 = e−˜s+i˜θ.
+(12.56)
+with
+˜s = s + 2 ln |λ|,
+˜θ = θ + i ln λ
+¯λ.
+(12.57)
+If s ∈ R+, the corresponding range of ˜s is
+˜s ∈ [s0, ∞),
+s0 := 2 ln |λ|.
+(12.58)
+This shows that rescaling the local coordinates by a constant parameter is equivalent to
+change the stub parameter.
+Note also how performing a global phase rotation in (12.57) is equivalent to shift the
+twist parameter. Working in ˆPg,n forces to take λ ∈ R+.
+12.4
+Summary
+In this chapter, we have explained how to parametrize the fibre bundle Pg,n, that is, appro-
+priate coordinates for the moduli space and the local coordinate systems. This was realized
+195
+
+by introducing different coordinate patches and encoding all the informations of Pg,n in
+the transition functions. Then, this description lead to a simple description of the tangent
+vectors through the Schiffer variation.
+In the next chapter, we will continue the program by building the p-forms required to
+describe off-shell amplitudes.
+12.5
+Suggested readings
+• Plumbing fixture [193, sec. 9.3].
+196
+
+Chapter 13
+Off-shell amplitudes
+Abstract
+While the previous chapter was purely geometrical, this one makes contact
+with string theory through the worldsheet CFT. We continue the description of Pg,n by
+constructing p-forms.
+The reason why we need to consider the CFT is that ghosts are
+necessary to build the p-forms: this can be understood from Chapter 2, where we found
+that the ghosts must be interpreted as part of the measure on the moduli space. Then, we
+build the off-shell amplitudes and discuss some properties.
+13.1
+Cotangent spaces and amplitudes
+In this section, we construct the p-forms on Pg,n which are needed for the amplitudes. We
+first motivate the expressions from general ideas, and check later that they have the correct
+properties.
+13.1.1
+Construction of forms
+A p-form ω(g,n)
+p
+∈ �p T ∗Pg,n is a multilinear antisymmetric map from �p TPg,n to a function
+of the moduli parameters. The superscript on the form is omitted when there is no ambiguity
+about the space considered. The components ωi1···ip of the p-form are defined by inserting
+p basis vectors ∂s1, . . . , ∂sp
+ωi1···ip := ωp(∂s1, . . . , ∂sp),
+(13.1)
+where ∂s =
+∂
+∂xs and xs are the coordinates (12.13). It is antisymmetric in any pair of two
+indices
+ωi1i2···ip = −ωi2i1···ip,
+(13.2)
+and multilinearity implies that
+ωp
+�
+V (1), . . . , V (p)�
+= ωp
+�
+V (1)
+s1 ∂s1, . . . , V (p)
+sp ∂sp
+�
+= ωi1···ipV (1)
+s1 · · · V (p)
+sp ,
+(13.3)
+given vectors V (α) = V (α)
+s
+∂s.
+The p-forms which are needed to define off-shell amplitudes depend on the external states
+Vi (i = 1, . . . , n) inserted at the punctures zi. They are maps from �p TPg,n × Hn to a
+function on Pg,n. The dependence on the states is denoted equivalently as
+ωp(V1, . . . , Vn) := ωp(⊗iVi).
+(13.4)
+The simplest way to get a function on Pg,n from the states Vi is to compute a CFT correlation
+function of the operators inserted at the points zi = fi(0) on the surface Σg,n described by
+the point in Mg,n.
+197
+
+The 0-form is just a function and is defined by:
+ω0 = (2πi)−Mc
+g,n
+� n
+�
+i=1
+fi ◦ Vi(0)
+�
+Σg,n
+.
+(13.5)
+For simplicity, the dependence in the local coordinates fi is kept implicit in the rest of the
+chapter.
+A natural approach for constructing p-forms is to build them from elementary 1-forms
+and to use ghosts to enforce the antisymmetry.
+Remembering the Beltrami differentials
+found in Chapter 2, the contour integral of ghosts b(z) weighted by some vector field is a
+good starting point. In the current language, it is defined by its contraction with a vector
+V = (v, C) ∈ TPg,n defined in (12.23):
+B(V ) :=
+�
+C
+dz
+2πi b(z)v(z) +
+�
+C
+d¯z
+2πi
+¯b(¯z)¯v(¯z),
+(13.6)
+where b(z) and ¯b(¯z) are the b-ghost components, and v is the vector field on Σg,n defining
+V . The contours run anti-clockwise. If the contour C includes several circles (C = ∪αCα),
+B(V ) is defined as the sum of the contour integral on each circle:
+B(V ) :=
+�
+α
+�
+Cα
+dz
+2πi b(z)v(z) + c.c.
+(13.7)
+It is also useful to define another object built from the energy–momentum tensor:
+T(V ) :=
+�
+C
+dz
+2πi T(z)v(z) +
+�
+C
+d¯z
+2πi
+¯T(¯z)¯v(¯z),
+(13.8)
+where T and ¯T are the components of the energy–momentum tensor. It is defined such that
+T(V ) = {QB, B(V )}.
+(13.9)
+Considering the coordinate system (12.13), the Beltrami form can be decomposed as:
+B = Bsdxs,
+Bs := B(∂s),
+(13.10a)
+Bs =
+�
+α
+�
+Cα
+dσα
+2πi b(σα) ∂Fα
+∂xs
+�
+F −1
+α (σα)
+�
++
+�
+α
+�
+Cα
+d¯σα
+2πi
+¯b(¯σα) ∂ ¯Fα
+∂xs
+� ¯F −1
+α (¯σα)
+�
+,
+(13.10b)
+where the contour orientations are defined by having the σα coordinate system on the left.
+We define the p-form contracted with a set of vectors V (1), . . . , V (p) by
+ωp
+�
+V (1), . . . , V (p)�
+(V1, . . . , Vn) := (2πi)−Mc
+g,n
+�
+B(V (1)) · · · B(V (p))
+n
+�
+i=1
+Vi
+�
+Σg,n
+,
+(13.11)
+and the corresponding p-form reads
+ωp = ωp,s1···sp dxs1 ∧ · · · ∧ dxsp
+(13.12a)
+= (2πi)−Mc
+g,n
+�
+Bs1dxs1 ∧ · · · ∧ Bspdxsp
+n
+�
+i=1
+Vi
+�
+Σg,n
+.
+(13.12b)
+In this expression, the form contains an infinite numbers of components ωp,s1···sp since there
+is an infinite number of coordinates. Note that the normalization is independent of p.
+198
+
+In practice, one is not interested in Pg,n, but rather in a subspace of it. Given a q-
+dimensional subspace S of Pg,n parametrized by q real coordinates t1, . . . , tq
+xs = xs(t1, . . . , tq),
+(13.13)
+the restriction of a p-form to this subspace is obtained by the chain rule:
+∀p ≤ q :
+ωp|S = (2πi)−Mc
+g,n
+�
+Br1
+∂xs1
+∂tr1
+dtr1 ∧ · · · ∧ Brp
+∂xsp
+∂trp
+dtrp
+n
+�
+i=1
+Vi
+�
+Σg,n
+,
+∀p > q :
+ωp|S = 0.
+(13.14)
+We will often write the expression directly in terms of the coordinates of S and abbreviate
+the notation as:
+Br := ∂xs
+∂tr
+Bs.
+(13.15)
+13.1.2
+Amplitudes and surface states
+It is now possible to write the amplitude more explicitly. An on-shell amplitude is defined
+as an integral over Mg,n. Off-shell, one needs to consider local coordinates around each
+puncture, that is, a point of the fibre for each point of the base Mg,n. This defines a Mg,n-
+dimensional section Sg,n of Pg,n (Figure 11.2). The g-loop n-point off-shell amplitude of the
+states V1, . . . , Vn reads:
+Ag,n(V1, . . . , Vn)Sg,n :=
+�
+Sg,n
+ωg,n
+Mg,n(V1, . . . , Vn)
+��
+Sg,n,
+(13.16a)
+ωg,n
+Mg,n(V1, . . . , Vn)
+��
+Sg,n = (2πi)−Mc
+g,n
+�Mg,n
+�
+λ=1
+Bs
+∂xs
+∂tλ
+dtλ
+n
+�
+i=1
+fi ◦ Vi(0)
+�
+Σg,n
+,
+(13.16b)
+where the choice of the fi is dictated by the section Sg,n. From now on, we stop to write the
+restriction of the form to the section. We also restrict to the cases where χg,n = 2−2g−n < 0.
+The complete (perturbative) n-point amplitude is the sum of contributions from all loops:
+An(V1, . . . , Vn) :=
+�
+g≥0
+Ag,n(V1, . . . , Vn).
+(13.17)
+More generally, we define the integral over a section Rg,n which projection on the base
+is a subspace of Mg,n (and not the full space as for the amplitude) as:
+Rg,n(V1, . . . , Vn) :=
+�
+Rg,n
+ωg,n
+Mg,n(V1, . . . , Vn),
+(13.18)
+For simplicity, we will sometimes use the same notation for the section of Pg,n and its
+projection on the base Mg,n. For this reason, the reader should assume that some choice of
+local coordinates around the punctures is made except otherwise stated.
+Given sections Rg,n, the sum over all genus contribution is written formally as
+Rn :=
+�
+g≥0
+Rg,n,
+(13.19)
+such that
+Rn(V1, . . . , Vn) :=
+�
+g≥0
+Rg,n(V1, . . . , Vn) =
+�
+g≥0
+�
+Rg,n
+ωg,n
+Mg,n(V1, . . . , Vn).
+(13.20)
+199
+
+A surface state is defined as a n-fold bra which reproduces the expression of a given
+function when contracted with n states Ai. The surface ⟨Σg,n|, form ⟨ωg,n|, section ⟨Rg,n|
+and amplitude ⟨Ag,n| n-fold states are defined by the following expressions:
+⟨Σg,n| Bs1 · · · Bsp | ⊗i Vi⟩ := ωs1···sp(V1, . . . , Vn),
+(13.21a)
+⟨ωg,n
+p
+| ⊗i Vi⟩ := ωp(V1, . . . , Vn),
+(13.21b)
+⟨Ag,n| ⊗i Vi⟩ := Ag,n(V1, . . . , Vn).
+(13.21c)
+The last relation is generalized to any section Rg,n:
+⟨Rg,n| ⊗i Vi⟩ := Rg,n(V1, . . . , Vn).
+(13.21d)
+The reason for introducing these objects is that the form (13.12) is a linear map from H⊗n
+to a form on Mg,n – see (13.4). Thus, there is always a state ⟨Σg,n| such that its BPZ
+product with the states reproduces the form. In particular, the state ⟨Σg,n| contains all the
+information about the local coordinates and the moduli (the dependence is kept implicit).
+The definition of the other states follow similarly. These states are defined as bras, but they
+can be mapped to kets.
+One finds the obvious relations:
+⟨ωg,n
+p
+| =⟨Σg,n| Bs1dxs1 · · · Bspdxsp,
+⟨Ag,n| =
+�
+Mg,n
+⟨ωg,n
+p
+| .
+(13.22)
+The surface states don’t contain information about the matter CFT: they collect the
+universal data (like local coordinates) needed to describe amplitudes. Hence, it is an im-
+portant step in the description of off-shell string theory to characterize this data. However,
+note that the relation between a surface state and the corresponding form does depend on
+the CFT.
+Example 13.1 – On-shell amplitude A0,4
+The transition functions are given by (see Figure 12.1):
+C1 : w1 = z1 − y1,
+C3 : w3 = z2 − y3,
+C5 : z1 = z2,
+C2 : w2 = z1 − y2,
+C4 : w4 = z2 − y4.
+(13.23)
+Three of the parameters (y1, y2 and y3) are fixed while the single complex modulus
+of M0,4 is taken to be y4. Since we are interested in the on-shell amplitude, it is not
+necessary to introduce local coordinates and the associated parameters.
+A variation of the modulus
+y4 −→ y4 + δy4,
+¯y4 −→ ¯y4 + δ¯y4
+(13.24)
+is equivalent to a change in the transition function of C4. This translates in turn into
+a transformation of z2:
+z′
+2 = z2 + δy4,
+¯z′
+2 = ¯z2 + δ¯y4.
+(13.25)
+Then, the tangent vector V = ∂y4 is associated to the vector field
+v = 1,
+¯v = 0,
+(13.26)
+with support on C4. For V = ∂¯y4, one finds
+v = 0,
+¯v = 1.
+(13.27)
+200
+
+The Beltrami 1-form for the unit vectors are
+B(∂y4) =
+�
+C4
+dz2 b(z2)(+1),
+B(∂¯y4) =
+�
+C4
+d¯z2 ¯b(¯z2)(+1),
+(13.28)
+with both contours running anti-clockwise.
+The components of the 2-form reads
+ω2(∂y4, ∂¯y4) =
+1
+2πi
+�
+B(∂y4)B(∂¯y4)
+4
+�
+i=1
+Vi
+�
+Σ0,4
+=
+1
+2πi
+��
+C4
+dz2 b(z2)
+�
+C4
+d¯z2 ¯b(¯z2)
+4
+�
+i=1
+Vi
+�
+Σ0,4
+.
+For on-shell states Vi = c¯cVi(yi, ¯yi), this becomes
+ω2(∂y4, ∂¯y4) =
+1
+2πi
+� 3
+�
+i=1
+c¯cVi(yi, ¯yi)
+�
+C4
+dz2 b(z2)
+�
+C4
+d¯z2 ¯b(¯z2)¯c(¯y4)c(y4)V4(y4, ¯y4)
+�
+Σ0,4
+.
+The first three operators could be moved to the left because they are not encircled by the
+integration contour. Note the difference with the example discussed in Section 11.1.2:
+here, the contour encircles z3, while it was encircling y3 for the s-channel.
+Using the OPE
+�
+C4
+dz2 b(z2)c(y4) ∼
+�
+C4
+dz2
+1
+z2 − y4
+(13.29)
+to simplify the product of b and c gives the amplitude
+A0,4 =
+1
+2πi
+�
+dy4 ∧ d¯y4
+� 3
+�
+i=1
+c¯cVi(yi) V4(y4)
+�
+Σ0,4
+.
+(13.30)
+This is the standard formula for the 4-point function derived from the Polyakov path
+integral.
+13.2
+Properties of forms
+In this section, we check that the form (13.12) has the correct properties:
+• antisymmetry under exchange of two vectors;
+• given a trivial vector of (a subspace of) Pg,n (Section 12.1), its contraction with the
+form vanishes: ωp(V (1), . . . , V (p)) = 0 if any of the V (i) generates:
+– reparametrizations of za for V (i) ∈ TPg,n,
+– rotation wi → (1 + iαi)wi for V (i) ∈ T ˆPg,n,
+– reparametrizations of wi keeping wi = 0 if the states are on-shell for V (i) ∈
+TMg,n;
+• BRST identity, which is necessary to prove several properties of the amplitudes.
+The first property is obvious. Indeed, the form is correctly antisymmetric under the
+exchange of two vectors V (i) and V (j) due to the ghost insertions.
+201
+
+13.2.1
+Vanishing of forms with trivial vectors
+Reparametrization of za
+Consider the sphere Sa with coordinate za, and denote by C1,
+C2 and C3 the three boundaries. Then, a reparametrization
+za −→ za + φ(za)
+(13.31)
+is generated by a vector field φ(z) which is regular on Sa. This transformation modifies
+the transition functions on the three circles and is thus associated to a tangent vector V
+described by a vector field v with support on the three circles:
+Ci :
+v(i) = φ|Ci.
+(13.32)
+The Beltrami form then reads
+B(V ) =
+3
+�
+i=1
+�
+Ci
+dza b(za)φ(za) + c.c.
+(13.33)
+where the orientations of the contours are such that Sa is on the left. Since the vector
+field φ is regular in Sa, two of the contours can be deformed until they merge together.
+The resulting orientation is opposite to the one of the last contour (Figure 13.1). As a
+consequence, both cancel and the integral vanishes.
+Figure 13.1: Deformation of the contour of integration defining the Beltrami form for a
+reparametrization of za.
+The figure is drawn for two circles at a hole, but the proof is
+identical for other types of circles.
+Rotation of wi
+Consider an infinitesimal phase rotation of the local coordinate wi in the
+disk Di:
+wi −→ (1 + iαi)wi,
+¯wi −→ (1 − iαi) ¯wi,
+(13.34)
+with αi ∈ R. The tangent vector is defined by the circle Ci and the vector field by
+v = iwi,
+¯v = −i ¯wi.
+(13.35)
+The Beltrami form for this vector is
+B(V ) = i
+�
+Ci
+dwi wi b(wi) − i
+�
+Ci
+d ¯wi ¯wi ¯b( ¯wi),
+(13.36)
+where Di is kept to the left.
+In the p-form (13.11), the ith operator Vi is inserted in Di and encircled by Ci. Because
+there is no other operator inside Di, the contribution of this disk to the form is
+B(V )Vi(0) = i
+�
+Ci
+dwi wi b(wi)Vi(0) − i
+�
+Ci
+d ¯wi ¯wi ¯b( ¯wi)Vi(0),
+(13.37)
+202
+
+The state–operator correspondence allows to rewrite this result as
+i(b0 − ¯b0) |Vi⟩ ,
+(13.38)
+since the contour integral picks the zero-modes of b and of ¯b.
+Requiring that the form
+vanishes implies the ghost counter-part of the level-matching condition:
+b−
+0 |Vi⟩ = 0.
+(13.39)
+Hence, consistency of off-shell amplitudes imply that
+Vi ∈ H−,
+(13.40)
+where H− is defined in (11.38).
+Reparametrization of wi
+A reparametrization of the local coordinate wi keeping the
+origin of Di fixed reads:
+wi −→ f(wi),
+f(0) = 0.
+(13.41)
+The function can be expanded in series:
+f(wi) =
+�
+m≥0
+pmwm+1
+i
+.
+(13.42)
+Because the transformation is holomorphic, it can be extended on Ci. Each parameter pm
+provides a coordinate of Pg,n and whose deformation corresponds to a vector field:
+vm = wm+1
+i
+,
+¯vm = 0.
+(13.43)
+The corresponding Beltrami differential is
+B(∂pm) =
+�
+Ci
+dwi b(wi)wm+1
+i
+.
+(13.44)
+Since only the operator Vi is inserted in the disk, the state–operator correspondence gives
+bm |Vi⟩. Requiring that the form vanishes on Mg,n for all m and also for the anti-holomorphic
+vectors gives the conditions:
+∀m ≥ 0 :
+bm |Vi⟩ = 0,
+¯bm |Vi⟩ = 0.
+(13.45)
+This holds automatically for on-shell states Vi = c¯cVi.
+13.2.2
+BRST identity
+The BRST identity for the p-form (13.12) reads
+ωp
+� �
+i
+Q(i)
+B ⊗i Vi
+�
+= (−1)pdωp−1(⊗Vi),
+(13.46)
+using the notation (13.4). The BRST operator acting on the ith Hilbert space is written as
+Q(i)
+B = 1i−1 ⊗ QB ⊗ 1n−i
+(13.47)
+and acts as
+QBVi(z, ¯z) =
+1
+2πi
+�
+dw jB(w)Vi(z, ¯z) + c.c.
+(13.48)
+203
+
+More explicitly, the LHS corresponds to
+ωp
+� �
+i
+Q(i)
+B ⊗i Vi
+�
+= ωp(QBV1, V2, . . . , Vn) + (−1)|V1|ωp(V1, QBV2, . . . , Vn)
++ · · · + (−1)|V1|+···+|Vn−1|ωp(V1, V2, . . . , QBVn).
+(13.49)
+We give just an hint of this identity, the complete proof can be found in [262, pp. 85–89,
+215, sec. 2.5].
+The contour of the BRST current around each puncture can be deformed, picking singu-
+larities due to the presence of the Beltrami forms. Using (13.9), we find that anti-commuting
+the BRST charge with the Beltrami form Bs leads to an insertion of
+Ts = {QB, Bs}.
+(13.50)
+The energy–momentum tensor generates changes of coordinates. Hence, Ts = T∂s is precisely
+the generator associated to an infinitesimal change of the coordinate xs on Pg,n. The latter
+is given by the vector ∂s. For this reason, one can write:
+dxs {QB, Bs} = dxs Ts = dxs ∂s = d,
+(13.51)
+where d is the exterior derivative on Pg,n. The minus signs arise if the states Vi are Grass-
+mann odd.
+13.3
+Properties of amplitudes
+In order for the p-form (13.12) to be non-vanishing, its total ghost number should match
+the ghost number anomaly:
+Ngh
+�
+ωp(V1, . . . , Vn)
+�
+=
+n
+�
+i=1
+Ngh(Vi) − p = 6 − 6g,
+(13.52)
+using Ngh(B) = −1. For an amplitude, one has p = Mg,n = 6g − 6 + 2n and thus:
+Ngh(ωMg,n) = 6 − 6g
+=⇒
+n
+�
+i=1
+Ngh(Vi) = 2n.
+(13.53)
+This condition holds automatically for on-shell states since Ngh(c¯cVi) = 2.
+13.3.1
+Restriction to ˆPg,n
+The goal of this section is to explain why amplitudes must be described in terms of a section
+of ˆPg,n (12.10) instead of Pg,n.
+This means that one should identify local coordinates
+differing by a global phase rotation.
+The off-shell amplitudes (13.16) are multi-valued on Pg,n. Indeed, the amplitude depends
+on the local coordinates1 and changes by a factor under a global phase rotation of any local
+coordinate wi → eiαwi. However, such a global rotation leaves the surface unchanged, since
+the flat metric |dwi|2 is invariant. This means that the same surface leads to different values
+for the amplitude. To prevent this multi-valuedness of the amplitudes, it is necessary to
+identify local coordinates differing by a constant phase.
+A second way to obtain this condition is to require that the section Sg,n is globally
+defined: every point of the section should correspond to a single point of the moduli space
+1The current argument does not apply for on-shell amplitudes.
+204
+
+Mg,n. However, there is a topological obstruction which prevents finding a global section in
+Pg,n in general. One hint [56, sec. 2, 71, sec. 3] is to exhibit a nowhere vanishing 1-form if
+Sg,n is globally defined: this leads to a contradiction since such a 1-form does not generally
+exist (see for example [59, sec. 6.3.2, ch. 7]). Then, consider a closed curve in the moduli
+space (such curves exist since Mg,n is compact). Starting at a given point Σ of the curve,
+one finds that the local coordinates typically change by a global phase when coming back
+to the point Σ (Figure 13.2), since this describes the same surface and there is no reason
+to expect the phase to be invariant. Up to this identification, it is possible to find a global
+section. The latter corresponds to a section of ˆPg,n.
+(a) Closed curve in Mg,n.
+(b) Change in the phase of wi.
+Figure 13.2: Schematic plot of the change in the phase of the local coordinate wi as one
+follows a closed curve in Mg,n. If the original phase at Σ is α0 and if the phase varies
+continuously along the path, then α1 ̸= α0 when returning back to Σ by continuity.
+Remark 13.1 (Degeneracy of the antibracket) It is possible to define a BV structure
+on Riemann surfaces [233, 234]. The antibracket is degenerate in Pg,n but not in ˆPg,n [234].
+Global phase rotations of the local coordinates are generated by L−
+0 . Hence, identifying
+the local coordinates wi → eiαiwi amounts to require that the amplitude is invariant under
+L−
+0 . This is equivalent to imposing the level-matching condition
+L−
+0 |Vi⟩ = 0
+(13.54)
+on the off-shell states. This condition was interpreted in Section 3.2.2 as a gauge-fixing
+condition for translations along the S1 of the string. This shows, in agreement with earlier
+comments, that the level-matching condition should also be imposed off-shell because no
+gauge symmetry is introduced for the corresponding transformation.
+If the generator L−
+0 is trivial, this means that the ghost associated to the corresponding
+tangent vector must be decoupled. According to Section 13.2.1, this corresponds to the
+constraint:
+b−
+0 |Vi⟩ = 0.
+(13.55)
+This can be interpreted as a gauge fixing condition (Section 10.5), which could in principle
+be relaxed. However, the decoupling of physical states (equivalent to gauge invariance in
+SFT) happens only after integrating over the moduli space. This requires having a globally
+defined section.
+As a consequence, off-shell states are elements of the semi-relative Hilbert space
+Vi ∈ H− ∩ ker L−
+0 ,
+(13.56)
+and the amplitudes are defined by integrating the form ωMg,n over a section Sg,n ⊂ ˆPg,n.
+205
+
+Computation – Equation (13.54)
+The operator associated to the state through |Ai⟩ = Ai(0) |0⟩ transforms as
+Vi(0) −→ (eiαi)h(e−iαi)
+¯hVi(0)
+(13.57)
+which translates into
+|Vi⟩ −→ eiαi(L0−¯L0) |Vi⟩
+(13.58)
+for the state, using the fact that the vacuum is invariant under L0 and ¯L0.
+Then,
+requiring the invariance of the state leads to (13.54).
+13.3.2
+Consequences of the BRST identity
+Two important properties of the on-shell amplitudes can be deduced from the BRST identity
+(13.46): the independence of physical results on the choice of local coordinates and the
+decoupling of pure gauge states.
+Given BRST closed states, the LHS of (13.46) vanishes identically
+∀i :
+QB |Vi⟩ = 0
+=⇒
+dωp−1(V1, . . . , Vn) = 0.
+(13.59)
+Using this result, one can compare the on-shell amplitudes computed for two different sec-
+tions S and S′:
+�
+S
+ωMg,n −
+�
+S′ ωMg,n =
+�
+∂T
+ωMg,n−1 =
+�
+T
+dωMg,n−1 = 0,
+(13.60)
+using Stokes’ theorem and where T is the surface delimited by the two sections (Figure 13.3).
+This implies that on-shell amplitudes do not depend on the section, and thus on the local
+coordinates. In obtaining the result, one needs to assume that the vertical segments do
+not contribute. The latter correspond to boundary contributions of the moduli space. In
+general, many statements hold up to this condition, which we will not comment more in this
+book.
+Figure 13.3: Two sections S and S′ of Pg,n delimiting a surface T .
+Next, we consider a pure gauge state together with BRST closed states:
+|V1⟩ = QB |Λ⟩ ,
+QB |Vi⟩ = 0.
+(13.61)
+The BRST identity (13.46) reads:
+ωMg,n(QBΛ, V2, . . . , Vn) = dωMg,n−1(Λ, V2, . . . , Vn),
+(13.62)
+206
+
+which gives the amplitude
+�
+S
+ωMg,n(QBΛ, V2, . . . , Vn) =
+�
+S
+dωMg,n−1(Λ, V2, . . . , Vn) =
+�
+∂S
+ωMg,n−1(Λ, V2, . . . , Vn)
+(13.63)
+where the last equality follows from Stokes’ theorem.
+Assuming again that there is no
+boundary contribution, this vanishes:
+�
+S
+ωMg,n(QBΛ, V2, . . . , Vn) = 0.
+(13.64)
+This implies that pure gauge states decouple from the physical states.
+13.4
+Suggested readings
+• Definition of the forms [71, 73, 215, 262].
+• Global phase rotation of local coordinates [56, sec. 2, 71, sec. 3, 174, 262, p. 54].
+207
+
+Chapter 14
+Amplitude factorization and
+Feynman diagrams
+Abstract
+In the previous chapter, we built the off-shell amplitudes by integrating forms
+on sections of Pg,n. Studying their factorizations lead to rewrite them in terms of Feynman
+diagrams, which allows to identify the fundamental interactions vertices. We will then be
+able to write the SFT action in the next chapter.
+14.1
+Amplitude factorization
+We have seen how to write off-shell amplitudes. The next step is to rewrite them as a sum
+of Feynman diagrams through factorization of amplitudes.
+Factorization consists in writing a g-loop n-point amplitude in terms of lower-order
+amplitudes in both g and n connected by propagators. Since an amplitude corresponds
+to a sum over all possible processes, which corresponds to integrating over the moduli space,
+it is natural to associate Feynman diagrams to different subspaces of the moduli space. One
+can expect that the plumbing fixture (Section 12.3) is the appropriate translation of the
+factorization at the level of Riemann surfaces. We will assume that it is the case and check
+that it is correct a posteriori.
+To proceed, we consider the contribution to the amplitude Ag,n of the family of surfaces
+obtained by the plumbing fixture of two surfaces (separating case) or a surface with itself
+(non-separating case).
+14.1.1
+Separating case
+In this section, we consider the separating plumbing fixture where part of the moduli space
+Mg,n is covered by Mg1,n1#Mg2,n2 with g = g1 + g2 and n = n1 + n2 − 2 (Section 12.3.1).
+The local coordinates read w(1)
+i
+and w(2)
+j
+for i = 1, . . . , n1 and j = 1, . . . , n2. By convention,
+the last coordinate of each set is used for the plumbing fixture:
+w(1)
+n1 w(2)
+n2 = q.
+(14.1)
+The g-loop n-point amplitude with external states {V (1)
+1
+, . . . , V (1)
+n1−1, V (2)
+1
+, . . . , V (2)
+n2−1} is
+denoted as:
+Ag,n =
+�
+Sg,n
+ωg,n
+Mg,n
+�
+V (1)
+1
+, . . . , V (1)
+n1−1, V (2)
+1
+, . . . , V (2)
+n2−1
+�
+.
+(14.2)
+208
+
+We need to study the form ωg,n
+Mg,n on Mg1,n1#Mg2,n2, which means to rewrite it in terms
+of the data from Mg1,n1 and from Mg2,n2. This corresponds to the degeneration limit where
+the two groups of punctures denoted by V (1)
+i
+and V (2)
+j
+(i = 1, . . . , n1 − 1, j = 1, . . . , n2 − 1)
+together with g1 and g2 holes move apart from each other (Figure 14.1).
+Figure 14.1: Degeneration limit of Σg,n where the punctures V (1)
+i
+and V (2)
+j
+move apart from
+each other.
+Since q is a coordinate of Pg,n, its variation is associated with a tangent vector and a
+Beltrami 1-form. The latter has to be inserted inside ωg,n
+Mg,n. A change q → q + δq translates
+into a change of coordinate
+w′(1)
+n1 = w(1)
+n1 + w(1)
+n1
+q
+δq,
+(14.3)
+where w(2)
+n2 is kept fixed (obviously, this choice is conventional as explained in Section 12.2).
+Thus, the vector field and the Beltrami form are
+vq = w(1)
+n1
+q
+,
+Bq = 1
+q
+�
+Cq
+dw(1)
+n1 b
+�
+w(1)
+n1
+�
+w(1)
+n1 .
+(14.4)
+Computation – Equation (14.3)
+Starting from (14.1), vary q → q + δq while keeping w(2)
+n2 fixed:
+w′(1)
+n1 w(2)
+n2 = q + δq
+w′(1)
+n1 = w(1)
+n1
+q
+(q + δq) = w(1)
+n1 +
+q
+w(1)
+n1
+δq.
+The second line follows by replacing w(2)
+n2 using (14.1).
+The Mg,n-form for the moduli described by the plumbing fixture can be expressed as:
+ωMg,n
+�
+V (1)
+1
+, . . . , V (1)
+Mg1,n1, ∂q, ∂¯q, V (2)
+1
+, . . . , V (2)
+Mg2,n2
+�
+(14.5)
+= (2πi)−Mc
+g,n
+�Mg1,n1
+�
+λ=1
+B
+�
+V (1)
+λ
+�
+B(∂q)B(∂¯q)
+Mg2,n2
+�
+κ=1
+B
+�
+V (2)
+κ
+�n1−1
+�
+i=1
+V (1)
+i
+n2−1
+�
+j=1
+V (2)
+j
+�
+Σg,n
+.
+We introduce the surface states Σn1 and Σn2 such that the BPZ inner product with the
+209
+
+new states V (1)
+n1
+and V (2)
+n2
+reproduce the Mg1,n1- and Mg2,n2-forms:
+⟨Σn1|V (1)
+n1 ⟩ := ωMg1,n1(V (1)
+1
+, . . . , V (1)
+n1 ) = (2πi)−Mc
+g1,n1
+�Mg1,n1
+�
+λ=1
+B
+�
+V (1)
+λ
+� n1−1
+�
+i=1
+V (1)
+i
+�
+Σg1,n1
+,
+(14.6a)
+⟨Σn2|V (2)
+n2 ⟩ := ωMg2,n2(V (2)
+1
+, . . . , V (2)
+n2 ) = (2πi)−Mc
+g2,n2
+�Mg1,n1
+�
+λ=1
+B
+�
+V (2)
+λ
+� n2−1
+�
+j=1
+V (2)
+j
+�
+Σg2,n2
+.
+(14.6b)
+As described in Section 13.1.2, these states exist since the p-form are linear in each of the
+external state and the BPZ inner-product is non-degenerate. Each of the surface states
+corresponds to an operator
+⟨Σn1| =⟨0| I ◦ Σn1(0),
+⟨Σn2| =⟨0| I ◦ Σn2(0),
+(14.7)
+defined from (6.136). Then, the forms can be interpreted as 2-point functions on the complex
+plane:
+⟨Σn1|V (1)
+n1 ⟩ = ⟨I ◦ Σn1(0)Vn1(0)⟩w(1)
+n1 ,
+⟨Σn2|V (2)
+n2 ⟩ = ⟨I ◦ Σn2(0)Vn2(0)⟩w(2)
+n2 .
+(14.8)
+All the complexity of the amplitudes has been lumped into the definitions of the surface
+states which contain information about the surface moduli (including the ghost insertions)
+and about the n1 − 1 remaining states (including the local coordinate systems). The local
+coordinates around V (1)
+n1
+and V (2)
+n2
+are denoted respectively as w(1)
+n1 and w(2)
+n2 . Correspond-
+ingly, the surface operators are inserted in the local coordinates w1 and w2 which are related
+to w(1)
+n1 and w(2)
+n2 through the inversion:
+w1 = I
+�
+w(1)
+n1
+�
+,
+w2 = I
+�
+w(2)
+n2
+�
+.
+(14.9)
+In order to rewrite (14.5) in terms of Σ1 and Σ2, it is first necessary to express all
+operators in one coordinate system, for example w(1)
+n1 . Hence, we need to find its relation to
+w2. Using the plumbing fixture (14.1), the relation between w(1)
+n1 and w2 is:
+w(1)
+n1 =
+q
+w(2)
+n2
+=
+q
+I(w2) = qw2 := f(w2).
+(14.10)
+Then, the form (14.5) becomes
+ωMg,n =
+1
+2πi ⟨I ◦ Σn1(0)BqB¯q f ◦ Σn2(0)⟩w(1)
+n1 =
+1
+2πi ⟨Σn1| BqB¯q qL0 ¯q
+¯L0 |Σ2⟩ ,
+(14.11)
+using that Σ2 has a well-defined scaling dimension. The factor of 2πi arises by comparing the
+contribution from Σn1 and Σn2 with the factor in (14.5). The expression can be simplified
+by using the relation
+⟨Σn1| BqB¯q |V (1)
+n1 ⟩ = 1
+q¯q ⟨Σn1| b0¯b0 |V (1)
+n1 ⟩
+(14.12)
+using the expression (14.4) for Bq and the state–operator correspondence:
+BqV (1)
+n1 (z, ¯z) = 1
+q
+�
+Cq
+dw(1)
+n1 b
+�
+w(1)
+n1
+�
+w(1)
+n1 V (1)
+n1 (z, ¯z) −→ 1
+q b0 |V (1)
+n1 ⟩ .
+(14.13)
+210
+
+Ultimately, the form (14.5) reads
+ωMg,n =
+1
+2πi
+1
+q¯q ⟨Σn1| b0¯b0 qL0 ¯q
+¯L0 |Σn2⟩ .
+(14.14)
+It is important to remember that the plumbing fixture describes only a patch of the
+moduli space, and the form defined in this way is valid only locally. As a consequence, the
+integration over all moduli of Mg1,n1#Mg2,n2 does not describe Mg,n, but only a part of
+it (Section 12.3.3). Every degeneration limit with a different puncture distribution in two
+different groups contributes to a different part of the amplitude.
+We denote the contribution to the total amplitude (14.2) from the region of the moduli
+space connected to this degeneration limit as:
+Fg,n
+�
+V (1)
+i
+|V (2)
+j
+�
+:=
+1
+2πi
+�
+Mg1,n1
+�
+λ=1
+dt(1)
+λ
+Mg2,n2
+�
+κ=1
+dt(2)
+κ ∧ dq
+q ∧ d¯q
+¯q ⟨Σn1| b0¯b0 qL0 ¯q
+¯L0 |Σn2⟩ . (14.15)
+To proceed, we introduce a basis {φα(k)} of eigenstates of L0 and ¯L0, where kµ is the D-
+dimensional momentum and α denotes the remaining quantum number. Then, introducing
+twice the resolution of the identity (11.36) gives:
+Fg,n
+�
+V (1)
+i
+|V (2)
+j
+�
+=
+1
+2πi
+�
+dDk
+(2π)D
+dDk′
+(2π)D (−1)|φα|
+×
+� dq
+q ∧ d¯q
+¯q ⟨φα(k)c| b0¯b0 qL0 ¯q
+¯L0 |φβ(k′)c⟩
+(14.16)
+×
+�
+Mg1,n1
+�
+λ=1
+dt(1)
+λ ⟨Σn1|φα(k)⟩
+�
+Mg2,n2
+�
+κ=1
+dt(2)
+κ ⟨φβ(k′)|Σn2⟩
+(with implicit sums over α and β). In the last line, one recognizes the expressions of the
+g1-loop n1-point amplitude with external states {V (1)
+1
+, . . . , V (1)
+n1−1, φα} and of the g2-loop
+and n2-point amplitudes with external states {V (2)
+1
+, . . . , V (2)
+n2−1, φβ}:
+Ag1,n1
+�
+V (1)
+1
+, . . . , V (1)
+n1−1, φα(k)
+�
+=
+�
+Sg1,n1
+ωMg1,n1
+�
+V (1)
+1
+, . . . , V (1)
+n1−1, φα(k)
+�
+=
+�
+Sg1,n1
+Mg1,n1
+�
+λ=1
+dt(1)
+λ ⟨Σn1|φα(k)⟩ ,
+(14.17a)
+Ag2,n2
+�
+V (2)
+1
+, . . . , V (2)
+n2−1, φβ(k′)
+�
+=
+�
+Sg2,n2
+ωMg2,n2
+�
+V (2)
+1
+, . . . , V (2)
+n2−2, φβ(k′)
+�
+=
+�
+Sg2,n2
+Mg2,n2
+�
+λ=1
+dt(2)
+λ ⟨Σn2|φβ(k′)⟩ .
+(14.17b)
+The property (B.27) has been used to reverse the order of the BPZ product for the second
+Riemann surface, and this cancels the factor (−1)|φα|.
+Defining the second line of (14.16) as
+∆αβ(k, k′) := ∆
+�
+φα(k)c, φβ(k′)c�
+:=
+1
+2πi
+� dq
+q ∧ d¯q
+¯q ⟨φα(k)c| b0¯b0 qL0 ¯q
+¯L0 |φβ(k′)c⟩ , (14.18)
+one has:
+Fg,n
+�
+V (1)
+i
+|V (2)
+j
+�
+=
+�
+dDk
+(2π)D
+dDk′
+(2π)D Ag1,n1
+�
+V (1)
+1
+, . . . , V (1)
+n1−1, φα(k)
+�
+∆αβ(k, k′)
+× Ag2,n2
+�
+V (2)
+1
+, . . . , V (2)
+n2−1, φβ(k′)
+�
+.
+(14.19)
+211
+
+We recover the expressions from Section 11.1.2, but for a more general amplitude.
+We
+had found that ∆ corresponds to the propagator: its properties are studied further in
+Section 14.2.2.
+Hence, the object (14.16) corresponds to the product of two amplitudes
+connected by a propagator (Figure 14.2).
+There are several points to mention about this amplitude:
+• We will find that the propagator depends only on one momentum because ⟨k|k′⟩ ∼
+δ(D)(k + k′), which removes one of the integral. Then, both amplitudes Ag1,n1 and
+Ag2,n2 contain a delta function for the momenta:
+Ag1,n1 ∼ δ(D)�
+k(1)
+1 +· · ·+k(1)
+n1−1+k
+�
+,
+Ag2,n2 ∼ δ(D)�
+k(2)
+1 +· · ·+k(2)
+n2−1+k′�
+. (14.20)
+As a consequence, the second momentum integral can be performed and yields a delta
+function:
+Fg,n ∼ δ(D)�
+k(1)
+1
++ · · · + k(1)
+n1−1 + k(2)
+1
++ · · · + k(2)
+n2−1
+�
+.
+(14.21)
+Hence, the momentum flowing in the internal line is fixed and this ensures the overall
+momentum conservation as expected.
+• The ghost numbers of the states φα and φβ are also fixed (in terms of the external
+states). Indeed, because of the ghost number anomaly, the amplitudes on Mg1,n1 and
+Mg2,n2 are non-vanishing only if the ghost numbers of these states satisfy:
+Ngh(φα) = 2n1 −
+n1−1
+�
+i=1
+Ngh
+�
+V (1)
+i
+�
+,
+Ngh(φβ) = 2n2 −
+n2−1
+�
+j=1
+Ngh
+�
+V (2)
+j
+�
+.
+(14.22)
+The non-vanishing of Fg,n also gives another relation:
+Ngh(φα) + Ngh(φβ) = 4.
+(14.23)
+In particular, if the external states are on-shell with Ngh = 2, we find:
+Ngh(φα) = Ngh(φβ) = 2.
+(14.24)
+As indicated in Chapter 10, such states are appropriate at the classical level since they
+do not contain spacetime ghosts.
+• The sum over α and β is over an infinite number of states and could diverge. In fact,
+the sum can be made convergent by tuning the stub parameter (Section 14.2.4).
+Properties of Feynman graphs and amplitudes in the momentum space will be discussed
+further in Chapter 18.
+14.1.2
+Non-separating case
+Next, we consider the non-separating plumbing fixture (Section 12.3.2). The computations
+are almost identical to the separating case, thus we outline only the general steps.
+Part of the moduli space Mg,n is covered by #Mg1,n1, with g = g1 + 1 and n = n1 − 2.
+The local coordinates are denoted as wi for i = 1, . . . , n1 and the plumbing fixture reads:
+wn1−1wn1 = q.
+(14.25)
+The g-loop n-point amplitude with external states {V (1)
+1
+, . . . , V (1)
+n1−2} is denoted as:
+Ag,n =
+�
+Sg,n
+ωg,n
+Mg,n
+�
+V (1)
+1
+, . . . , V (1)
+n1−2
+�
+.
+(14.26)
+212
+
+Figure 14.2: Factorization of the amplitude into two sub-amplitudes connected by a propag-
+ator (dashed line).
+When the n1 − 2 punctures and g1 = g − 1 holes move lose to each other, the form can
+be written as:
+ωMg,n
+�
+V (1)
+1
+, . . . , V (1)
+Mg1,n1, ∂q, ∂¯q
+�
+= (2πi)−Mc
+g,n
+�Mg1,n1
+�
+λ=1
+B
+�
+V (1)
+λ
+�
+B(∂q)B(∂¯q)
+n1−2
+�
+i=1
+V (1)
+i
+�
+Σg,n
+.
+(14.27)
+To proceed, one needs to introduce the surface state Σn1−1,n1:
+⟨Σn1−1,n1|V (1)
+n1−1 ⊗ V (1)
+n1 ⟩ := ωMg1,n1(V (1)
+1
+, . . . , V (1)
+n1 ).
+(14.28)
+Following the same step as in the previous section leads to:
+Fg,n
+�
+V (1)
+i
+|
+�
+=
+�
+dDk
+(2π)D
+dDk′
+(2π)D Ag1,n1
+�
+V (1)
+1
+, . . . , V (1)
+n1−2, φα(k), φβ(k′)
+�
+∆αβ(k, k′),
+(14.29)
+where the propagator is given in (14.18). This is equivalent to an amplitude for which two
+external legs are glued together with a propagator, giving a loop (Figure 14.3).
+Since both states φα and φβ are inserted on the same surface, their ghost numbers are
+not fixed, even if the external states are physical. The non-vanishing of Fg,n only leads to
+the constraint:
+Ngh(φα) + Ngh(φβ) = 2n1 −
+n1−2
+�
+i=1
+Ngh
+�
+V (1)
+i
+�
+= 4.
+(14.30)
+As a consequence, loop diagrams force to introduce states of every ghost number. Internal
+states with Ngh ̸= 2 correspond to spacetime ghosts.
+Since the propagator contains a delta function δ(D)(k − k′), the integral over k′ can be
+removed by setting k′ = −k. However, the integral over k remains since
+Ag1,n1
+�
+V (1)
+1
+, . . . , V (1)
+n1−2, φα(k), φβ(−k)
+�
+∼ δ(D)�
+k(1)
+1
++ · · · + k(1)
+n1−2
+�
+.
+(14.31)
+Hence, the loop momentum k is not fixed, as expected in QFT.
+Remark 14.1 Not all values of the moduli associated to the holes can be associated to loops
+in Feynman diagrams. Only the values close to the degeneration limit can be interpreted in
+this way, the other being just standard (quantum) vertices.
+14.2
+Feynman diagrams and Feynman rules
+In the standard QFT approach, Feynman graphs compute Green functions, and scattering
+amplitudes are obtained by amputating the external propagators through the LSZ prescrip-
+tion. For connected tree-level processes, this requires n ≥ 3 (corresponding to χ0,n < 0).
+213
+
+Figure 14.3: Factorization of the amplitude into two sub-amplitudes connected by a propag-
+ator (dashed line). The propagator connects two punctures of the same surface, which is
+equivalent to a loop.
+Given a theory, there is a minimal set of Feynman diagrams – the Feynman rules – from
+which every other diagram can be constructed. These rules include the definitions of the
+fundamental vertices – the fundamental interactions – and of the propagator – how states
+propagate between two interactions (or, how to glue vertices together). In this section, we
+describe these different elements.
+14.2.1
+Feynman graphs
+The amplitude factorization described in Section 14.1 gives a natural separation of amp-
+litudes into several contributions. Considering all the possible degeneration limits lead to
+a set of diagrams with amplitudes of lower order connected by propagators (Figure 14.2,
+Figure 14.3). This corresponds exactly to the idea behind Feynman graphs. Then, the goal
+is to find the Feynman rules of the theory: since the propagator has already been identified
+(further studied in Section 14.2.2), it is sufficient to find the interaction vertices.
+Let’s make this more precise by considering an amplitude Ag,n(V1, . . . , Vn). The index
+of an amplitude is defined to be the index (12.50) of the corresponding Riemann surfaces
+r(Ag,n) := r(Σg,n) = 3g + n − 2.
+(14.32)
+Contributions to an amplitude with a given r(Ag,n) can be described in terms of amp-
+litudes Ag′,n′ with r(Ag′,n′) < r(Ag,n). But, the moduli space Mg,n cannot (generically) be
+completely covered with the plumbing fixture of lower-dimensional moduli spaces, i.e. with
+r(Mg′,n′) < r(Mg,n) (Section 12.3.3). Then, the same must be true for the amplitudes,
+such that Ag,n cannot be uniquely expressed in terms of amplitudes Ag′,n′.
+The g-loop n-point fundamental vertex is defined by:
+Vg,n(V1, . . . , Vn) :=
+:=
+�
+Rg,n
+ωg,n
+Mg,n(V1, . . . , Vn),
+(14.33)
+The form defined in (13.16) is integrated over a sub-section Rg,n ⊂ Sg,n of ˆPg,n. Its pro-
+jection on the base is the region Vg,n ⊂ Mg,n which cannot be described by the plumbing
+214
+
+fixture, see (12.42b). In general, we will keep the choice of local coordinates implicit and
+always write Vg,n to avoid surcharging the notations.
+It corresponds to the remaining contribution of the amplitude once all graphs containing
+propagators have been taken into account:
+=
+�
+0≤h≤g
+0≤m<n−1
++
++ perms +
+(14.34)
+where the permutations are taken over all legs exiting the amplitudes in the first two terms
+(this includes the two legs glued together in the second term), including if necessary a weight
+to avoid overcounting. In the RHS, the amplitudes Ag,1 are tadpoles and have no external
+vertices Vi (from Ag,n); this corresponds to the terms for m = 0 and m = n − 2.
+In general, the fundamental vertex is non-vanishing for every value g, n ∈ N such that
+χg,n < 0. For this reason, the index g helps to distinguish between graphs with identical
+values of n. It may look strange that one needs vertices at every loop: the interpretation will
+be made clearer when translating this in the language of string field theory (Chapter 15). We
+stress again that the definition of the fundamental vertex (and the region covered) depends on
+the choice of local coordinates for all lower-order vertices Vg′,n′ such that r(Vg′,n′) < r(Vg,n).
+Remark 14.2 There are different alternative notations for (14.33):
+Vg,n(V1, . . . , Vn) := Vg,n(⊗iVi) := {V1, . . . , Vn}g.
+(14.35)
+Example 14.1 – Scalar QFT
+Consider a scalar field theory with a cubic and a quartic interaction. The 4-point amp-
+litude contains four contributions, three from gluing 3-point vertices with a propagator,
+and one from the fundamental quartic vertex. The mismatch between the amplitude
+and the three graphs with a propagator hints at the existence of the quartic interac-
+tions. This example gives an idea of how one can identify the fundamental interactions
+recursively.
+The definition (14.34) of the vertex shows that it can also be interpreted as an ampu-
+tated Green function without internal propagator (i.e. there is no propagator at all). This
+definition is expected from the definition of the interactions vertices from an action, as will
+be exemplified in Chapter 15. Before describing (and generalizing) the vertices, we describe
+first the properties of the propagator.
+14.2.2
+Propagator
+The propagator has been defined in (14.18):
+∆ =
+1
+2πi
+� dq
+q ∧ d¯q
+¯q b0¯b0 qL0 ¯q
+¯L0.
+(14.36)
+215
+
+The plumbing modulus q is parametrized by (12.31)
+q = e−s+iθ,
+s ∈ R+,
+θ ∈ [0, 2π),
+(14.37)
+such that the integration measure becomes
+dq
+q ∧ d¯q
+¯q = −2i ds ∧ dθ.
+(14.38)
+Using the variables L±
+0 = L0 ± ¯L0 and b±
+0 = b0 ± ¯b0, the propagator can be recast as:
+∆ = 1
+2π b+
+0 b−
+0
+� ∞
+0
+ds e−sL+
+0
+� 2π
+0
+dθ eiθL−
+0 .
+(14.39)
+The form of the first integral is recognized as the Schwinger parametrization of the propag-
+ator, while the second is the Fourier transformation of the discrete delta function:
+� ∞
+0
+ds e−sL+
+0 =
+1
+L+
+0
+,
+� 2π
+0
+dθ eiθL−
+0 = 2π δL−
+0 ,0.
+(14.40)
+In fact, the first integral converges only if L+
+0 > 0. As argued in the introduction, divergences
+for L+
+0 ≤ 0 are either non-physical or IR divergences which can be cured by renormalization.
+For this reason, we take the RHS as a definition of the integral, which would be the correct
+result if one starts with a field theory action instead of a first-quantized formalism.
+In this case, the propagator becomes
+∆ = b+
+0
+L+
+0
+b−
+0 δL−
+0 ,0.
+(14.41)
+This is the standard expression for the propagator. For completeness, the form in terms of
+the holomorphic and anti-holomorphic components is:
+∆ = −2b0¯b0
+1
+L0 + ¯L0
+δL0,¯L0.
+(14.42)
+The delta function restricts the amplitude to states satisfying the level-matching condition,
+that is, annihilated by L−
+0 .
+Considering a basis {φα(k)} of eigenstates of both L0 and ¯L0:
+L+
+0 |φα(k)⟩ = α′
+2 (k2 + m2
+α) |φα(k)⟩ ,
+L−
+0 |φα(k)⟩ = 0
+(14.43)
+leads to the following momentum-space kernel for the propagator:
+∆αβ(k, k′) :=⟨φα(k)c| ∆ |φβ(k′)c⟩ := (2π)Dδ(D)(k + k′) ∆αβ(k),
+(14.44a)
+∆αβ(k) := Mαβ(k)
+k2 + m2α
+,
+Mαβ(k) := 2
+α′ ⟨φc
+α(k)| b+
+0 b−
+0 |φc
+β(−k)⟩ ,
+(14.44b)
+with Mαβ a finite-dimensional matrix giving the overlap of states of identical masses (because
+the number of states at a given level is finite).
+For the propagator to be well-defined, it must be invertible (in particular, to define a
+kinetic term). The propagator (14.41) is non-vanishing if the states it acts on satisfy:
+b+
+0 |φc
+α⟩ ̸= 0,
+b−
+0 |φc
+α⟩ ̸= 0.
+(14.45)
+216
+
+Necessary and sufficient conditions for this to be true are
+c+
+0 |φc
+α⟩ = 0,
+c−
+0 |φc
+α⟩ = 0.
+(14.46)
+Indeed, decomposing the state on the ghost zero-modes
+|φc
+α⟩ = |φ1⟩ + b±
+0 |φ2⟩ ,
+c±
+0 |φ1⟩ = c±
+0 |φ2⟩ = 0
+(14.47)
+gives
+c±
+0 |φc
+α⟩ = 0
+=⇒
+|φ2⟩ = 0,
+(14.48)
+and one has correctly b±
+0 |φ1⟩ ̸= 0.
+These conditions are given for the dual states: translating them on the normal states
+reverses the roles of b0 and c0. Hence, the states must satisfy the conditions:
+b+
+0 |φα⟩ = 0,
+b−
+0 |φα⟩ = 0.
+(14.49)
+The second condition is satisfied automatically because the Hilbert space is H− when work-
+ing with ˆPg,n (Section 13.3.1). However, the first condition further restricts the states which
+propagate in internal lines. This leads to postulate that the external states should also be
+taken to satisfy this condition
+b+
+0 |Vi⟩ = 0,
+(14.50)
+since external states are usually a subset of the internal states. This provides another mo-
+tivation of the statement in Section 3.2.2 that scattering amplitudes for the states not anni-
+hilated by b+
+0 must be trivial. A field interpretation of this condition is given in Chapters 10
+and 15.
+Under these constraints on the states, the propagator can be inverted:
+∆−1 = c+
+0 c−
+0 L+
+0 δL−
+0 ,0.
+(14.51)
+14.2.3
+Fundamental vertices
+The vertices (14.33) can be constructed recursively assuming that all amplitudes are known.
+The starting point is the tree-level cubic amplitude A0,3: since it does not contain any
+internal propagator, it is equal to the fundamental vertex V0,3.
+The fist thing to extract from the recursion relations are the background independent
+data. This amounts to find local coordinates and a characterization of the subspaces Vg,n ⊂
+Mg,n, starting with P0,3 and iterating.
+In the rest of this section, we show how this works schematically.
+Recursive definition: tree-level vertices
+The description of tree-level amplitudes A0,n is the simplest since only the separating plumb-
+ing fixture is used and Feynman graphs are trees. The possible factorizations of the amp-
+litude correspond basically to all the partitions of the set {Vi} into subsets.
+Tree-level cubic vertex
+Since M0,3 = 0, the moduli space of the 3-punctured sphere
+Σ0,3 reduces to a point, and so does the section S0,3 of P0,3 (Figure 14.4a):
+V0,3(V1, V2, V3) := A0,3(V1, V2, V3) = ω0,3
+0 (V1, V2, V3).
+(14.52)
+The corresponding graph is indicated in Figure 14.4b.
+217
+
+(a) A section S0,3 over P0,3 reduces to
+a point.
+(b) Fundamental cubic vertex.
+Figure 14.4: Section of P0,3 and cubic vertex.
+Tree-level quartic vertex
+Part of the contributions to the 4-point amplitude A0,4 with
+external states Vi (i = 1, . . . , 4) comes from gluing two cubic vertices. Because there are four
+external states, there are three different partitions 2|2 which are described in Figure 14.5
+(see also Figure 12.7). The sum of these three diagrams does not reproduce A0,4: the moduli
+space M0,4 is not completely covered by the three amplitudes. Equivalently, the projection
+of the section over P0,4 does not cover all of M0,4. The missing contribution is defined by
+the quartic vertex (Figure 14.7)
+V0,4(V1, V2, V3, V4) :=
+�
+R0,4
+ω0,4
+2 (V1, . . . , V4),
+(14.53)
+and the corresponding section is denoted by R0,4 (Figure 14.6). Denoting by F(s,t,u)
+0,4
+the
+graphs 14.5 in the s-, t- and u-channels, one has the relation
+A0,4 = F(s)
+0,4 + F(t)
+0,4 + F(u)
+0,4 + V0,4.
+(14.54)
+Tree-level quintic vertex
+The amplitude A0,5 can be factorized in a greater number
+of channels, the two types being 2|2|1 and 2|3.
+The possible Feynman graphs are built
+either from three cubic vertices and two propagators (Figure 14.8a and permutations), or
+from one cubic and one quartic vertices together with one propagator (Figure 14.8b and
+permutations). The remaining contribution is the fundamental vertex (Figure 14.8c):
+V0,5(V1, . . . , V5) :=
+�
+R0,5
+ω0,5
+4 (V1, . . . , V5).
+(14.55)
+The construction to higher-order follows exactly this scheme.
+Recursive definition: general vertices
+Next, one needs to consider Feynman diagrams with loops. The first amplitude which can
+be considered is the one-loop tadpole A1,1(V1).
+The factorization region corresponds to
+the graph obtained by gluing two legs of the cubic vertex (Section 14.2.3). The remaining
+contribution is the fundamental tadpole vertex V1,1(V1) (Section 14.2.3) – note the index
+g = 1 on the vertex, indicating that it is a 1-loop effect.
+Next, the 1-loop 2-point amplitude can be obtained using the cubic and quartic tree-level
+vertices V0,3 and V0,4, but also the one-loop tadpole V1,1. Iterating, the number of loops
+218
+
+(a) s-channel.
+(b) t-channel.
+(c) u-channel.
+Figure 14.5: Factorization of the quartic amplitude A0,4 in the s-, t- and u-channels.
+Figure 14.6: A section S0,4 over P0,4, the contribution from the s-, t- and u-channels (Fig-
+ure 14.5) are indicated by the corresponding indices. The fundamental vertex is defined by
+the section V0,4.
+219
+
+Figure 14.7: Fundamental quartic vertex.
+(a) Factorization 12|3|45.
+(b) Factorization 12|345.
+(c) Fundamental vertex.
+Figure 14.8: Factorization of the amplitude G0,5 in channels and fundamental quintic vertex.
+Only the cases where V1 and V2 factorize on one side is indicated, the other cases follow by
+permutations of the external states.
+220
+
+can be increased either by gluing together two external legs of a graph, or by gluing two
+different graphs with loops together.
+For g ≥ 2, the recursion implies the existence of vertices with no external states Vg,0:
+they should be interpreted as loop corrections to the vacuum energy density.
+It is important to realize that, in this language, a handle in the Riemann surface is
+not necessarily mapped to a loop in the Feynman graph: only handles described by the
+region Fg,n = Mg,n − Vg,n do. The higher-order vertices – corresponding to surfaces with
+small handles only and described by Vg,n – should be regarded as quantum fundamental
+interactions.
+In Chapter 15, it will be explained that they really correspond to (finite)
+counter-terms: the measure is not invariant under the gauge symmetry of the theory and
+these terms must be introduced to restore it.
+(a) Internal loop.
+(b) Fundamental vertex.
+Figure 14.9: Factorization of the amplitude G1,1 and fundamental tadpole at 1-loop.
+Other vertices
+The definition given at the end of (14.2.1) suggests to introduce additional vertices. The
+previous recursive definition gives only vertices with χg,n = 2 − 2g − n < 0, but, in fact, it
+makes sense to consider the additional cases: g = 0 and n = 0, 1, 2, and g = 1, n = 0.
+The definition of the vertices as amputated Green function without internal propagators
+provides a hint for the tree-level quadratic vertex V0,2. We define the latter as the amputated
+tree-level 2-point Green function:
+V0,2 := ∆−1∆∆−1 = ∆−1.
+(14.56)
+Hence, we have
+V0,2(V1, V2) :=⟨V1| c+
+0 c−
+0 L+
+0 δL−
+0 ,0 |V2⟩ .
+(14.57)
+Note that V0,2 is not the 2-point scattering amplitude.
+We denote the tree-level 1-point and 0-point vertices as V0,1(V1) and V0,0. The first can
+be interpreted as a classical source in the action, while the second is a classical vacuum
+energy. They are set to zero in most applications and can be safely ignored. However, they
+appear when formulating the theory on a background which does not solve the equation of
+motion [263].
+Finally, the 1-loop vacuum energy V1,0 can also be defined as the partition function of
+the worldsheet CFT integrated over the torus modulus.
+This allows to define the vertices Vg,n for all g, n ∈ N. We define the sum of all loop
+contributions for a fixed n as:
+Vn(V1, . . . , Vn) :=
+�
+g≥0
+(ℏg2
+s)g Vg,n(V1, . . . , Vn).
+(14.58)
+221
+
+14.2.4
+Stubs
+In Section 12.3.4, we have indicated that the plumbing fixture can be modified by adding
+stubs or, equivalently, by rescaling the local coordinates. This amounts to introduce a cut-off
+(12.53) on the variable s such that
+q = e−s+iθ,
+s ∈ [s0, ∞),
+θ ∈ [0, 2π).
+(14.59)
+instead of (14.37). In this case, the s-integral in the propagator (14.36) is modified to
+� ∞
+s0
+ds e−sL+
+0 = e−s0L+
+0
+L+
+0
+.
+(14.60)
+This leads to a new expression for the propagator:
+∆(s0) = b+
+0
+e−s0L+
+0
+L+
+0
+b−
+0 δL−
+0 ,0.
+(14.61)
+In momentum space, this reads
+∆αβ(k) := e− α′s0
+2
+(k2+m2
+α)
+k2 + m2α
+Mαβ(k).
+(14.62)
+It is more convenient to work with the canonical propagator (14.41). This can be achieved
+by absorbing e− s0
+2 L+
+0 in the interaction vertex: a n-point interaction will get n such factors.1
+Since s0 changes the local coordinates, this means that it also changes the region Vg,n
+(Figure 12.10). The freedom in the choice of s0 translates into a freedom to choose which part
+of the amplitude is described by propagator graphs Fg,n(s0), and which part is described by
+a fundamental vertex Vg,n(s0). The amplitude Ag,n is independent of s0 since it is described
+in terms of the complete moduli space Mg,n. This also means that the parameter s0 must
+disappear when summing over the contributions from Vg,n(s0) and Fg,n(s0). This indicates
+that the value of s0 is not relevant, even off-shell: it can be taken to any convenient value.
+The possibility of adding stubs solves the problem that the sum over all states could
+diverge (see Section 14.1.1). Indeed, the expression (14.62) in momentum space shows that
+the propagator includes an exponential suppression for very massive particle propagating as
+intermediate states. Since the mass of a particle increases with the level, this shows that the
+sum converges for a sufficiently large value of s0 thanks to the factor e−α′s0m2. A second
+interesting aspect is the exponential momentum suppression e−α′s0k2: this is responsible for
+the nice UV behaviour of string theory. Since the value of s0 is not physical, this means
+that all Feynman graphs must share these properties. These two points will be made more
+precise in Chapter 18.
+14.2.5
+1PI vertices
+We can follow the same procedure as before, but considering only the separating plumb-
+ing fixture. In this case, the Feynman diagrams are all 1PR (1-particle reducible): if the
+propagator line is cut, then the graphs split in two disconnected components. The region
+of the moduli space covered by these graphs is written as F1PR
+g,n
+(12.43a). The complement
+1To make this identification precise for vertices involving external states, one has to consider the non-
+amputated Green functions.
+222
+
+defines the 1PI region V1PI
+g,n (12.43b). Then, the 1PI g-loop n-point fundamental vertices are
+defined as:
+V1PI
+g,n (V1, . . . , Vn) :=
+:=
+�
+R1PI
+g,n
+ωg,n
+Mg,n(V1, . . . , Vn),
+(14.63)
+where R1PI
+g,n is a section of Pg,n which projection on the base is V1PI
+g,n .
+14.3
+Properties of fundamental vertices
+14.3.1
+String product
+Following the definition of surfaces states (Section 13.1.2), the vertex state is defined as:
+⟨Vg,n| ⊗i Vi⟩ := Vg,n(⊗iVi).
+(14.64)
+The vertex is a map Vg,n : H⊗n → C where C ≃ H⊗0. We will find very useful to
+introduce the string products ℓg,n : H⊗n → H through the closed string inner product:
+Vg,n+1(V0, V1, . . . , Vn) :=⟨V0| c−
+0 |ℓg,n(V1, . . . , Vn)⟩ .
+(14.65)
+An alternative notation is:
+ℓg,n(V1, . . . , Vn) := [V1, . . . , Vn]g
+(14.66)
+The advantage of the second notation is to show that the products with n ≥ 3 are direct
+generalization of the 2-product, which is very similar to a super-Lie bracket. These products
+play a central role in SFT – in fact, the description of SFT is more natural using the ℓg,n
+rather than the Vg,n.
+Note that the products with n = 0 are maps C → H, which means that they correspond
+to a particular fixed state.
+ℓg,0 := [·]g ∈ H.
+(14.67)
+The ghost number of the product (14.65) is
+Ngh
+�
+ℓg,n(V1, . . . , Vn)
+�
+= 3 − 2n +
+n
+�
+i=1
+Ngh(Vi) = 3 +
+n
+�
+i=1
+�
+Ngh(Vi) − 2
+�
+,
+(14.68)
+and it is independent of the genus g. As a consequence, the parity of the product is
+|ℓg,n(V1, . . . , Vn)| = 1 +
+n
+�
+i=1
+|Vi|
+mod 2,
+(14.69)
+and the string product itself is always odd.
+The vertices satisfy the following identity for g ≥ 0 and n ≥ 1 [262, pp. 41–42]
+0 =
+�
+g1,g2≥0
+g1+g2=g
+�
+n1,n2≥0
+n1+n2=n
+n!
+n1! n2!Vg1,n1+1
+�
+Ψn1, ℓg2,n2(Ψn2)
+�
++ (−1)|φs|Vg−1,n+2
+�
+φs, b−
+0 φc
+s, Ψn�
+.
+(14.70)
+The last term is absent for g = 0. It is a consequence of the definition of the vertices as the
+missing region from gluing lower-order vertices.
+223
+
+14.3.2
+Feynman graph interpretation
+The vertices must satisfy a certain number of conditions to be interpreted as Feynman
+diagrams. The first is that they must be symmetric under permutations of the states. Not
+every choice of local coordinates satisfies this requirement: this can be solved by defining
+the vertex over a generalized section. In this case, the vertex is defined as the average of
+the integrals over N sections S(a)
+g,n of Pg,n:
+Vg,n(V1, . . . , Vn) = 1
+N
+N
+�
+a=1
+�
+R(a)
+g,n
+ωg,n
+Mg,n(V1, . . . , Vn).
+(14.71)
+Example 14.2 – 3-point vertex
+The cubic vertex must be symmetric under permutations
+V0,3(V1, V2, V3) = V0,3(V3, V1, V2) + · · ·
+(14.72)
+Taking the vertex to be given by a section S0,3 with local coordinates fi
+V0,3(V1, V2, V3) = ω0,3
+0 (V1, V2, V3)|S0,3 = ⟨f1 ◦ V1(0)f2 ◦ V2(0)f3 ◦ V3(0)⟩,
+(14.73)
+one finds that a permutation looks different
+V0,3(V3, V1, V2) = ⟨f1 ◦ V3(0)f2 ◦ V1(0)f3 ◦ V2(0)⟩ ̸= V0,3(V1, V2, V3),
+(14.74)
+unless the local coordinates satisfy special properties (remember that the local co-
+ordinates are specified by the vertex state V and not by the external states Vi, so a
+permutation of them does not permute the local maps). Obviously, both amplitudes
+agree on-shell since the dependence in the local coordinates cancel (equivalently one
+can rotate the punctures using SL(2, C)).
+Writing zi = fi(0), there is a SL(2, C) transformation g(z) such that
+g(z1) = z2,
+g(z2) = z3,
+g(z3) = z1
+(14.75)
+such that
+V0,3(V3, V1, V2) = ⟨g ◦ f1 ◦ V3(0)g ◦ f2 ◦ V1(0)g ◦ f3 ◦ V2(0)⟩.
+(14.76)
+While the state Vi is correctly inserted at the puncture zi in this expression, this is not
+sufficient to guarantee the equality of the amplitudes. Indeed the fibre is defined by the
+complete functions fi(w) and not only by their values at w = 0. For this reason the
+amplitudes can be equal only if
+g ◦ f1 = f2,
+g ◦ f2 = f3,
+g ◦ f3 = f1.
+(14.77)
+This provides constraints on the functions fi, but it is often not possible to solve them.
+If the constraints cannot be solved, then one must introduce a general section. In
+this case a generalized section will be made of 6 sections S(a) (a = 1, . . . , 6) because
+there are 6 permutations. Then the amplitude reads
+V0,3(V1, V2, V3) = 1
+6
+6
+�
+a=1
+ω0,3
+0 (V1, V2, V3)|S(a)
+0,3 .
+(14.78)
+224
+
+Figure 14.10: A generalized section {S(a)
+0,3} (a = 1, . . . , 6) of P0,3 for the 3-point vertex. This
+is to be compared with Figure 14.4a.
+When computing the Feynman graphs by gluing lower-dimensional amplitudes, it is
+possible that parts of the section overlap, meaning that several graphs cover the same part
+of the moduli space. In this case, the fundamental vertex should be defined as a negative
+contribution in the overlap region. This procedure is perfectly well-defined since all graphs
+are finite and there is no ambiguity. In practice, it is always simpler to work with non-
+overlapping sections (i.e. a single covering of the moduli space). A simple way to prevent
+overlaps is to tune the stub parameter s0 to a large value.
+By construction, the integral over Vg,n should be finite. If this is not the case, it means
+that the propagator graphs also diverge and that the parametrization is not good. This can
+also be solved by considering a sufficiently large value of the stub parameter s0.
+14.4
+Suggested readings
+• Plumbing fixture and amplitude factorization [193, sec. 9.3, 9.4, 256, sec. 6].
+225
+
+Chapter 15
+Closed string field theory
+Abstract
+We bring together the elements from the previous chapters in order to write
+the closed string field action. We first study the gauge fixed theory before reintroducing the
+gauge invariance. We then prove that the action satisfies the BV master equation meaning
+that closed SFT is completely consistent at the quantum level. Finally, we describe the 1PI
+effective action.
+15.1
+Closed string field expansion
+In Chapters 11, 13 and 14, constraints on the external and internal states were found to
+be necessary. But, to provide another perspective and decouples the properties of the field
+from the ones of the state, we assume that the string field does not obey any constraint.
+They will be derived later in order to reproduce the scattering amplitudes from the action
+and to make the latter well-defined.
+The string field is expanded on a basis {φr} of the CFT Hilbert space H (see Section 11.2
+for more details)
+|Ψ⟩ =
+�
+r
+ψr |φr⟩ .
+(15.1)
+Using the decomposition (11.40) of the Hilbert space according to the ghost zero-modes, the
+string field can also be expanded as
+|Ψ⟩ =
+�
+r
+�
+ψ↓↓,r |φ↓↓,r⟩ + ψ↓↑,r |φ↓↑,r⟩ + ψ↑↓,r |φ↑↓,r⟩ + ψ↑↑,r |φ↑↑,r⟩
+�
+,
+(15.2)
+where we recall that the basis states satisfy
+b0 |φ↓↓,r⟩ = ¯b0 |φ↓↓,r⟩ = 0,
+b0 |φ↓↑,r⟩ = ¯c0 |φ↓↑,r⟩ = 0,
+c0 |φ↑↓,r⟩ = ¯b0 |φ↑↓,r⟩ = 0,
+c0 |φ↑↑,r⟩ = ¯c0 |φ↑↑,r⟩ = 0.
+(15.3)
+We recall the definition of the dual basis {φc
+r} through the BPZ inner product
+⟨φc
+r|φs⟩ = δrs.
+(15.4)
+In terms of the ghost decomposition, the components of the dual states satisfy:
+⟨φc
+↓↓,r| c0 =⟨φc
+↓↓,r| ¯c0 = 0,
+⟨φc
+↓↑,r| c0 =⟨φc
+↓↑,r|¯b0 = 0,
+⟨φc
+↑↓,r| b0 =⟨φc
+↑↓,r| ¯c0 = 0,
+⟨φc
+↑↑,r| b0 =⟨φc
+↑↑,r|¯b0 = 0,
+⟨φc
+x,r|φy,s⟩ = δxyδrs,
+(15.5)
+226
+
+where x, y =↓↓, ↑↓, ↓↑, ↑↑. The spacetime ghost number of the fields ψr is defined by
+G(ψr) = 2 − nr.
+(15.6)
+Remember that the ghost number of the basis states are denoted by
+nr = Ngh(φr),
+nc
+r = Ngh(φc
+r) = 6 − nr.
+(15.7)
+15.2
+Gauge fixed theory
+Having built the kinetic term (Chapter 9), one needs to construct the interactions. For
+the same reason – our ignorance of SFT first principles – that forced us to start with the
+free equation of motion to derive the quadratic action (Chapter 10), we also need to infer
+the interactions from the scattering amplitudes. Preparing the stage for this analysis was
+the goal of Chapter 14, where we introduced the factorization of amplitudes to derive the
+fundamental interactions.
+Scattering amplitudes are expressed in terms of gauge fixed states since only them are
+physical. This allows to give an alternative derivation of the kinetic term by defining it as
+the inverse of the propagator, which is well-defined for gauge fixed states.1 The price to
+pay by constructing interactions in this way is that the SFT action itself is gauge fixed. To
+undercover its deeper structure it is necessary to release the gauge fixing condition. In view
+of the analysis of the quadratic action in Chapter 10, we can expect that the BV formalism
+is required. Another possibility is to consider directly the 1PI action.
+In this section, we first derive the kinetic term by inverting the propagator. For this to
+be possible, the string field must obey some constraints: we will find that they correspond
+to the level-matching and Siegel gauge conditions. Then, we introduce the interactions into
+the action.
+15.2.1
+Kinetic term and propagator
+In Chapter 14, it was found that the propagator reads (14.41):
+∆ = b+
+0 b−
+0
+1
+L+
+0
+δL−
+0 ,0,
+∆rs =⟨φc
+r| b+
+0 b−
+0
+1
+L+
+0
+δL−
+0 ,0 |φc
+s⟩ .
+(15.8)
+The most natural guess for the kinetic term is
+S0,2 = 1
+2 ⟨Ψ| K |Ψ⟩ = 1
+2 ψrKrsψs
+(15.9)
+where
+K = c−
+0 c+
+0 L+
+0 δL−
+0 ,0
+Krs =⟨φr| c−
+0 c+
+0 L+
+0 δL−
+0 ,0 |φs⟩ .
+(15.10)
+Indeed, it looks like K∆ = 1 using the identities c±
+0 b±
+0 ∼ 1 and it matches (10.115). In
+terms of the holomorphic and anti-holomorphic modes, we have
+K = 1
+2 c0¯c0L+
+0 δL−
+0 ,0.
+(15.11)
+But, when writing c±
+0 b±
+0 ∼ 1, the second part of the anti-commutator {b±
+0 , c±
+0 } = 1 is
+missing. The relation c±
+0 b±
+0 ∼ 1 is correct only when acting on basis dual states annihilated
+1This step is not necessary because the propagator corresponding to the plumbing fixture (Section 14.2.2)
+matches the one found in Section 10.5 by considering the simplest gauge fixing. However, this would have
+been necessary if the factorization had given another propagator, or if the structure of the theory was more
+complicated, for example for the superstring.
+227
+
+by c±
+0 .
+The problem stems from the fact that Ψ is not yet subject to any constraint.
+Moreover, some of the string field components will not appear in the expression since they
+are annihilated by the ghost zero-mode. As a consequence, the kinetic operator in (15.10)
+(or equivalently the propagator) is not invertible in the Hilbert space H because its kernel
+is not empty:
+ker K|H ̸= ∅.
+(15.12)
+This can be seen by writing φr as a 4-vector and Krs as a 4 × 4-matrix:
+Krs = 1
+2
+�
+�
+�
+�
+⟨φ↓↓,r|
+⟨φ↓↑,r|
+⟨φ↑↓,r|
+⟨φ↑↑,r|
+�
+�
+�
+�
+t �
+�
+�
+�
+c0¯c0L+
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+�
+�
+�
+�
+�
+�
+�
+�
+|φ↓↓,s⟩
+|φ↓↑,s⟩
+|φ↑↓,s⟩
+|φ↑↑,s⟩
+�
+�
+�
+� .
+(15.13)
+The matrix is mostly empty because the states φx,r with different x =↓↓, ↑↓, ↓↑, ↑↑ are
+orthogonal (no non-diagonal terms) and the states with x ̸=↓↓ are annihilated by c0 or ¯c0.
+The same consideration applies for the delta-function: if the field does not satisfy L−
+0 = 0,
+then the kinetic operator is non-invertible.
+To summarize the string field must satisfy three conditions in order to have an invertible
+kinetic term
+L−
+0 |Ψ⟩ = 0,
+b−
+0 |Ψ⟩ = 0,
+b+
+0 |Ψ⟩ = 0.
+(15.14)
+This means that the string field is expanded on the H0 ∩ ker L−
+0 Hilbert space:
+|Ψ⟩ =
+�
+r
+ψ↓↓,r |φ↓↓,r⟩ .
+(15.15)
+Ill-defined kinetic terms are expected in the presence of a gauge symmetry: this was already
+discussed in Sections 10.1.4 and 10.5 for the free theory, and this will be discussed further
+later in this chapter for the interacting case.
+Computation
+Let’s check that Krs is correctly the inverse of ∆rs when Ψ is restricted to H0:
+Krs∆st =⟨φr| c−
+0 c+
+0 L+
+0 δL−
+0 ,0 |φs⟩⟨φc
+s| b+
+0 b−
+0
+1
+L+
+0
+δL−
+0 ,0 |φc
+t⟩
+=⟨φr| c−
+0 c+
+0 L+
+0 δL−
+0 ,0b+
+0 b−
+0
+1
+L+
+0
+δL−
+0 ,0 |φc
+t⟩
+=⟨φr| {c−
+0 , b−
+0 }{c+
+0 , b+
+0 } |φc
+t⟩
+= ⟨φr|φc
+t⟩ = δrt.
+The second equality follows from the resolution of the identity (11.36): due to the zero-
+mode insertions, the resolution of the identity collapses to a sum over the ↓↓ states
+1 =
+�
+r
+|φr⟩⟨φc
+r| =
+�
+r
+|φ↓↓,r⟩⟨φc
+↓↓,r| .
+(15.16)
+The third equality uses that L+
+0 commutes with the ghost modes, that φr is annihilated
+by b±
+0 , and that (δL−
+0 ,0)2 = δL−
+0 ,0 = 1 on states with L−
+0 = 0.
+Finally, we find that the kinetic term matches the classical quadratic vertex V0,2 defined
+in (14.56) such that
+S0,2 = 1
+2 V0,2(Ψ2) = 1
+2 ⟨Ψ| c−
+0 c+
+0 L+
+0 δL−
+0 ,0 |Ψ⟩ .
+(15.17)
+228
+
+15.2.2
+Interactions
+The second step to build the action is to write the interaction terms from the Feynman rules.
+Before proceeding to SFT, it is useful to remember how this works for a standard QFT.
+Example 15.1 – Feynman rules for a scalar field
+Consider a scalar field with a standard kinetic term and a n-point interaction:
+S =
+�
+dDx
+�1
+2 φ(x)(−∂2 + m2)φ(x) + λ
+n! φ(x)n
+�
+.
+(15.18)
+First, one needs to find the physical states, which correspond to solutions of the lin-
+earised equation of motion. In the current case, they are plane-waves (in momentum
+representation):
+φk(x) = eik·x.
+(15.19)
+Then, the vertex (in momentum representation) Vn(k1, . . . , kn) is found by replacing
+in the interaction each occurrence of the field by a different state, and summing over
+all the different contributions. Here, this means that one considers states φki(x) with
+different momenta:
+Vn(k1, . . . , kn) = λ
+n!
+�
+dDx n!
+n
+�
+i=1
+φki(x) = λ
+�
+dDx ei(k1+···+kn)x
+= λ(2π)D δ(D)(k1 + · · · + kn).
+(15.20)
+The factor n! comes from all the permutations of the n states in the monomial of order
+n. Reversing the argument, one sees how to move from the vertex Vn(k1, . . . , kn) written
+in terms of states to the interaction in the action in terms of the field.
+Obviously, if the field has more states (for example if it has a spin or if it is in a
+representation of a group), then one needs to consider all the different possibilities. The
+above prescription also yields directly the insertion of the momentum necessary if the
+interaction contains derivatives.
+In Section 14.2, the Feynman rule for a g-loop n-point fundamental vertex of states
+(V1, . . . , Vn) was found to be given by (14.33):
+Vg,n(V1, . . . , Vn) =
+�
+Rg,n
+ωg,n
+Mg,n(V1, . . . , Vn) =
+(15.21)
+where Rg,n is a section over the fundamental region Vg,n ⊂ Mg,n (12.42b) which cannot be
+covered from the plumbing fixture of lower-dimensional surfaces.
+From the example Example 15.1, it should be clear that the g-loop n-point contribution
+to the action can be obtained simply by replacing every state with a string field in Vg,n:
+Sg,n = ℏg g2g−2+n
+s
+n!
+Vg,n(Ψn).
+(15.22)
+where Ψn := Ψ⊗n.
+The power of the coupling constant has been reinstated: it can be
+motivated by the fact that it should have the same power as the corresponding amplitude
+(Section 3.1.1). Note that the interactions are defined only when the power of gs is positive:
+229
+
+χg,n = 2 − 2g − n < 0. We have also written explicitly the power of ℏ, which counts the
+number of loops.
+Before closing this section, we need to comment on the effect of the constraints (15.14)
+on the interactions. Building a Feynman graph by gluing two m- and n-point interactions
+with a propagator, one finds that the states proportional to φx,r for x ̸=↓↓ do not propagate
+inside internal legs
+Vg,m(V1, . . . , Vm−1, φr)⟨φc
+r| b+
+0 b−
+0
+1
+L+
+0
+|φc
+s⟩ Vg′,n(W1, . . . , Wn−1, φs)
+= Vg,m(V1, . . . , Vm−1, φ↓↓,r)⟨φc
+↓↓,r| b+
+0 b−
+0
+1
+L+
+0
+|φc
+↓↓,s⟩ Vg′,n(W1, . . . , Wn−1, φ↓↓,s).
+(15.23)
+Thus, they do not contribute to the final result even if the interactions contain them. While
+the conditions L−
+0 = b−
+0 = 0 were found to be necessary for defining off-shell amplitudes, the
+condition b+
+0 = 0 does not arise from any consistency requirement. But, it is also consistent
+with the interactions, since only fundamental vertices have a chance to give a non-vanishing
+result for states which do not satisfy (15.14). Hence, the interactions (15.22) are compatible
+with the definition of the kinetic term and the restriction of the string field.
+15.2.3
+Action
+The interacting gauge-fixed action is built from the kinetic term V0,2 (15.17) and from the
+interactions Vg,n (15.22) with χg,n < 0. However, this is not sufficient: we have seen in
+Section 14.3 that it makes sense to consider the vertices with χg,n ≥ 0. First, we should
+consider the 1-loop cosmological constant V1,0. Then, we can also add the classical source
+V0,1 and the tree-level cosmological constant V0,0. With all the terms together, the action
+reads:
+S =
+�
+g,n≥0
+ℏg g2g−2+n
+s
+n!
+Vg,n(Ψn)
+:= 1
+2 ⟨Ψ| c−
+0 c+
+0 L+
+0 δL−
+0 ,0 |Ψ⟩ +
+�′
+g,n≥0
+ℏg g2g−2+n
+s
+n!
+Vg,n(Ψn).
+(15.24)
+where Vn was defined in (14.58). A prime on the sum indicates that the term g = 0, n = 2
+is removed, such that one can single out the kinetic term. We will often drop the delta
+function imposing L−
+0 = 0 because the field are taken to satisfy this constraint.
+Rewriting the vertices in terms of the products ℓg,n defined in (14.65)
+Vg,n(Ψn) :=⟨Ψ| c−
+0
+��ℓg,n−1(Ψn−1)
+�
+(15.25)
+leads to the alternative form
+S =
+�
+g,n≥0
+ℏg g2g−2+n
+s
+n!
+⟨Ψ| c−
+0
+��ℓg,n−1(Ψn−1)
+�
+.
+(15.26)
+The definition (14.56) leads to the following explicit expression for ℓ0,1:
+ℓ0,1(Ψcl) = c+
+0 L+
+0 |Ψcl⟩ .
+(15.27)
+In most cases, the terms g = 0, n = 0, 1 vanish such that the action reads:
+S =
+�
+g,n≥0
+χg,n≤0
+ℏg g2g−2+n
+s
+n!
+Vg,n(Ψn).
+(15.28)
+230
+
+However, we will often omit the condition χg,n ≤ 0 to simplify the notation, except when the
+distinction is important, and the reader can safely assumes V0,0 = V0,1 = 0 if not otherwise
+stated. The classical action is obtained by setting ℏ = 0:
+Scl = 1
+2 ⟨Ψcl| c−
+0 c+
+0 L+
+0 |Ψcl⟩ +
+�
+n≥3
+gn
+s
+n! V0,n(Ψn
+cl).
+(15.29)
+Rescaling the string field by g−1
+s
+gives the more canonical form of the action (using the
+same symbol):
+S =
+�
+g,n≥0
+ℏgg2g−2
+s
+1
+n! Vg,n(Ψn)
+:=
+1
+2g2s
+⟨Ψ| c−
+0 c+
+0 L+
+0 δL−
+0 ,0 |Ψ⟩ + 1
+g2s
+�′
+g,n≥0
+(ℏg2
+s)g
+n!
+Vg,n(Ψn).
+(15.30)
+In the path integral, the action is divided by ℏ such that
+S
+ℏ =
+�
+g,n≥0
+(ℏg2
+s)g−1 1
+n! Vg,n(Ψn).
+(15.31)
+This shows that there is a single coupling constant ℏg2
+s, instead of two (ℏ and gs separately)
+as it looks at the first sight. This makes sense because gs is in fact the expectation value
+of the dilaton field (2.166) and its value can be changed by deforming the background with
+dilatons [16, 17, 197].
+The previous remark also allows to easily change the normalization of the action, for
+example, to perform a Wick rotation, to normalize canonically the action in terms of space-
+time fields, or reintroduce ℏ. Rescaling the action by α is equivalent to rescale g2
+s by α−1:
+S → α S
+=⇒
+g2
+s → g2
+s
+α .
+(15.32)
+The linearized equation of motion is:
+L+
+0 |Ψ⟩ = 0,
+(15.33)
+which corresponds to the Siegel gauge equation of motion of the free theory (10.116). Hence,
+this equation is not sufficient to determine the physical states (cohomology of the BRST op-
+erator, Chapter 8), as discussed in Chapter 10, and additional constraints must be imposed.
+One can interpret this by saying that the action (15.24) provides only the Feynman rules,
+not the physical states. Removing the gauge fixing will be done in Sections 15.3 and 15.4.
+The action (15.24) looks overly more complicated than a typical QFT theory: instead
+of few interaction terms for low n (n ≤ 4 in d = 4 renormalizable theories), it has contact
+interactions of all orders n ∈ N. The terms with g ≥ 1 are associated to quantum corrections
+as indicate the power of ℏ, which means that they can be interpreted as counter-terms. But,
+how is it that one needs counter-terms despite the claim that every Feynman graphs (includ-
+ing the fundamental vertices) in SFT are finite? The role of renormalization is not only to
+cure UV divergences, but also IR divergences (due to vacuum shift and mass renormaliza-
+tion). Equivalently, this can be understood by the necessity to correct the asymptotic states
+of the theory, or to consider renormalized instead of bare quantities. Indeed, the asymptotic
+states obtained from the linearized classical equations of motion are idealization: turning
+on interactions modify the states. In typical QFTs, these corrections are infinite and renor-
+malization is crucial to extract a number; however, even if the effect is finite, it is needed to
+describe correctly the physical quantities [250, p. 411]. There is a second reason for these
+231
+
+additional terms: when relaxing the gauge fixing condition, the path integral is anomalous
+under the gauge symmetry, and the terms with g > 0 are necessary to cancel the anomaly
+(this will be discussed more precisely in Section 15.4). It may thus seem that SFT cannot
+be predictive because of the infinite number of counter-terms. Fortunately, this is not the
+case: the main reason for the loss of predictability in non-renormalizable theory is that
+the renormalization procedure introduces an infinite number2 of arbitrary parameters (and
+thus making a prediction would require to have already made an infinite number of observa-
+tions to determine all the parameters). These parameters come from the subtraction of two
+infinities: there is no unique way to perform it and thus one needs to introduce a new para-
+meter. The case of SFT is different: since every quantity is finite, the renormalization has
+no ambiguity because one subtracts two finite numbers, and the result is unambiguous. As
+a consequence, renormalization does not introduce any new parameter and there is a unique
+coupling constant gs in the theory, which is determined by the tree-level cubic interaction.
+The coupling constants of higher-order and higher-loop interactions are all determined by
+powers of gs, and thus a unique measurement is sufficient to make predictions.
+Another important point is that the action (15.24) is not uniquely defined. The definition
+of the vertices depends on the choice of the local coordinates and of the stub parameter s0.
+Changing them modifies the vertices, and thus the action.
+But, one can show that the
+different theories are related by field redefinitions and are thus equivalent.
+15.3
+Classical gauge invariant theory
+In the previous section, we have found the gauge fixed action (15.24). Since the complete
+gauge invariant quantum action has a complicated structure, it is instructive to first focus
+on the classical action (15.29). The full action is discussed in Section 15.4.
+The gauge fixing is removed by relaxing the b+
+0 = 0 constraint on the field (the other
+constraints must be kept in order to have well-defined the interactions). The classical field
+Ψcl is then defined by:
+Ψcl ∈ H− ∩ ker L−
+0 ,
+Ngh(Ψcl) = 2.
+(15.34)
+The restriction on the ghost number translates the condition that the field is classical, i.e.
+that there are no spacetime ghosts at the classical level. The relation (15.6) implies that all
+components have vanishing spacetime ghost number.
+In the free limit, the gauge invariant action should match (10.105)
+S0,2 = 1
+2 ⟨Ψ| c−
+0 QB |Ψ⟩ .
+(15.35)
+and lead to the results from Section 10.5. A natural guess is that the form of the interactions
+is not affected by the gauge fixing (the latter usually modifies the propagator but not the
+interactions). This leads to the gauge invariant classical action:
+Scl = 1
+2 ⟨Ψcl| c−
+0 QB |Ψcl⟩ + 1
+g2s
+�
+n≥3
+gn
+s
+n! V0,n(Ψn
+cl),
+(15.36)
+where the vertices V0,n with n ≥ 3 are the ones defined in (14.33) (we consider the case
+where V0,0 = V0,1 = 0). It is natural to generalize the definition of V0,2 as:
+V0,2(Ψ2
+cl) :=⟨Ψcl| c−
+0 QB |Ψcl⟩
+(15.37)
+2In practice, this number does not need to be infinite to wreck predictability, it is sufficient that it is
+very large.
+232
+
+such that
+Scl = 1
+g2s
+�
+n≥2
+gn
+s
+n! V0,n(Ψn
+cl) = 1
+g2s
+�
+n≥2
+gn
+s
+n! ⟨Ψcl| c−
+0
+��ℓ0,n−1(Ψn−1
+cl
+)
+�
+,
+(15.38)
+where (15.37) implies:
+ℓ0,1(Ψcl) = QB |Ψcl⟩ .
+(15.39)
+The equation of motion is
+Fcl(Ψcl) :=
+�
+n≥1
+gn−1
+s
+n!
+ℓ0,n(Ψn
+cl) = QB |Ψcl⟩ +
+�
+n≥2
+gn−1
+s
+n!
+ℓ0,n(Ψn
+cl) = 0.
+(15.40)
+Computation – Equation (15.40)
+δScl = 1
+g2s
+�
+n≥2
+gn
+s
+n! n{δΨcl, Ψn−1
+cl
+}0 = 1
+g2s
+�
+n≥2
+gn
+s
+(n − 1)! ⟨δΨcl| c−
+0
+��ℓ0,n−1(Ψn−1
+cl
+�
+.
+(15.41)
+The first equality follows because the vertex is completely symmetric. Simplifying and
+shifting n, one obtains c−
+0 |Fcl⟩. The factor c−
+0 is invertible because of the constraint
+b−
+0 = 0 imposed on the field.
+The action is invariant
+δΛScl = 0
+(15.42)
+under the gauge transformation
+δΛΨcl =
+�
+n≥0
+gn
+s
+n! ℓ0,n+1(Ψn
+cl, Λ) = QB |Λ⟩ +
+�
+n≥1
+gn
+s
+n! ℓ0,n+1(Ψn
+cl, Λ).
+(15.43)
+The gauge algebra is [262, sec. 4]:
+[δΛ2, δΛ1]Ψcl = δΛ(Λ1,Λ2,Ψcl) |Ψcl⟩ +
+�
+n≥0
+gn+2
+s
+n!
+ℓ0,n+3
+�
+Ψn
+cl, Λ2, Λ1, Fcl(Ψcl)
+�
+,
+(15.44a)
+where Fcl is the equation of motion (15.40), and Λ(Λ1, Λ2, Ψcl) is a field-dependent gauge
+parameter:
+Λ(Λ1, Λ2, Ψcl) =
+�
+n≥0
+gn+1
+s
+n!
+ℓ0,n+2(Λ1, Λ2, Ψn
+cl)
+= gs ℓ0,2(Λ1, Λ2) +
+�
+n≥1
+gn+1
+s
+n!
+ℓ0,n+2(Λ1, Λ2, Ψn
+cl).
+(15.44b)
+The classical gauge algebra is complicated which explains why a direct quantization (for
+example through the Faddeev–Popov procedure) cannot work: the second term in (15.44a)
+indicates that the algebra is open (it closes only on-shell), while the first term is a gauge
+transformation with a field-dependent parameter. As reviewed in Appendix C.3, both prop-
+erties require using the BV formalism for the quantization, and the latter is performed in
+Section 15.4. An important point is that if the theory had only cubic interactions, i.e. if
+∀n ≥ 4 :
+V0,4(V1, . . . , Vn) = 0,
+ℓg,n−1(V1, . . . , Vn−1) = 0,
+(cubic theory), (15.45)
+then the algebra closes off-shell and Λ(Λ1, Λ2, Ψcl) becomes field-independent.
+233
+
+Computation – Equation (15.42)
+δΛScl =
+�
+n≥2
+gn−2
+s
+n!
+nV0,n(δΨcl, Ψn−1
+cl
+) =
+�
+m,n≥0
+gm+n−1
+s
+m! n!
+V0,n+1
+�
+ℓ0,m+1(Ψm
+cl , Λ), Ψn
+cl
+�
+=
+�
+m≥0
+m
+�
+n=0
+gm−1
+s
+(m − n)! n!
+�
+ℓm−n+1(Ψm−n
+cl
+, Λ)
+�� c−
+0 |ℓ0,n(Ψn
+cl)⟩ .
+For simplicity we have extended the sum up to n = 0 and m = 0 by using the fact that
+lower-order vertices vanish. The bracket can be rewritten as
+= ⟨ℓ0,n(Ψn
+cl)| c−
+0
+��ℓ0,m−n+1(Ψm−n
+cl
+, Λ)
+�
+= V0,m−n+2
+�
+ℓ0,n(Ψn
+cl), Ψm−n
+cl
+, Λ
+�
+= −V0,m−n+2
+�
+Λ, ℓ0,n(Ψn
+cl), Ψm−n
+cl
+�
+= ⟨Λ| c−
+0
+��ℓ0,m−n+1
+�
+ℓ0,n(Ψn
+cl), Ψm−n
+cl
+��
+.
+Then, one needs to use the identity (defined for all m ≥ 0)
+0 =
+m
+�
+n=0
+m!
+(m − n)! n! ℓ0,m−n+1
+�
+ℓ0,n(Ψn
+cl), Ψm−n
+cl
+�
+,
+(15.46)
+which comes from (14.70). Multiplying this by gm−1
+s
+/m! and summing over m ≥ 0
+proves (15.42).
+Remark 15.1 (L∞ algebra) The identities satisfied by the products ℓ0,n from the gauge
+invariance of the action implies that they form a L∞ homotopy algebra [74, 169, 262] (for
+more general references, see [106, 108, 148, 149]). The latter can also be mapped to a BV
+structure, which explains why the BV quantization Section 15.4 is straightforward. This
+interplay between gauge invariance, covering of the moduli space, BV and homotopy algebra
+is particularly beautiful. It has also been fruitful in constructing super-SFT.
+15.4
+BV theory
+As indicated in the previous section (Section 15.3), the classical gauge algebra is open and
+has field-dependent structure constants. The BV formalism (Appendix C.3) is necessary to
+define the theory.
+In the BV formalism, the classical action for the physical fields is extended to the
+quantum master action by solving the quantum master equation (C.114).
+It is generic-
+ally difficult to build this action exactly, but the discussion of Section 10.3 can serve as a
+guide: it was found that the free quantum action (with the tower of ghosts) has exactly the
+same form as the free classical action (without ghosts). Hence, this motivates the ansatz
+that it should be of the same form as the classical action (15.36) to which are added the
+234
+
+counter-terms from (15.24):
+S = 1
+g2s
+�
+g≥0
+ℏgg2g
+s
+�
+n≥0
+gn
+s
+n! Vg,n(Ψn)
+(15.47a)
+= 1
+2 ⟨Ψ| c−
+0 QB |Ψ⟩ +
+�′
+g,n≥0
+ℏgg2g−2+n
+s
+n!
+Vg,n(Ψn)
+(15.47b)
+= 1
+g2s
+�
+g,n≥0
+ℏgg2g−2+n
+s
+n!
+⟨Ψ| c−
+0
+��ℓg,n−1(Ψn−1)
+�
+,
+(15.47c)
+but without any constraint on the ghost number of Ψ:
+Ψ ∈ H− ∩ ker L−
+0 .
+(15.48)
+In order to show that (15.47) is a consistent quantum master action, it is necessary to
+show that it solves the master BV equation (C.114):
+(S, S) − 2ℏ∆S = 0.
+(15.49)
+The first step is to introduce the fields and antifields. In fact, because the CFT ghost number
+induces a spacetime ghost number, there is a natural candidate set.
+The string field is expanded as (15.1)
+|Ψ⟩ =
+�
+r
+ψr |φr⟩ ,
+(15.50)
+where the {φr} forms a basis of H−. The string field can be further separated as:
+Ψ = Ψ+ + Ψ−,
+(15.51)
+where Ψ− (Ψ+) contains only states which have negative (positive) cylinder ghost numbers
+(this gives an offset of 3 when using the plane ghost number):
+Ψ− =
+�
+r
+�
+nr≤2
+|φr⟩ ψr,
+Ψ+ =
+�
+r
+�
+ncr>2
+b−
+0 |φc
+r⟩ ψ∗
+r.
+(15.52)
+The order of the basis states and coefficients matter if they anti-commute. The sum in Ψ+
+can be rewritten as a sum over nr ≤ 2 like the first term since nr +nc
+r = 6. Correspondingly,
+the spacetime ghost numbers (15.6) for the coefficients in Ψ− (Ψ+) are positive (negative)
+G(ψr) ≥ 0,
+G(ψ∗
+r) < 0.
+(15.53)
+Moreover, one finds that the ghost numbers of ψr and ψ∗
+r are related as:
+G(ψ∗
+r) = −1 − G(ψr),
+(15.54)
+which also implies that they have opposite parity. Comparing with Appendix C.3, this shows
+that the ψr (ψ∗
+r) contained in Ψ− (Ψ+) can be identified with the fields (antifields).
+Computation – Equation (15.54)
+G(ψ∗
+r) = 2 − Ngh(b−
+0 φc
+r) = 2 + 1 − nc
+r
+= 3 − (6 − nr) = −3 + (2 − G(ψr)) = −1 − G(ψr).
+235
+
+In terms of fields and antifields, the master action is
+∂RS
+∂ψr
+∂LS
+∂ψ∗r
++ ℏ ∂R∂LS
+∂ψr∂ψ∗r
+= 0.
+(15.55)
+Plugging the expression (15.47) of S inside and requiring that the expression vanishes order
+by order in g and n give the set of equations:
+�
+g1,g2≥0
+g1+g2=g
+�
+n1,n2≥0
+n1+n2=n
+∂RSg1,n1
+∂ψr
+∂LSg2,n2
+∂ψ∗r
++ ℏ ∂R∂LSg−1,n
+∂ψr∂ψ∗r
+= 0,
+(15.56)
+where Sg,n was defined in (15.22). This holds true due to the identity (14.70) (the complete
+proof can be found in [262, pp. 42–45]). The fact that the second term is not identically zero
+means that the measure is not invariant under the classical gauge symmetry (anomalous
+symmetry): corrections need to be introduced to cancel the anomaly. It is a remarkable
+fact that one can construct directly the quantum master action in SFT and that it takes
+the same form as the classical action.
+15.5
+1PI theory
+The BV action is complicated: instead, it is often simpler and sufficient to work with the
+1PI effective action. The latter incorporates all the quantum corrections in 1PI vertices
+such that scattering amplitudes are expressed only in terms of tree Feynman graphs (there
+are no loops in diagrams since they correspond to quantum effects, already included in the
+definitions of the vertices).
+A 1PI graph is a Feynman graph which stays connected if one cuts any single internal
+line. On the other hand, a 1PR graph splits in two disconnected by cutting one of the line.
+The scattering amplitudes Ag,n are built by summing all the different ways to connect two
+1PI vertices with a propagator: diagrams connecting two legs of the same 1PI vertex are
+forbidden by definition.
+The g-loop n-point 1PR and 1PI Feynman diagrams are associated to some regions of
+the moduli space Mg,n. Comparing the previous definitions with the gluing of Riemann
+surfaces (Section 12.3), 1PR diagrams are obtained by gluing surfaces with the separating
+plumbing fixture (Section 14.1.1). Thus, the 1PR and 1PI regions F1PR
+g,n
+and V1PI
+g,n can be
+identified with the regions defined in (12.43a) and (12.43b). In particular, the n-point 1PI
+interaction is the sum over g of the g-loop n-point 1PI interactions (14.63):
+V1PI
+n
+(V1, . . . , Vn) :=
+:=
+�
+g≥0
+(ℏg2
+s)g V1PI
+g,n (V1, . . . , Vn),
+V1PI
+g,n (V1, . . . , Vn) :=
+�
+R1PI
+g,n
+ωg,n
+Mg,n(V1, . . . , Vn),
+(15.57)
+where R1PI
+g,n is a section of Pg,n over V1PI
+g,n .
+Given the interactions vertices, it is possible to follow the same reasoning as in Sec-
+tions 15.2 and 15.3.
+236
+
+The gauge fixed 1PI effective action reads:
+S1PI = 1
+g2s
+�
+n≥0
+gn
+s
+n! V1PI
+n
+(Ψn) := 1
+2 ⟨Ψ| c−
+0 c+
+0 L+
+0 |Ψ⟩ + 1
+g2s
+�′
+n≥0
+gn
+s
+n! V1PI
+n
+(Ψn).
+(15.58)
+Here, the prime means again that the term g = 0, n = 2 is excluded from the definition of
+V1PI
+2
+. The action has the same form as the classical gauge fixed action (15.29), which is
+logical since it generates only tree-level Feynman graphs. For this reason the vertices V1PI
+n
+have exactly the same properties as the brackets V0,n. This fact can be used to write the
+1PI gauge invariant action:
+S1PI = 1
+2 ⟨Ψ| c−
+0 QB |Ψ⟩ + 1
+g2s
+�′
+n≥0
+gn
+s
+n! V1PI
+n
+(Ψn),
+(15.59)
+which mirrors the classical gauge invariant action (15.36). Then it is straightforward to see
+that it enjoys the same gauge symmetry upon replacing the tree-level vertices by the 1PI
+vertices. But, since this action incorporates all quantum corrections this also proves that
+the quantum theory is correctly invariant under a quantum gauge symmetry.
+Remark 15.2 The 1PI action (15.59) can also be directly constructed from the BV action
+(15.47).
+15.6
+Suggested readings
+• Gauge fixed and classical gauge invariant closed SFT [262] (see also [138, 139]).
+• BV closed SFT [262] (see also [245]).
+• Construction of the open–closed BV SFT [264].
+• 1PI SFT [213, 214, 42, sec. 4.1, 5.2].
+237
+
+Chapter 16
+Background independence
+Abstract
+Spacetime background independence is a fundamental property of any candidate
+quantum gravity theory. In this chapter, we outline the proof of background independence
+for the closed SFT by proving that the equations of motion of two background related by a
+marginal deformation are equivalent after a field redefinition.
+16.1
+The concept of background independence
+Background independence means that the formalism does not depend on the background
+– if any – used to write the theory. A dependence in the background would imply that
+there is a distinguished background among all possibilities, which seems in tension with the
+dynamics of spacetime and the superposition principle from quantum mechanics. Moreover,
+one would expect a fundamental theory to tell which backgrounds are consistent and that
+they could be derived instead of postulated. Background independence allows spacetime to
+emerge as a consequence of the dynamics of the theory and of its defining fundamental laws.
+Background independence can be manifest or not. In the second case, one needs to fix a
+background to define the theory, but the dynamics on different backgrounds are physically
+equivalent.1 This implies that two theories with different backgrounds can be related, for
+example by a field redefinition.
+While fields other than the metric can also be expanded around a background, no diffi-
+culty is expected in this case. Indeed, the topic of background independence is particularly
+sensible only for the metric because it provides the frame for all other computations – and
+in particular for the questions of dynamics and quantization. Generally, these questions are
+subsumed into the problem of the emergence of time in a generally covariant theory. In
+the previous language, QFTs without gravity are (generically) manifestly background inde-
+pendent after minimal coupling.2 For example, a classical field theory is defined on a fixed
+Minkowski background and a well-defined time is necessary to perform its quantization and
+to obtain a QFT, but it is not needed to choose a background for the other fields. For this
+reason, the extension of a QFT on a curved background is generally possible if the space-
+time is hyperbolic, implying that there is a distinguished time direction. But the coupling
+to gravity is difficult and restricted to a (semi-)classical description.
+What is the status of background independence in string theory? The worldsheet for-
+mulation requires to fix a background (usually Minkowski) to quantize the theory and to
+compute scattering amplitudes. Thus, the quantum theory is at least not manifestly back-
+1This does not mean that the physics in all backgrounds are identical, but that the laws are. Hence, a
+computation made in one specific background can be translated into another background.
+2However, non-minimal coupling terms may be necessary to make the theory physical.
+238
+
+ground independent. On the other hand, the worldsheet action can be modified to a generic
+CFT including a generic non-linear sigma model describing an arbitrary target spacetime.
+Conformal invariance reproduces (at leading order) Einstein equations coupled to various
+matter and gauge field equations of motion. From this point of view, the classical theory can
+be written as a manifestly background independent theory, and this provides hopes that the
+quantum theory may be also background independent, even if non-manifestly. This idea is
+supported by other definitions of string theory (e.g. through the AdS/CFT conjecture – and
+other holographic realizations – or through matrix models) which provide, at least partially,
+background independent formulations.
+Ultimately, the greatest avenue to establish the background independence is string field
+theory. Indeed, the form of the SFT action and of its properties (gauge invariance, equa-
+tion of motion. . . ) are identical irrespective of the background [234]. This provides a good
+starting point. The background dependence enters in the precise definition of the string
+products (BRST operator and vertices). The origin of this dependence lies in the derivation
+of the action (Chapters 14 and 15): one begins with a particular CFT describing a given
+background (spacetime compactifications, fluxes, etc.) and defines the vertices from correl-
+ation functions of vertex operators, and the Hilbert space from the CFT operators. As a
+consequence, even though it is clear that no specific property of the background has been
+used in the derivation – and that the final action describes SFT for any background –, this
+is not sufficient to establish background independence. Since the theory assumes implicitly
+a background choice, one cannot guarantee that the physical quantities have no residual
+dependence in the background, even if the action looks superficially background independ-
+ent. Background independence in SFT is thus the statement that theories characterized by
+different CFTs can be related by a field redefinition.
+In this chapter, we will sketch the proof of background independence for backgrounds
+related by marginal deformations.3 It is possible to prove it at the level of the action [232,
+233], or at the level of the equations of motion [226]. The advantage of the second approach
+is that one can use the 1PI theory, which simplifies vastly the analysis. It also generalizes
+directly to the super-SFT.
+Remark 16.1 (Field theory on the CFT space) As mentioned earlier, the string field
+is defined as a functional on the state space of a given CFT and not as a functional on the
+field theory space (off-shell states would correspond to general QFTs, only on-shell states are
+CFTs). In this case, background independence would amount to reparametrization invariance
+of the action in the theory space, and would thus almost automatically hold. A complete
+formulation of SFT following this line is currently out of reach, but some ideas can be found
+in [255].
+16.2
+Problem setup
+Given a SFT on a background, there are two ways to describe it on another background:
+• deform the worldsheet CFT and express the SFT on the new background;
+• expand the original action around the infinitesimal classical solution (to the linearised
+equations of motion) corresponding to the deformation.
+Background independence amounts to the equivalence of both theories up to a field redefin-
+ition. The derivation can be performed at the level of the action or of the equations of
+motion. To prove the background independence at the quantum level, one needs to take
+into account the changes in the path integral measure or to work with the 1PI action.
+3An alternative approach based on morphism of L∞ algebra is followed in [169].
+239
+
+The simplest case is when the two CFTs are related by an infinitesimal marginal deform-
+ation
+δScft = λ
+2π
+�
+d2z ϕ(z, ¯z),
+(16.1)
+with ϕ a (1, 1)-primary operator and λ infinitesimal. The two CFTs are denoted by CFT1
+and CFT2, and quantities associated to each CFTs is indexed with the appropriate number.
+Establishing background independence in this case also implies it for finite marginal
+deformation since they can be built from a series of successive deformations. In the latter
+case, the field redefinition may be singular, which reflects that the parametrization of one
+CFT is not adapted for the other (equivalently, the coordinate systems for the string field
+breaks down), which is expected if both CFTs are far in the field theory space.
+Remember the form of the 1PI action (15.59):
+S1[Ψ1] = 1
+g2s
+�
+�1
+2 ⟨Ψ1| c−
+0 QB |Ψ1⟩ +
+�′
+n≥0
+1
+n! V1PI
+n
+(Ψn
+1)
+�
+� ,
+(16.2)
+where the prime indicates that vertices with n < 3 do not include contributions from the
+sphere. In all this chapter, we remove the index 1PI to lighten the notations. The equation
+of motion is:
+F1(Ψ1) = QB |Ψ1⟩ +
+�
+n
+1
+n! ℓn(Ψn
+1) = 0.
+(16.3)
+16.3
+Deformation of the CFT
+Consider the case where the theory CFT1 is described by an action Scft,1[ψ1] given in terms
+of fields ψ1. Then, the deformation of this action by (16.1) gives an action for CFT2:
+Scft,2[ψ1] = Scft,1[ψ1] + λ
+2π
+�
+d2z ϕ(z, ¯z).
+(16.4)
+Correlation functions on a Riemann surface Σ in both theories can be related by expanding
+the action to first order in λ in the path integral:
+��
+i
+Oi(zi, ¯zi)
+�
+2
+=
+�
+exp
+�
+− λ
+2π
+�
+d2z ϕ(z, ¯z)
+� �
+i
+Oi(zi, ¯zi)
+�
+1
+(16.5a)
+≈
+��
+i
+Oi(zi, ¯zi)
+�
+1
+− λ
+2π
+�
+Σ
+d2z
+�
+ϕ(z, ¯z)
+�
+i
+Oi(zi, ¯zi)
+�
+1
+,
+(16.5b)
+where the Oi are operators built from the matter fields ψ1. This expression presents two
+obvious problems. First, the correlation function may diverge when ϕ collides with one of the
+insertions, i.e. when z = zi in the integration. Second, there is an inherent ambiguity: the
+correlation functions are written in terms of operators in the Hilbert space of CFT1, which
+is different from the CFT2 Hilbert space, and there is no canonical isomorphism between
+both spaces.
+Seeing the Hilbert space as a vector bundle over the CFT theory space, the second
+problem can be solved by introducing a connection on this bundle. This allows to relate
+Hilbert spaces of neighbouring CFTs. In fact, the choice of a non-singular connection also
+regularizes the divergences.
+The simplest definition of a connection corresponds to cut unit disks around each operator
+insertions [30, 198, 199, 210, 238]. This amounts to define the variation between the two
+240
+
+correlation functions as:
+δ
+��
+i
+Oi(zi, ¯zi)
+�
+1
+= − λ
+2π
+�
+Σ−∪iDi
+d2z
+�
+ϕ(z, ¯z)
+�
+i
+Oi(zi, ¯zi)
+�
+1
+.
+(16.6)
+The integration is over Σ minus the disks Di = {|wi| ≤ 1} where wi is the local coordinate
+for the insertion Oi. The divergences are cured because ϕ never approaches another operator
+since the corresponding regions have been removed. The changes in the correlation functions
+induce a change in the string vertices denoted by δVn(V1, . . . , Vn).
+The next step consists in computing the deformations of the operator modes. Since it
+involves only a matter operator, the modes in the ghost sector are left unchanged. The
+Virasoro generators change as:
+δLn = λ
+�
+|z|=1
+d¯z
+2πi zn+1ϕ(z, ¯z),
+δ ¯Ln = λ
+�
+|z|=1
+dz
+2πi ¯zn+1ϕ(z, ¯z).
+(16.7)
+As a consequence, the BRST operator changes as
+δQB = λ
+�
+|z|=1
+d¯z
+2πi c(z)ϕ(z, ¯z) + λ
+�
+|z|=1
+d¯z
+2πi ¯c(¯z)ϕ(z, ¯z).
+(16.8)
+One can prove that
+{QB, δQB} = O(λ2)
+(16.9)
+such that the BRST charge QB + δQB in CFT2 is correctly nilpotent if QB is nilpotent in
+CFT1.
+For the deformation to provide a consistent SFT, the conditions b−
+0 = 0 and L−
+0 = 0 must
+be preserved. The first is automatically satisfied since the ghost modes are not modified.
+Considering an weight-(h, h) operator O, one finds
+δL−
+0 |O⟩ = λ
+�
+|z|=1
+d¯z
+2πi z
+�
+p,q
+zp−1¯zq−1 |Op,q⟩ − λ
+�
+|z|=1
+d¯z
+2πi
+�
+p,q
+zp−1¯zq−1 |Op,q⟩ ,
+(16.10)
+where Op,q are the fields appearing in the OPE with ϕ:
+ϕ(z, ¯z)O(0, 0) =
+�
+p,q
+zp−1¯zq−1Op,q(0, 0).
+(16.11)
+The terms with p ̸= q vanish because the contour integrals are performed around circles of
+unit radius centred at the origin. Moreover, the terms p = q are identical and cancel with
+each other, showing that δL−
+0 = 0 when acting on states satisfying L−
+0 = 0.
+The SFT action S2[Ψ1] in the new background reads
+S2[Ψ1] = S1[Ψ1] + δS1[Ψ1]
+(16.12)
+where the change δS1 in the action is induced by the changes in the string vertices:
+δS1[Ψ1] = 1
+g2s
+�
+�1
+2 ⟨Ψ1| c−
+0 δQB |Ψ1⟩ +
+�
+n≥0
+1
+n! δVn(Ψn
+1)
+�
+� .
+(16.13)
+The equation of motion is:
+F2(Ψ1) = F1(Ψ1) + λ δF1(Ψ1) = 0,
+(16.14)
+where F1 is given in (16.3) and
+λ δF1(Ψ1) = δQB |Ψ1⟩ +
+�
+n
+1
+n! δℓn(Ψn
+1).
+(16.15)
+241
+
+16.4
+Expansion of the action
+Given a (1, 1) primary ϕ, a BRST invariant operator is c¯cϕ. Hence the field
+|Ψ1⟩ = λ |Ψ0⟩ ,
+|Ψ0⟩ = c1¯c1(0) |ϕ⟩
+(16.16)
+is a classical solution to first order in λ since the interactions on the sphere are at least
+cubic.
+Separating the string field as the contribution from the (fixed) background and a fluctu-
+ation Ψ′
+|Ψ1⟩ = λ |Ψ0⟩ + |Ψ′⟩ ,
+(16.17)
+the action expanded to first order in λ reads:
+S1[Ψ1] = S1[Ψ0] + S′[Ψ′],
+(16.18)
+where
+S′[Ψ′] = 1
+g2s
+�
+1
+2 ⟨Ψ′| c−
+0 QB |Ψ′⟩ +
+�
+n
+1
+n!
+�
+Vn(Ψ′n) + λ Vn+1(Ψ0, Ψ′n)
+�
+�
+.
+(16.19)
+The equation of motion is:
+F′(Ψ′) := F1(Ψ′) + λ δF′(Ψ′) = 0,
+(16.20)
+where F1 is given in (16.3) and
+δF′(Ψ′) =
+�
+n
+1
+n! ℓn+1(Ψ0, Ψ′n).
+(16.21)
+16.5
+Relating the equations of motion
+In the previous section, we have derived the equations of motion for two different descriptions
+of a SFT obtained after shifting the background: (16.14) arises by deforming the CFT and
+computing the changes in the BRST operator and string products, while (16.20) arises by
+expanding the SFT action around the new background. The theory is background independ-
+ent if both sets of equations (16.14) and (16.20) are related by a (possibly field-dependent)
+linear transformation M(Ψ′) after a field redefinition of Ψ1 = Ψ1(Ψ′):
+F1(Ψ1) + λ δF1(Ψ1) =
+�
+1 + λM(Ψ′)
+��
+F1(Ψ′) + λ δF′(Ψ′)
+�
+,
+(16.22a)
+|Ψ1⟩ = |Ψ′⟩ + λ |δΨ′⟩ .
+(16.22b)
+The zero-order equation is automatically satisfied. To first order, this becomes
+d
+dλF1(Ψ′ + λδΨ′)
+����
+λ=0
++ δF1(Ψ1) − δF′(Ψ′) = M(Ψ′)F1(Ψ′).
+(16.23)
+Taking Ψ′ to be a solution of the original action removes the RHS, such that:
+λ QB |δΨ′⟩ + λ
+�
+n
+1
+n! ℓn+1(δΨ′, Ψ′n) + δQB |Ψ′⟩
++
+�
+n
+1
+n! δℓn(Ψ′n) − λ
+�
+n
+1
+n! ℓn+1(Ψ0, Ψ′n) = 0.
+(16.24)
+242
+
+To simplify the computations, it is simpler to consider the inner product of this quantity
+with an arbitrary state A (assumed to be even):
+∆ := λ ⟨A| c−
+0 QB |δΨ′⟩ + λ
+�
+n
+1
+n! Vn+2(A, δΨ′, Ψ′n) +⟨A| c−
+0 δQB |Ψ′⟩
++
+�
+n
+1
+n! δVn+1(A, Ψ′n) − λ
+�
+n
+1
+n! Vn+2(A, Ψ0, Ψ′n).
+(16.25)
+The goal is to prove the existence of δΨ′ such that ∆ = 0 up to the zero-order equation of
+motion F1(Ψ′) = 0.
+16.6
+Idea of the proof
+In this section, we give an idea of how the proof ends, referring to [226] for the details.
+The first step is to introduce new vertices V′
+0,3 and V′
+n parametrizing the variations of
+the string vertices:
+⟨A| c−
+0 δQB |B⟩ = λ V′
+0,3(Ψ0, B, A),
+δVn(Ψ′n) = λ V′
+n+1(Ψ0, Ψ′n),
+(16.26)
+where the notation (13.19) has been used. Each subspace V′
+g,n is defined such that the LHS
+is recovered upon integrating the appropriate ωg,n over this section segment. Next, the field
+redefinition δΨ′ is parametrized as:
+⟨A| c−
+0 |δΨ′⟩ =
+�
+n
+1
+n! Bn+2(Ψ0, Ψ′n, A).
+(16.27)
+The objective is to prove the existence (and if possible the form) of the subspaces Bn+2.
+Both the vertices V′
+n and Bn admit a genus expansion:
+V′
+n =
+�
+g≥0
+V′
+g,n,
+Bn =
+�
+g≥0
+Bg,n.
+(16.28)
+Plugging the new expressions in (16.25) give:
+∆ = −
+�
+n
+1
+n! Bn+2(Ψ0, Ψ′n, QBA) +
+�
+m,n
+1
+m!n! Bn+2(Ψ0, Ψ′m, ℓn+1(A, Ψ′n))
++
+�
+n
+1
+n! V′
+n+2(A, Ψ0, Ψ′n) −
+�
+n
+1
+n! Vn+2(A, Ψ0, Ψ′n).
+(16.29)
+Next, the BRST identity (13.46) and the equation of motion F1(Ψ′) = 0 allow to rewrite
+the first term as:
+Bn+2(Ψ0, Ψ′n, QBA) = ∂Bn+2(Ψ0, Ψ′n, A) + n Bn+2(Ψ0, Ψ′n−1, QBΨ′, A)
+(16.30a)
+= ∂Bn+2(Ψ0, Ψ′n, A) −
+�
+m
+n
+m! Bn+2(Ψ0, Ψ′n−1, ℓm(Ψ′m), A).
+(16.30b)
+In the second term, the sum over n is shifted. Combining everything together gives:
+∆ =
+�
+n
+1
+n! ∂Bn+2(Ψ0, Ψ′n, A) −
+�
+m,n
+1
+m!n! Bn+3(Ψ0, Ψ′n, ℓm(Ψ′m), A)
++
+�
+m,n
+1
+m!n! Bn+2(Ψ0, Ψ′m, ℓn+1(A, Ψ′n)) +
+�
+n
+1
+n! V′
+n+2(A, Ψ0, Ψ′n)
+−
+�
+n
+1
+n! Vn+2(A, Ψ0, Ψ′n).
+(16.31)
+243
+
+Solving for ∆ = 0 requires that each term with a different power of Ψ′ vanishes independ-
+ently:
+∂Bn+2(Ψ0, Ψ′n, A) = − V′
+n+2(A, Ψ0, Ψ′n) + Vn+2(A, Ψ0, Ψ′n)
++
+�
+m1,m2
+m1+m2=n
+n!
+m1!m2! Bm1+3(Ψ0, Ψ′m1, ℓm2(Ψ′m2), A)
+−
+�
+m1,m2
+m1+m2=n
+n!
+m1!m2! Bm1+2
+�
+Ψ0, Ψ′m1, ℓm2+1(A, Ψ′m2)
+�
+.
+(16.32)
+In order to proceed, one needs to perform a genus expansion of the various spaces:
+this allows to solve recursively for all Bg,n starting from B0,3. One can then build |δΨ′⟩
+recursively, which provides the field redefinition. Indeed, the RHS of this equation contains
+only Bg′,n′ for g′ < g or n′ < n and the equation for B0,3 contains no Bg,n in the RHS.
+It should be noted that the field redefinition is not unique, but there is the freedom of
+performing (infinite-dimensional) gauge transformations. Finding an obstruction to solve
+these equations mean that the field redefinition does not exist, and thus that the theory is
+not background independent
+The form of the equation
+∂B0,3 = V0,3 − V′
+0,3
+(16.33)
+suggests to use homology theory. The interpretation of B0,3 is that it is a space interpolating
+between V0,3 and V′
+0,3. A preliminary step is to check that there is no obstruction: since the
+LHS is already a boundary one has ∂2B0,3 = 0 and one should check that ∂(RHS) = 0 as
+well. It can be shown that it is indeed true. It was proved in [226] that this equation admits
+a solution and that the equations for higher g and n can all be solved. Hence, there exists
+a field redefinition and SFT is background independent.
+16.7
+Suggested readings
+• Proof of the background independence under marginal deformations [226, 232, 233]
+(see also [209–211] for earlier results laying foundations for the complete proof).
+• L∞ perspective [169, sec. 4] (see also [168, 167, sec. III.B].
+• Connection on the space of CFTs [30, 198, 199, 210, 238].
+244
+
+Chapter 17
+Superstring
+Abstract
+Superstring theory is generally the starting point for physical model building.
+It has indeed several advantages over the bosonic string, most importantly, the removal
+of the tachyon and the inclusion of fermions in the spectrum. The goal of this chapter is
+to introduce the most important concepts needed to generalize the bosonic string to the
+superstring, both for off-shell amplitudes and string field theory. We refer to the review [42]
+for more details.
+17.1
+Worldsheet superstring theory
+There are five different superstring theories with spacetime supersymmetry: the types I, IIA
+and IIB, and the E8 × E8 and SO(32) heterotic models.
+In the Ramond–Neveu–Schwarz formalism (RNS), the left- and right-moving sectors of
+the superstring worldsheet are described by a two-dimensional super-conformal field theory
+(SCFT), possibly with different numbers of supersymmetries. The prototypical example is
+the heterotic string with N = (1, 0) and we will focus on this case: only the left-moving
+sector is supersymmetric, while the right-moving is given by the same bosonic theory as in
+the other chapters. Up to minor modifications, the type II theory follows by duplicating the
+formulas of the left-moving sector to the right-moving one.
+17.1.1
+Heterotic worldsheet
+The ghost super-CFT is characterized by anti-commuting ghosts (b, c) (left-moving) and
+(¯b, ¯c) (right-moving) with central charge c = (−26, −26), associated to diffeomorphisms, and
+by commuting ghosts (β, γ) with central charge c = (11, 0), associated to local supersym-
+metry. As a consequence the matter SCFT must have a central charge c = (15, 26). If
+spacetime has D non-compact dimensions, then the matter CFT is made of:
+• a free theory of D scalars Xµ and D left-moving fermions ψµ (µ = 0, . . . , D − 1) such
+that cfree = 3D/2 and ¯cfree = D;
+• an internal theory with cint = 15 − 3D/2 and ¯cint = 26 − D.
+The critical dimension is reached when cint = 0 which corresponds to D = 10.
+The diffeomorphisms are generated by the energy–momentum tensor T(z); correspond-
+ingly, supersymmetry is generated by its super-partner G(z) (sometimes also denoted by
+245
+
+TF ). The OPEs of the algebra formed by T(z) and G(z) is:
+T(z)T(w) ∼
+c/2
+(z − w)4 +
+2T(w)
+(z − w)2 + ∂T(w)
+z − w ,
+(17.1a)
+G(z)G(w) ∼
+2c/3
+(z − w)3 + 2T(w)
+(z − w),
+(17.1b)
+T(z)G(w) ∼ 3
+2
+G(w)
+(z − w)2 + ∂G(w)
+(z − w).
+(17.1c)
+The superconformal ghosts form a first-order system (see Section 7.2) with ϵ = −1 and
+λ = 3/2. Hence, they have conformal weights
+h(β) =
+�3
+2, 0
+�
+,
+h(γ) =
+�
+−1
+2, 0
+�
+(17.2)
+and OPEs
+γ(z)β(w) ∼
+1
+z − w,
+β(z)γ(w) ∼ −
+1
+z − w.
+(17.3)
+The expressions of the ghost energy–momentum tensors are
+T gh = −2b ∂c + c∂b,
+T βγ = 3
+2 β∂γ + 1
+2 γ ∂β.
+(17.4)
+The ghost numbers of the different fields are
+Ngh(b) = Ngh(β) = −1,
+Ngh(c) = Ngh(γ) = 1.
+(17.5)
+The worldsheet scalars satisfy periodic boundary conditions. On the other hand, fermions
+can satisfy anti-periodic or periodic conditions: this leads to two different sectors, called
+Neveu–Schwarz (NS) and Ramond (R) respectively.
+βγ system
+The βγ system can be bosonized as
+γ = η eφ,
+β = ∂ξ e−φ,
+(17.6)
+where (ξ, η) are fermions with conformal weights 0 and 1 (this is a first-order system with
+ϵ = 1 and λ = 1), and φ is a scalar field with a background charge (Coulomb gas). This
+provides an alternative representation of the delta functions:
+δ(γ) = e−φ,
+δ(β) = eφ.
+(17.7)
+Introducing these operators is necessary to properly define the path integral with bosonic
+zero-modes. They play the same role as the zero-modes insertions for fermionic fields needed
+to obtain a finite result (see also Appendix C.1.3):
+�
+dc0 = 0
+=⇒
+�
+dc0 c0 = 1,
+(17.8)
+because c0 = δ(c0). For a bosonic path integral, one needs a delta function:
+�
+dγ0 = ∞
+=⇒
+�
+dγ0 δ(γ0) = 1.
+(17.9)
+246
+
+By definition of the bosonization, one has:
+T βγ = T ηξ + T φ,
+(17.10)
+where
+T ηξ = −η ∂ξ,
+T φ = −1
+2 (∂φ)2 − ∂2φ.
+(17.11)
+The OPE between the new fields are:
+ξ(z)η(w) ∼
+1
+z − w,
+eq1φ(z)eq2φ(w) ∼ e(q1+q2)φ(w)
+(z − w)q1q2 ,
+∂φ(z)∂φ(w) ∼ −
+1
+(z − w)2 .
+(17.12)
+The simplest attribution of ghost numbers to the new fields is:
+Ngh(η) = 1,
+Ngh(ξ) = −1,
+Ngh(φ) = 0.
+(17.13)
+To the scalar field φ is associated another U(1) symmetry whose quantum number is
+called the picture number Npic. The picture number of η and ξ are assigned1 such that β
+and γ have Npic = 0:
+Npic(eqφ) = q,
+Npic(ξ) = 1,
+Npic(η) = −1.
+(17.14)
+Because of the background charge, this symmetry is anomalous and correlation functions
+are non-vanishing if the total picture number (equivalently the number of φ zero-modes) is:
+Npic = 2(g − 1) = −χg.
+(17.15)
+For the same reason, the vertex operators eqφ are the only primary operators:
+h(eqφ) = −q
+2(q + 2),
+(17.16)
+and the Grassmann parity of these operators is (−1)q. Special values are
+h(eφ) = 3
+2,
+h(e−φ) = 1
+2.
+(17.17)
+The superstring theory features a Z2 symmetry called the GSO symmetry. All fields are
+taken to be GSO even, except β and γ which are GSO odd and eqφ whose parity is (−1)q.
+Physical states in the NS sector are restricted to be GSO even: it is required to remove
+the tachyon of the spectrum and to get a spacetime with supersymmetry. In type II, the
+Ramond sector can be projected in two different ways, leading to the type IIA and type IIB
+theories.
+The components of the BRST current are:
+jB = c(T m + T βγ) + γG + bc∂c − 1
+4 γ2b,
+(17.18a)
+¯ȷB = ¯c ¯T m + ¯b¯c¯∂¯c.
+(17.18b)
+From there, it is useful to define the picture changing operator (PCO):
+X(z) = {QB, ξ(z)} = c∂ξ + eφG − 1
+4 ∂η e2φ b − 1
+4 ∂(η e2φb),
+(17.19)
+which is a weight-(0, 0) primary operator which carries a unit picture number. It is obviously
+BRST exact. This operator will be necessary to saturate the picture number condition: the
+1Any linear combination of both U(1) could have been used. The one given here is conventional, but
+also the most convenient.
+247
+
+naive insertion of eφ ∼ δ(β) breaks the BRST invariance. The PCO zero-mode is obtained
+from the contour integral:
+X0 =
+1
+2πi
+� dz
+z X(z).
+(17.20)
+It can be interpreted as delocalizing a PCO insertion from a point to a circle, which decreases
+the risk of divergence.
+17.1.2
+Hilbert spaces
+The description in terms of the (η, ξ, φ) fields leads to a subtlety: the bosonization involves
+only the derivative ∂ξ and not the field ξ itself, meaning that the zero-mode ξ0 is absent from
+the original Hilbert space defined from (β, γ). In the bosonized language, the Hilbert space
+without the ξ zero-mode is called the small Hilbert space and is made of state annihilated
+by η0 (the η zero-mode)
+Hsmall =
+�
+|ψ⟩ | η0 |ψ⟩ = 0
+�
+.
+(17.21)
+Removing this condition leads to the large Hilbert space:2
+Hsmall = Hlarge ∩ ker η0.
+(17.22)
+A state in Hsmall contains ξ with at least one derivative acting on it.
+A correlation function defined in terms of the (η, ξ, φ) system is in the large Hilbert space
+and will vanish since there is no ξ factor to absorb the zero-mode of the path integral. As
+a consequence, correlation functions (and the inner product) are defined with a ξ0 insertion
+(by convention at the extreme left) or, equivalently, ξ(z). The position does not matter since
+only the zero-mode contribution survives, and the correlation function is independent of z.
+Sometimes it is more convenient to work in the large Hilbert space and to restrict later to
+the small Hilbert space.
+The SL(2, C) invariant vacuum is normalized as
+⟨k| c−1¯c−1c0¯c0c1¯c1 e−2φ(z) |k′⟩ = (2π)Dδ(D)(k + k′).
+(17.23)
+Remark 17.1 (Normalization in type II) In type II theory, the SL(2, C) is normalised
+as:
+⟨k| c−1¯c−1c0¯c0c1¯c1 e−2φ(z)e− ¯φ( ¯
+w) |k′⟩ = −(2π)Dδ(D)(k + k′).
+(17.24)
+The sign difference allows to avoid sign differences between type II and heterotic string
+theories in most formulas [42].
+The Hilbert space of GSO even states satisfying the b−
+0 = 0 and L−
+0 = 0 conditions is
+denoted by HT (ghost and picture numbers are arbitrary). This Hilbert space is the direct
+sum of the NS and R Hilbert spaces:
+HT = HNS ⊕ HR.
+(17.25)
+The subspace of states with picture number Npic = n is written Hn. The picture number
+of NS and R states are respectively integer and half-integer. Two special subspaces of HT
+play a distinguished role:
+�
+HT = H−1 ⊕ H−1/2,
+�
+HT = H−1 ⊕ H−3/2.
+(17.26)
+2The relation between the small and large Hilbert spaces is similar to the one between the H and
+H0 = b0H Hilbert space from the open string since the (b, c) and (η, ξ) are both fermionic first-order
+systems.
+248
+
+To understand this, consider the vacuum |p⟩ of the φ field with picture number p:
+|p⟩ = epφ(0) |0⟩ .
+(17.27)
+Then, acting on the vacuum with the βn and γn modes implies
+∀n ≥ −p − 1
+2 :
+βn |p⟩ = 0,
+∀n ≥ p + 3
+2 :
+γn |p⟩ = 0.
+(17.28)
+For p = −1, all positive modes (starting with n = 1/2) annihilate the vacuum in the NS
+sector. This is a positive asset because positive modes which do not annihilate the vacuum
+can create states with arbitrary negative energy (since it is bosonic).3 For p = −1/2 or
+p = −3/2, the vacuum is annihilated by all positive modes, but not by one of the zero-mode
+γ0 or β0. Nonetheless, one can show that the propagator in the R sector allows to propagate
+only a finite number of states if one chooses H−1/2; the role of H−3/2 will become apparent
+when discussing how to build the superstring field theory.
+Basis states are introduced as in the bosonic case:
+�
+HT = Span{|φr⟩},
+�
+HT = Span{|φc
+r⟩}
+(17.29)
+such that
+⟨φc
+r|φs⟩ = δrs.
+(17.30)
+The completeness relations are
+1 =
+�
+r
+|φr⟩⟨φc
+r|
+(17.31)
+�
+HT , and
+1 =
+�
+r
+(−1)|φr| |φc
+r⟩⟨φr|
+(17.32)
+on �
+HT .
+Finally, the operator G is defined as:
+G =
+�
+1
+NS sector,
+X0
+R sector.
+(17.33)
+Note the following properties
+[G, L±
+0 ] = [G, b±
+0 ] = [G, QB] = 0.
+(17.34)
+It will be appear in the propagator and kinetic term of the superstring field theory.
+17.2
+Off-shell superstring amplitudes
+In this section, we are going to build the scattering amplitudes.
+The procedure is very
+similar to the bosonic case, except for the PCO insertions and of the Ramond sector. For
+this reason, we will simply state the result and motivate the modifications with respect to
+the bosonic case.
+3This is not a problem on-shell since the BRST cohomology is independent of the picture number.
+However, this matters off-shell since such states would propagate in loops and make the theory inconsistent.
+249
+
+17.2.1
+Amplitudes
+External states can be either NS or R: the Riemann surface corresponding to the g-loop
+scattering of m external NS states and n external R states is denoted by Σg,m,n. R states
+must come in pairs because they correspond to fermions. As in the bosonic case, the amp-
+litude is written as the integration of an appropriate p-form Ω(g,m,n)
+p
+over the moduli space
+Mg,m,n (or, more precisely, of a section of a fibre bundle with this moduli space as a basis).
+From the geometric point of view, nothing distinguishes the punctures and thus:
+Mg,m,n := dim Mg,m,n = 6g − 6 + 2m + 2n.
+(17.35)
+The form ΩMg,m,n is defined as a SCFT correlation function of the physical vertex operators
+together with ghost and PCO insertions.
+Remark 17.2 A simple way to avoid making errors with signs is to multiply every Grass-
+mann odd external state with a Grassmann odd number. These can be removed at the end
+to read the sign.
+The two conditions from the U(1) anomalies on the scattering amplitude are:
+Ngh = 6 − 6g,
+Npic = 2g − 2.
+(17.36)
+Given an amplitude with m NS states V NS
+i
+∈ H−1 and n R states V R
+j
+∈ H−1/2, the above
+picture number can be reached by introducing a certain number of PCO X(yA):
+npco := 2g − 2 + m + n
+2 .
+(17.37)
+These PCO are inserted at various positions: while the amplitude does not depend on these
+locations on-shell, off-shell it will (because the vertex operators are not BRST invariant).
+The choices of PCO locations are arbitrary except for several consistency conditions:
+1. avoid spurious poles (Section 17.2.3);
+2. consistent with factorization (each component of the surface in the degeneration limits
+must saturate the picture number condition).
+This parallels the discussion of the choices of local coordinates: as a consequence, the
+natural object is a fibre bundle �Pg,m,n with the local coordinate choices (up to global phase
+rotations) and the PCO locations as fibre, and the moduli space Mg,m,n as base. Forgetting
+about the PCO locations leads to a fibre bundle �Pg,m,n which is a generalization of the
+one found in the bosonic case. The coordinate system of the fibre bundle presented in the
+bosonic case is extended by including the PCO locations {yA}.
+With these information, the amplitude can be written as:
+Ag,m,n(V NS
+i
+, V R
+j ) =
+�
+Sg,m,n
+ΩMg,m,n(V NS
+i
+, V R
+j ),
+(17.38a)
+where
+ΩMg,m,n = (−2πi)−Mc
+g,m,n
+�Mg,m,n
+�
+λ=1
+Bλ dtλ
+npco
+�
+A=1
+X(yA)
+m
+�
+i=1
+V NS
+i
+n
+�
+j=1
+V R
+j
+�
+Σg,n
+.
+(17.38b)
+where Sg,m,n is a Mg,m,n-dimensional section of �Pg,m,n parametrized by coordinates tλ. The
+1-form B corresponds to a generalization of the bosonic 1-form.
+It has ghost number 1
+250
+
+and includes a correction to compensate the variation of the PCO locations in terms of the
+moduli parameters:
+Bλ =
+�
+α
+�
+Cα
+dσα
+2πi b(σα) ∂Fα
+∂tλ
+�
+F −1
+α (σα)
+�
++
+�
+α
+�
+Cα
+d¯σα
+2πi
+¯b(¯σα) ∂ ¯Fα
+∂tλ
+� ¯F −1
+α (¯σα)
+�
+−
+�
+A
+1
+X(yA)
+∂yA
+∂tλ
+∂ξ(yA).
+(17.39)
+The last factor amounts to consider the combination
+X(yA) − ∂ξ(yA) dyA
+(17.40)
+for each PCO insertion:4 the correction is necessary to ensure that the BRST identity
+(13.46) holds. This can be understood as follows: the derivative acting on the PCO gives a
+term dX(z) = ∂X(z)dz which must be cancelled. This is achieved by the second term since
+{QB, ∂ξ(z)} = ∂X(z).
+Remark 17.3 While it is sufficient to work with Mg,n for on-shell bosonic amplitudes,
+on-shell superstring amplitudes are naturally expressed in �Pg,m,n (with the local coordinate
+removed) since the positions of the PCO must be specified even on-shell.
+Remark 17.4 (Amplitudes on the supermoduli space) Following Polyakov’s appro-
+ach from Chapters 2 and 3 to the superstring would lead to replace the moduli space by the
+supermoduli space. The latter includes Grassmann-odd moduli parameters in addition to the
+moduli parameters from Mg,m+n (in the same way the superspace includes odd coordinates
+θ along with spacetime coordinates x). The natural question is whether it is possible to split
+the integration over the even and odd moduli, and to integrate over the latter such that only
+an integral over Mg,m+n remains. In view of (17.38a), the answer seems positive. However,
+this is incorrect: it was proven in [58] that there is no global holomorphic projection of the
+supermoduli space to the moduli space.
+This is related to the problem of spurious poles
+described below. But, this does not prevent to do it locally: in that case, implementing the
+procedure carefully should give the rules of vertical integration [73, 215, 230].
+17.2.2
+Factorization
+The plumbing fixture of two Riemann surfaces Σg1,m1,n1 and Σg2,m2,n2 can be performed in
+two different ways since two NS or two R punctures can be glued.
+If two NS punctures are glued, the resulting Riemann surface is Σ(NS)
+g1+g2,m1+m2−2,n1+n2.
+The number of PCO inherited from the two original surfaces is
+n(1)
+pco + n(2)
+pco = 2(g1 + g2) − 2 + (m1 + m2 − 2) + n1 + n2
+2
+= n(NS)
+pco ,
+(17.41)
+which is the required number for a non-vanishing amplitude. As a consequence, the propag-
+ator is the same as in the bosonic case:
+∆NS = b+
+0 b−
+0
+1
+L0 + ¯L0
+δ(L−
+0 ).
+(17.42)
+If two R punctures are glued, the numbers of PCO do not match by one unit:
+n(1)
+pco + n(2)
+pco = 2(g1 + g2) − 2 + (m1 + m2) + n1 + n2 − 2
+2
+− 1 = n(R)
+pco − 1.
+(17.43)
+4The sum is formal since it is composed of 0- and 1-forms.
+251
+
+This means that an additional PCO must be inserted in the plumbing fixture procedure:
+the natural place for it is in the propagator since this is the only way to keep both vertices
+symmetric as required for a field theory interpretation. Another way to see the need of
+this modification is to study the propagator (17.42) for Ramond states: since Ramond
+states carry a picture number −1/2, the conjugate states have Npic = −3/2 and thus the
+propagator has a total picture number −3 instead of −2 (the propagator graph is equivalent
+to a sphere). Then, to avoid localizing the PCO at a point of the propagator, one inserts
+the zero-mode which corresponds to smear the PCO:
+∆R = b+
+0 b−
+0
+X0
+L0 + ¯L0
+δ(L−
+0 ).
+(17.44)
+Delocalizing the PCO amounts to average the amplitude over an infinite number of points
+(i.e. to consider a generalized section): this is necessary to preserve the L−
+0 eigenvalue since
+X0 is rotationally invariant while X(z) is not.
+Note that the zero-mode can be written
+equivalently as a contour integral around one of the two glued punctures:
+X0 =
+1
+2πi
+� dw(1)
+n
+w(1)
+n
+X
+�
+w(1)
+n
+�
+=
+1
+2πi
+� dw(2)
+n
+w(2)
+n
+X
+�
+w(2)
+n
+�
+.
+(17.45)
+The equality of both expressions holds because X(z) has conformal weight 0.
+Using the operator G (17.33), the propagator can be written generically as
+∆ = b+
+0 b−
+0
+G
+L0 + ¯L0
+δ(L−
+0 ).
+(17.46)
+Remark 17.5 (Propagators) NS and R states correspond respectively to bosonic and fer-
+mionic fields: the operators L+
+0 and X0 can be interpreted as the (massive) Laplacian and
+Dirac operators, such that both propagators can be written
+∆NS ∼
+1
+k2 + m2 ,
+∆R ∼ i/∂ + m
+k2 + m2 .
+(17.47)
+To motivate the identification of X0 with the Dirac operator, remember that X(z) contains
+a term eφ(z)G(z) (this is the only term which contributes on-shell), where G(z) in turn
+contains ψµ∂Xµ. But, the zero-modes of ψµ and ∂Xµ correspond respectively to the gamma
+matrix γµ and momentum kµ when acting on a state.
+The PCO zero-mode insertion inside the propagator has another virtue. It was noted
+previously that states with Npic = −3/2 are infinitely degenerate since one can apply β0
+an arbitrary number of time. These states have large negative ghost numbers. Considering
+a loop amplitude, all these states would appear in the sum over the states and lead to a
+divergence. The problem is present only for loops because the ghost number is not fixed: in
+a tree propagator, the ghost number is fixed and only a finite number of β0 can be applied.
+But, the PCO insertion turns these states into Npic = −1/2 states. In this picture number,
+one cannot create an arbitrarily large negative ghost number since γn
+0 can only increase the
+ghost number.
+17.2.3
+Spurious poles
+A spurious pole corresponds to a singularity of the amplitude which cannot be interpreted
+as the degeneration limit of Riemann surfaces. As a consequence, they do not correspond to
+infrared divergences and don’t have any physical meaning; they must be avoided in order to
+define a consistent theory. To achieve this, the section Sg,m,n must be chosen such that it
+252
+
+avoids all spurious poles. However, while it is always possible to avoid these poles locally, it
+is not possible globally (this is related to the results from [58]). Poles can be avoided using
+vertical integration: two methods have been proposed, in the small (Sen–Witten) [215, 230]
+and large (Erler–Konopka) [73] Hilbert spaces respectively. Before describing the essence of
+both approaches, we review the origin of spurious poles.
+Origin
+Spurious poles arise in three different ways:
+• two PCOs collide;
+• one PCO and one matter vertex collide;
+�� other singularities of the correlation functions.
+The last source is the less intuitive one and we focus on it.
+A general correlation function of (η, ξ, φ) on the torus5 (satisfying the ghost number
+condition) reads
+C(xi, yj, zq) =
+�n+1
+�
+i=1
+ξ(xi)
+n
+�
+j=1
+η(yj)
+m
+�
+k=1
+eqkφ(zk)
+�
+=
+n�
+j′=1
+ϑδ
+�
+− yj′ + �
+i
+xi − �
+j
+yj + �
+k
+qkzk
+�
+n+1
+�
+i′=1
+ϑδ
+�
+− xi′ + �
+i
+xi − �
+j
+yj + �
+k
+qkzk
+� ×
+�
+i<i′ E(xi, xi′) �
+j<j′ E(yj, yj′)
+�
+i,j
+E(xi, yj) �
+k,ℓ
+E(zk, zℓ)qkqℓ .
+(17.48)
+The additional ξ insertion is necessary since it provides the ξ zero-mode, the correlation
+function being defined in the large Hilbert space. On the torus, the picture numbers must
+add to zero and thus the charges qk satisfy
+�
+k
+qk = 0.
+(17.49)
+The function E(x, y) is called the prime form and is a generalization of the function x − y
+on te torus:
+E(x, y) = ϑ1(x − y)
+ϑ′
+1(0)
+∼x→y x − y.
+(17.50)
+Its presence ensures that C vanishes or diverges appropriately when the operators collide (i.e.
+that the zeros and poles of C are the expected ones from the OPE). The theta functions are
+used to make sure that the correlation function satisfies the appropriate boundary conditions
+(specified by the spin structure δ) for each cycle of the surface.
+However, theta functions can also vanish and the ones in the denominator lead to addi-
+tional singularities (not implied by any OPE) for the correlation function. Since x1 can be
+chosen arbitrarily, the only theta function which can have poles is
+ϑδ
+� n+1
+�
+i=2
+xi −
+n
+�
+j=1
+yj +
+m
+�
+k=1
+qkzk
+�
+= 0.
+(17.51)
+This defines a complex codimension 1 curve in �Pg,m,n, depending on the vertex and PCO
+locations, but also on the moduli parameters (appearing in the definition of the theta func-
+tion). On the other hand, it does not depend on the local coordinate choice. If the section
+Sg,m,n intersects this curve, it will be ill-defined (even on-shell).
+5The discussion generalizes directly to higher-genus Riemann surfaces.
+253
+
+From this formula, several comments can be made. If an operator inserted at z contains
+nξ fields ∂ξ, nη fields η and a factor epφ, then the dependence in z of the theta function
+is of the form (nξ − nη + p)z = Npicz. Then, if the PCO locations are chosen as to avoid
+spurious poles for a given operator, this will also avoid them for any operator of the same
+picture number. This also implies that insertion of β and γ cannot lead to spurious poles
+since they have Npic = 0; this is important since they appear in the BRST current, thus
+insertion of the latter cannot lead to new poles.
+Since it is always possible to choose locally a distribution of PCO to avoid spurious
+poles, the idea is to discretize the moduli space in small pieces. But, since the PCO cannot
+be distributed continuously along the different components of the moduli space, correction
+terms are required. These can be generated in two different ways in both the small and
+large Hilbert spaces. The second is more general while the first may be more adapted since
+it keeps the amplitude in the small Hilbert space.
+Vertical integration: large Hilbert space
+Consider the n-point amplitude state⟨A(p)| which produces an amplitude with p PCO when
+contracted with n external states are specified. The BRST identity implies that the amp-
+litude state is closed (i.e. gauge invariant)
+⟨A(p)| Q = 0,
+(17.52)
+where
+Q = QB ⊗ 1⊗n−1 + · · · + 1⊗n−1 ⊗ QB.
+(17.53)
+Moreover, this state is in the small Hilbert space which implies that it is in the kernel of η0:
+⟨A(p)| η = 0,
+(17.54)
+where
+η = η0 ⊗ 1⊗n−1 + · · · + 1⊗n−1 ⊗ η0.
+(17.55)
+The BRST cohomology is trivial in the large Hilbert space: thus, if ⟨A(p)| is closed, it
+must be exact in this space:
+⟨A(p)| =⟨α(p)| Q,
+(17.56)
+where the state α(p) (called gauge amplitude) must be in the large Hilbert space. This is
+consistent with ⟨A(p)| η = 0 only if
+⟨α(p)| ηQ = 0
+(17.57)
+(Q and η anti-commute); It is then natural to interpret the state on which Q acts as an
+amplitude with one less PCO
+⟨A(p−1)| =⟨α(p)| η
+(17.58)
+since Npic(η) = −1.
+Continuing this procedure leads to an amplitude ⟨A(0)| without any PCO insertion, and
+thus without spurious singularities. Consistency with the picture number anomaly requires
+the external state to have non-canonical picture numbers. But, this should not be a puzzle
+since the amplitude states should be viewed as intermediate object to obtain the final amp-
+litude.
+Hence, the amplitude ⟨A(p)| can be constructed by starting with ⟨A(0)|: inserting ξ(z) in
+the amplitude leads to the gauge amplitude⟨α(1)|, whose BRST variation yields⟨A(2)|. Con-
+tinuing recursively helps to construct the desired amplitude. Moreover, Q and ξ insertions
+automatically take care of the corrections at the interfaces of the components.
+Showing that the amplitude is independent of the non-physical data (i.e. gauge invari-
+ance) is trivial since it is expressed as a BRST exact expression.
+254
+
+Vertical integration: small Hilbert space
+The section of �Pg,m,n is given by a series of discontinuous components linked by vertical
+segments. On the vertical segment, the PCO configuration interpolates continuously between
+the components and the integrand can encounter a spurious pole. Since the integrand is not
+a total derivative in terms of the fibre coordinates, its integration over a segment depends
+on the path followed and not only on the end points. This implies that it diverges when it
+encounters the spurious pole. However, there is a specific prescription which avoids these
+problems. When only one PCO varies, the integrand is a total derivative of the PCO location
+and can thus be integrated directly, giving a difference in the two end points. In this case,
+the result is independent from the specific path and from the presence of the spurious pole.
+To be more concrete, given a PCO insertion X(y1), the variation of its location inserts a
+factor −∂ξ(y1). Integrating this term between two components labelled by i and j – keeping
+everything else fixed – leads to a factor ξ(y(j)
+1 ) − ξ(y(j)
+1 ).
+When several PCOs are involved, it is not sufficient to integrate the vertical segment
+along a path where only one PCO varies at a time. Indeed, because a hole is left in the
+process of the vertical integration. Additional segments must be added and integrated over.
+This avoids the spurious poles and one can show that it yields a well-defined amplitude.
+Moreover, it agrees with the large Hilbert space approach.
+Finally, it remains to address the question of the Feynman diagrams construction. In
+this case, every graph obtained by plumbing fixture inherits its PCO locations from the
+lower-dimensional surfaces, and there is no control on the resulting distribution. It can be
+shown that no spurious singularity is generated in the gluing process if the lower-dimensional
+graphs have no spurious poles. Hence, it is sufficient to ensure that the fundamental graphs
+have no spurious poles.
+17.3
+Superstring field theory
+The construction of super-SFT has proceeded along different directions (for reviews, see [42,
+71, 178]). There are two main strategies for constructing the superstring vertices:
+1. brute-force construction: build the vertices recursively from amplitude factorization;
+2. dress the bosonic products with superconformal ghosts.
+While the second approach is simpler and preferred for explicit construction, the first allows
+to derive the general structure as was done for the bosonic string. There are two main
+strategies for dressing the vertices:
+1. Munich construction (homotopy algebra bootstrap): use the L∞ and A∞ structures
+to derive the superstring vertices from the bosonic vertices (small Hilbert space).
+2. Berkovits’ construction (WZW action): generalize Witten’s cubic bosonic open SFT
+(NS / R in large / small Hilbert space).
+As indicated in parenthesis, a super-SFT can be written in the small or large Hilbert space
+(or a combination).
+The different approaches have been shown to be equivalent at the
+classical level.
+The main difficulty in building a super-SFT is to properly describe the Ramond sector.
+This can be done following two different approaches:
+• constraining the Ramond string field;
+• using an auxiliary string field.
+255
+
+Berkovits’ original SFT cannot describe the Ramond sector in the large Hilbert space, but
+it is possible to couple Berkovits’ action for the NS field in the large Hilbert space to a
+Ramond field in the small Hilbert space. Another limitation of Berkovits’ approach is that
+it works only for the open and heterotic superstrings (but not for type II).
+We assume that the problems with PCO are absent (in Berkovits’ and supermoduli
+constructions) or that they have been defined using vertical integration.
+In the rest of this chapter, we will discuss the kinetic term for each of the first three
+approaches. At the level of the free action, the open and heterotic super-SFT differs only in
+the bosonic factors as in Chapter 10.
+17.3.1
+String field and propagator
+As in the bosonic case, it is natural to consider a string field gathering all possible states
+Ψ = Ψ−1 + Ψ−1/2,
+(17.59)
+where Ψ−1 and Ψ−1/2 are respectively the NS and R string fields. If the field is in the small
+Hilbert space, it satisfies:
+η0 |Ψ⟩ = 0.
+(17.60)
+The propagator was found in (17.46) to be
+∆ = b+
+0 b−
+0
+G
+L0 + ¯L0
+δ(L−
+0 ),
+G =
+�
+1
+NS,
+X0
+R.
+(17.61)
+As for the bosonic case, the constraints
+b−
+0 |Ψ⟩ = L−
+0 |Ψ⟩ = 0,
+b+
+0 |Ψ⟩ = 0
+(17.62)
+must be imposed on the field to ensure that the propagator is invertible.
+For similar reasons, the PCO insertion implies that the propagator is not invertible since
+X0 has zero-modes: this means equivalently that it has a non-empty kernel off-shell or that
+it contains derivatives. Two different solutions can be chosen to address this issue: imposing
+constraints as for the level-matching condition, or introducing auxiliary fields.
+17.3.2
+Constraint approach
+Two new PCO operators must be introduced:
+X = G0 δ(β0) + b0 δ′(β0),
+Y = −c0 δ′(γ0).
+(17.63)
+The first operator commutes with the BRST operator
+[QB, X] = 0.
+(17.64)
+The product of these operators is a projector
+XY X = X.
+(17.65)
+Then, the R string field is constrained to satisfy
+XY |Ψ−1/2⟩ = |Ψ−1/2⟩ .
+(17.66)
+A state satisfying this condition is said to be in the restricted Hilbert space. It can be shown
+that it reproduces the cohomology of QB on-shell.
+256
+
+Remark 17.6 Since G0 contains derivatives, the restriction is not purely algebraic as in
+the bosonic case. It prevents the degeneration due to γn
+0 .
+Remark 17.7 (Comparison with level-matching) The conditions b−
+0 = L−
+0 = 0 can
+be rephrased as the statement that the string field Ψ is invariant under the action of the
+projector Bc−
+0
+Bc−
+0 |Ψ⟩ = |Ψ⟩ ,
+(17.67)
+where
+B = b−
+0
+� 2π
+0
+dθ
+2π eiθL−
+0 = δ(b−
+0 )δ(L−
+0 ).
+(17.68)
+The kinetic term (after unfixing the gauge) reads:
+S0,2 = −1
+2 ⟨Ψ−1| c−
+0 QB |Ψ−1⟩ − 1
+2 ⟨Ψ−1/2| c−
+0 Y QB |Ψ−1/2⟩ .
+(17.69)
+The action is invariant under the gauge transformation
+δ |Ψ⟩ = QB |Λ⟩
+(17.70)
+where
+Λ = Λ−1 + Λ−1/2.
+(17.71)
+Each gauge parameter satisfies the same conditions as the associated field (in particular,
+Λ−1/2 is in the restricted Hilbert space).
+17.3.3
+Auxiliary field approach
+The disadvantage of the constraint approach is two-fold. First, it treats both components of
+the field on a different footing. Second, the constraint must be imposed by hand and does not
+follow from any fundamental principle. Another possibility is to embed the propagator in a
+higher-dimensional field space by introducing additional fields: in this way, the propagator
+can be inverted without introducing the inverse of X0.
+Let’s introduce the new field
+�Ψ = �Ψ−1 + �Ψ−3/2
+(17.72)
+which satisfies the same conditions as Ψ:
+b−
+0 |�Ψ⟩ = L−
+0 |�Ψ⟩ = 0,
+b+
+0 |�Ψ⟩ = 0.
+(17.73)
+A tentative kinetic term is then:
+S0,2 = 1
+2 ⟨�Ψ| c−
+0 c+
+0 L+
+0 G |�Ψ⟩ −⟨�Ψ| c−
+0 c+
+0 L+
+0 |Ψ⟩ .
+(17.74)
+The kinetic operator in matrix form for (�Ψ, Ψ) reads
+K = c−
+0 c+
+0 L+
+0
+�−G
+1
+1
+0
+�
+(17.75)
+and its inverse is
+∆ = b−
+0 b+
+0
+1
+L+
+0
+�0
+1
+1
+G
+�
+.
+(17.76)
+This reproduces the expected propagator for (Ψ, Ψ) without needing to invert X0.
+257
+
+What is the interpretation of the additional fields? The gauge invariance of the action
+is
+δ |Ψ⟩ = QB |Λ⟩ ,
+δ |�Ψ⟩ = QB |�Λ⟩ ,
+(17.77)
+where Λ satisfies the same constraints as Ψ (in particular, it contains more components than
+the Λ of the previous section). Then, the equations of motion are
+QB |Ψ⟩ = 0,
+QB |�Ψ⟩ = 0.
+(17.78)
+This shows that both fields are free and decoupled and that the spectrum is doubled. To
+push the interpretation further, one needs to consider the interactions.
+Amplitudes involve only the states contained in Ψ and thus the interactions are built
+solely in terms of Ψ. Then, the equations of motion have the form:
+QB
+�
+|Ψ⟩ − G |�Ψ⟩
+�
+= 0,
+QB |�Ψ⟩ = |J(Ψ)⟩ ,
+(17.79)
+where J(Ψ) is a source term due to the interactions. An equation for Ψ only is obtained by
+multiplying the second with G
+QB |Ψ⟩ = G |J(Ψ)⟩ .
+(17.80)
+Once Ψ is determined by solving this equation, the auxiliary field �Ψ is completely fixed by
+the second equation up to free field solutions. This shows that �Ψ describes only free fields
+even when Ψ is interacting. Note that this implies that the degrees of freedom contained
+in �Ψ do not even couple to the gravitational field! This can also be shown at the level of
+Feynman diagrams.
+Remark 17.8 The field �Ψ is not an auxiliary field strictly speaking since it is propagating
+(its equation of motion is not algebraic).
+17.3.4
+Large Hilbert space
+The last formulation of the kinetic term considers the NS string field to be in the large
+Hilbert space, i.e. η0 ̸= 0. The Ramond field must be described with one of the two previous
+approach.
+Writing the action requires to use a NS field Ψ0 with picture number 0. The kinetic term
+becomes
+S0,2 = −1
+2 ⟨⟨Ψ0, η0QBΨ0⟩⟩,
+(17.81)
+where ⟨⟨·, ·⟩⟩ is the inner product in the large Hilbert space (contains a ξ0 insertion). This
+action has an enlarged gauge invariance:
+δ |Ψ0⟩ = QB |Λ0⟩ + η0 |Ω1⟩ ,
+(17.82)
+and the equation of motion reads
+QBη0 |Ψ0⟩ = 0.
+(17.83)
+The η0 gauge invariance can be fixed with the condition
+ξ0 |Ψ0⟩ = 0,
+(17.84)
+and one can introduce a new field Ψ−1 such that
+|Ψ0⟩ = ξ0 |Ψ−1⟩
+(17.85)
+to satisfy automatically the condition. The equation of motion becomes
+QB |Ψ−1⟩ = 0,
+(17.86)
+and one recovers the small Hilbert space formulation.
+258
+
+17.4
+Suggested readings
+• General reviews [42, 71, sec. 6].
+• Spurious poles and vertical integration:
+– small Hilbert space [42, app. C, D, 215, 230]
+– large Hilbert space [73]
+• Constructions of super-SFT:
+– “Sen’s” amplitude factorization construction [42, 183, 213, 214, 216, 218]
+– “Munich” homotopy algebra bootstrap [74–76, 80, 131]
+– Berkovits’ SFT [18, 69, 80, 134, 144, 159]
+– supermoduli space [175, 241]
+– democratic SFT [132, 133]
+– light-cone SFT [114, 117]
+• Relations between different constructions [67, 68, 79, 80, 111].
+• Ramond string field:
+– constrained field [69, 80, 144, 241]
+– auxiliary field [42, 80, 214, 218]
+259
+
+Chapter 18
+Momentum-space SFT
+Abstract
+In this chapter, we describe the general properties of SFT actions in the mo-
+mentum space. This allows to make SFT more intuitive, but also to use standard QFT
+methods to prove various properties of string theory. We explain how the Wick rotation
+is generalized for theories with vertices diverging at infinite real energies (Lorentzian sig-
+nature). This allows to prove important properties of string theory, such as unitarity or
+crossing symmetry.
+18.1
+General form
+Since the explicit expressions of the string vertices are not known, it is not possible to write
+explicitly the SFT action. However, the general properties of the vertices are known: then,
+one can write a general QFT which contains SFT as a subcase. This is sufficient to already
+extract a lot of informations. The other advantage is that the QFT language is more familiar
+and intuitive in many situations. Hence, one can use this general form to built intuition
+before translating the results in a more stringy language. In a nutshell, SFT is a QFT:
+• with an infinite number of fields (of all spins);
+• with an infinite number of interactions;
+• with non-local interactions ∝ e−#k2;
+• which reproduces the worldsheet amplitudes (if the latter are well-defined).
+The non-locality of the interactions is the most salient property of SFT, beyond the
+infinite number of fields. This has a number of consequences:
+• the Wick rotation is ill-defined;
+• the position representation cannot be used, nor any property relying on it (micro-
+causality, largest time equation. . . );
+• standard assumptions from local QFT (in particular, from the constructive S-matrix
+program, such as micro-causality) break down.
+Together, these points imply that the usual arguments from QFTs must be improved. This
+has been an active topic in the recent years and the results will be summarized in Sec-
+tion 18.2.
+260
+
+We expand the string field in Fourier space using a basis {φα(k)} as (Chapter 9):
+|Ψ⟩ =
+�
+j
+�
+dDk
+(2π)D ψα(k) |φα(k)⟩ ,
+(18.1)
+where k is the D-dimensional momentum and α the discrete indices (Lorentz indices, group
+representation, KK modes. . . ) of the spacetime fields ψα(k). The action in momentum
+space takes the form (in Lorentzian signature):
+S = −
+�
+dDk ψα(k)Kαβ(k)ψβ(−k)
+−
+�
+n≥0
+�
+dDk1 · · · dDkn V (n)
+α1···αn(k1, . . . , kn) ψα1(k1) · · · ψαn(kn).
+(18.2)
+The kinetic matrix Kαβ is usually quadratic in the momentum. In the direct Fourier ex-
+pansion of the SFT action (15.24), it describes only the classical kinetic term: the quantum
+corrections are found in the vertex V (2).
+From the action, we can write the Feynman rules (for the path integral weight eiS and
+S-matrix S = 1 + iT). The propagator reads:
+= Kαβ(k)−1 = −i Mαβ
+k2 + m2α
+Qα(k),
+(18.3)
+where Mαβ is mixing matrix for states of equal mass and Qα a polynomial in k (there is
+no sum over α). The interactions are obtained by plugging the basis states {φα} inside the
+vertices Vn (14.58):
+= i V (n)
+α1···αn(k1, . . . , kn) := i Vn
+�
+φα1(k1), . . . , φαn(kn)
+�
+= i
+�
+dt e
+−g
+{αk}
+ij
+(t) ki·kj−λ�
+α
+m2
+α
+Pα1,...,αn
+�
+k1, . . . , kn; t
+�
+,
+(18.4)
+where t denotes collectively the moduli parameters, P{αi} is a polynomial in k, gij is a
+positive-definite matrix, λ > 0 is a number. There is an implicit sum over the momentum
+indices.
+The terms quadratic in the momenta inside the exponential arise from two sources:
+• The correlation functions of the vertex operators ⟨�
+i eiki·X(zi)⟩ is proportional to
+e−ki·kjG(zi,zj), where G is the Green function.
+Additional factors like ∂X contrib-
+ute to the polynomial Pα1,...,αn.
+• It is possible to add stubs to the vertices. The effect is to multiply each leg by a factor
+e−λ(k2
+i +m2
+i ) with λ > 0 (we take λ to be the same for all vertices for simplicity). The
+first term of the exponential contributes to the diagonal of the matrix gij. By taking
+λ sufficiently large, one can enforce that all eigenvalues are positive.
+Finally, the exponential term with the masses m2
+α ensures that the sum over all intermediate
+states converge despite an infinite number of states. Indeed, the number of states of mass
+261
+
+mα grows as ecmα, which is dominated by e−λm2
+α for sufficiently large λ. Hence, the addition
+of stubs make explicit the absence of divergences in SFT.1
+The vertices have no singularity for ki ∈ C finite. As the energy becomes infinite |k0
+i | →
+∞, they behave as:
+lim
+k0→±i∞ V (n) = 0,
+lim
+k0→±∞ V (n) = ∞.
+(18.5)
+The first property is responsible for the soft UV behaviour of string theory in Euclidean
+signature, while the second prevents from performing the Wick rotation (indeed, the pole
+at infinity implies that the arcs closing the contour contribute).
+The g-loop n-point amputated Green functions are sums of Feynman diagrams, each of
+the form:
+Fg,n(p1, . . . , pn) ∼
+�
+dT
+�
+s
+dDℓs e−Grs(T ) ℓr·ℓs−2Hri(T ) ℓr·pi−Fij(T ) pi·pj
+×
+�
+a
+1
+k2a + m2a
+P(pi, ℓr; T),
+(18.6)
+where {pi} are the external momenta, {ℓr} the loop momenta and {ki} the internal mo-
+menta, with the latter given by a linear combination of the other. Moreover, T denotes the
+dependence in the moduli parameters of all the internal vertices, and P is a polynomial in
+(pi, ℓr). The matrix Grs is positive definite, which implies that:
+• integrations over spatial loop momenta ℓr converge;
+• integrations over loop energies ℓ0
+r diverge.
+As a consequence, the Feynman diagrams in Lorentzian signature are ill-defined: we will
+explain in the next section how to fix this problem.
+18.2
+Generalized Wick rotation
+We have seen that loop integrals in Lorentzian signature are divergent because of the large
+energy behaviour of the interactions. But, this is not different from the usual QFT, where the
+loop integrals are also ill-defined in Lorentzian signature. Indeed, poles of the propagators
+sit on the real axis and also give divergent loop integrals (note that the same problem arise
+also here). In that case, the strategy is to define the Feynman diagrams in Euclidean space
+and to perform a Wick rotation: the latter matches the expressions in Lorentzian signature
+up to the iε-prescription. The goal of the latter is to move slightly the poles away from the
+real axis.
+Example 18.1 – Scalar field
+Consider a scalar field of mass m with a quartic interaction. The 1-loop 4-point Feyn-
+man diagram is given in Figure 18.1. The external momenta are pi, i = 1, . . . , 4. There
+are one loop momentum ℓ and two internal momenta k1 = ℓ and k2 = p − ℓ, where
+p = p1 + p2. The poles in the loop energy ℓ0 are located at:
+p± = ±
+�
+ℓ2 + m2,
+q± = p0 ±
+�
+(p − ℓ)2 + m2.
+(18.7)
+The graph is first defined in Euclidean signature, where the external and loop en-
+ergies are pure imaginary, p0
+i , ℓ0 ∈ iR. The poles are shown in Figure 18.2. Then,
+the external momenta are analytically continued to real values, p0
+i ∈ R. At the same
+1Remember that λ is not a physical parameter and disappears on-shell. This means that the cancellation
+of the divergences is independent of λ and must always happen on-shell.
+262
+
+time, the integration contour is also analytically continued thanks to the Wick rotation
+(Figure 18.3). The contour is closed with arcs, but they don’t contribute since there is
+no poles in the upper-right and lower-left quadrants, and no poles at infinity. However,
+one cannot continue the contour such that ℓ0 ∈ R because of the poles on the real axis.
+The Wick rotation is possible for ℓ0 in the upper-right quadrant, Re ℓ0 ≥ 0, Im ℓ0 > 0,
+which leads to the iε-prescription ℓ0 ∈ R + iε.
+Figure 18.1: 1-loop 4-point function for a scalar field theory.
+Figure 18.2: Integration contour for external Euclidean momenta.
+Since the Feynman diagram (18.6) is not defined in Lorentzian signature because of the
+poles at ℓ0
+r → ±∞, it is also necessary to start with Euclidean momenta. However, the
+same behaviour at infinity prevents from using the Wick rotation since the contribution
+from the arcs does not vanish. It is then necessary to find another prescription for defining
+the Feynman diagrams in SFT starting from the Euclidean Green functions. This is given
+by the following generalized Wick rotation (Pius–Sen [187]):
+1. Define the Green functions for Euclidean internal and external momenta.
+2. Perform an analytic continuation of the external energies and of the integration contour
+such that:
+• keep poles on the same side;
+• keep the contour ends fixed at ±i∞.
+One can show [187] that the Green functions are analytic in the upper-right quadrant Im p0
+a >
+0, Re p0
+a ≥ 0, for pa ∈ R, p0
+a. Moreover, the result is independent of the contour chosen as
+long as it satisfies the conditions described above. In fact, this generalized Wick rotation is
+263
+
+Figure 18.3: Integration contour for external Lorentzian momenta after Wick rotation (reg-
+ular vertices).
+valid even for normal QFT, which raises interesting questions. For example, it seems that
+the internal and external set of states have no intersection, which can be puzzling when
+trying to interpret the Cutkosky rules. Nonetheless, everything works as expected.
+Remark 18.1 (Timelike Liouville theory) It has been shown in [13] that this general-
+ized Wick rotation is also the correct way for defining the timelike Liouville theory.
+The fact that the amplitude is analytic only when the imaginary parts of the momenta
+are not zero, Im p0
+a > 0, is equivalent to the usual iε-prescription for QFT. Moreover, it has
+been shown [222] to be equivalent to the moduli space iε-prescription from [257]. Then, it
+has also been used to prove several important properties of string theory shared by local
+QFTs: Cutkosky rules [187, 188], unitarity [220, 221], analyticity in a subset of the primitive
+domain and crossing symmetry [43]. Finally, general soft theorems for string theory (and, in
+fact, any theory of quantum gravity) have been proven in [34, 150, 223, 224]. All together,
+these properties establish string theory as a very strong candidate for a consistent theory
+of everything.
+The next main question is how to obtain an expression of SFT which is
+amenable to explicit computations. This will certainly require to understand even better
+the deep structure of SFT, a goal which this book will hopefully help the reader to achieve.
+18.3
+Suggested readings
+• SFT momentum space action [42, 187, 225].
+• Consistency properties of string theory [42]:
+– generalized Wick rotation, Cutkosky rules and unitarity [187, 188, 219–222].
+– analyticity and crossing symmetry [43].
+– soft theorems [34, 150, 223, 224].
+264
+
+(a)
+(b)
+(c)
+Figure 18.4: Integration contour after analytic continuation to external Lorentzian momenta.
+Depending on the values of the external momenta, different cases can happen.
+265
+
+Appendix A
+Conventions
+Most of the book uses natural units where c = ℏ = 1, but the string length ℓs (or Regge
+slope α′) are kept.
+A bar is used to denote both complex conjugation and the anti-holomorphic operators.
+The symbol := (resp. =:) means that the LHS (RHS) is defined by the expression in the
+RHS (LHS).
+A.1
+Coordinates
+The number of spacetime (target-space) dimensions is denoted by D = d + 1, where d is
+the number of spatial dimensions. The corresponding spacetime and spatial coordinates are
+written with Greek and Latin indices:
+xµ = (x0, xi),
+µ = 0, . . . , D − 1 = d
+i = 1, . . . , d
+(A.1)
+When time is singled out, one writes x0 = t in Lorentzian signature and x0 = tE in Euclidean
+signature (or x0 = τ when there is no ambiguity with the worldsheet time).
+A p-brane is a (p+1)-dimensional object whose worldvolume is parametrized by coordin-
+ates:
+σa = (σ0, σα),
+a = 0, . . . , p − 1,
+α = 1, . . . , p.
+(A.2)
+The time coordinate can also be singled out as σ0 = τM in Lorentzian signature and as
+σ0 = τ in Euclidean signature. For the string, the index α is omitted since it takes only one
+value.
+The Lorentzian signature is taken to be mostly plus and the flat Minkowski metric reads
+ηµν = diag(−1, 1, . . . , 1
+� �� �
+d
+).
+(A.3)
+The flat Euclidean metric is
+δµν = diag(1, . . . , 1
+� �� �
+D
+).
+(A.4)
+Similar notations hold for the worldvolume metrics ηab and δab. The Levi–Civita (completely
+antisymmetric) tensor is normalized by
+ϵ01 = −ϵ01 = 1.
+(A.5)
+Wick rotation from Lorentzian time t to Euclidean time τ (either worldsheet or target
+spacetime) is defined by
+t = −iτ.
+(A.6)
+266
+
+Accordingly, contravariant (covariant) vector transforms with the same (opposite) factor:
+V 0
+M = −iV 0
+E,
+VM,0 = iVE,0.
+(A.7)
+Most computations are performed with both spacetime and worldsheet Euclidean signatures.
+Expressions are Wick rotated when needed.
+Light-cone coordinates are defined by
+x± = x0 ± x1.
+(A.8)
+A function depending only on x+ (x−) is said to be left-moving (right-moving) by ana-
+logy with the displacement of a wave. Under analytic continuation, the left-moving (right-
+moving) coordinate is mapped to the holomorphic1 (anti-holomorphic) coordinate z (¯z). In
+chiral theories, the left-moving value is written first.
+The worldsheet coordinates (τ, σ) on the cylinder are defined by
+τ ∈ R,
+σ ∈ [0, L),
+σ ∼ σ + L,
+(A.9)
+where typically L = 2π. The integration over the spatial coordinate is normalized such that
+the perimeter of spatial slice is normalized to 1 if L = 2π:
+L = 1
+2π
+� L
+0
+dσ = L
+2π .
+(A.10)
+This implies that 2d action, conserved charges, etc. are divided by an extra factor of 2π.
+The coordinates can be written in terms of complex coordinates
+w = τ + iσ,
+¯w = τ − iσ
+(A.11)
+such that the flat metric is
+ds2 = dτ 2 + dσ2 = dwd ¯w.
+(A.12)
+Under Wick rotation, the complex coordinates are mapped to light-cone coordinates as
+follows:
+w = iσ+,
+¯w = iσ−.
+(A.13)
+The cylinder can be mapped to the complex plane through
+z = e2πw/L,
+¯z = e2π ¯
+w/L.
+(A.14)
+The definition of the Levi–Civita tensor includes the √g factor, such that
+ϵz¯z = i
+2,
+ϵz¯z = −2i
+(A.15)
+on the complex plane with flat metric.
+1The terms of holomorphic is simply used to indicate that the object depends only on z, but not on ¯z.
+Typically, the objects have singularities and are really meromorphic in z.
+2In fact, the terms of “left”- and “right”-moving are interchanged in [193, p. 34] to get agreement with
+the literature. But, it means that the spatial axis orientation is reversed.
+Moreover, concerning [54], the first definition agrees with (6.1) but not with (6.53) since the definition of
+ξ (our w) is modified in-between. This explains why the definitions of left- and right-moving [54, p. 161] do
+not agree with the one given in the table.
+267
+
+refs
+cylinder
+plane
+light-cone
+left-moving
+here, Di Francesco et al.
+[25, 54, 124, 126, 128, 212, 252, 265]
+w = τ + iσ
+z = ew
+w = iσ+, ¯w = iσ−
+holomorphic
+Blumenhagen et al.
+[14, 24, 123, 205]
+w = τ − iσ
+z = ew
+w = iσ−, ¯w = iσ+
+anti-holomorphic
+Polchinski [176, 193, 246]
+w = σ + iτ
+z = e−iw
+w = −σ−, ¯w = σ+
+anti-holomorphic2
+Table A.1: Conventions for the coordinates.The notations are the following (they can slightly
+vary depending on the references): the Euclidean time is obtained by the analytic continu-
+ation τ = it (denoted also by τ = σ0 = σ2) the spatial direction is σ = σ1, and the light-cone
+coordinates are σ± = t ± σ.
+A.2
+Operators
+Commutators and anti-commutators are denoted by
+[A, B] := [A, B]− = AB − BA,
+{A, B} := [A, B]+ = AB + BA.
+(A.16)
+The Grassmann parity of a field A is denoted by |A|
+|A| =
+�
++1
+Grassmann odd,
+0
+Grassmann even.
+(A.17)
+Two (anti-)commuting operators satisfy
+AB = (−1)|A||B|BA.
+(A.18)
+A.3
+QFT
+Energy is defined as the first component of the momentum vector
+pµ := (E, pi).
+(A.19)
+The following notations are used to denote the number of supersymmetries:
+(NL, NR),
+N = NL + NR,
+(A.20)
+where NL and NR are the numbers of left- and right-chirality supersymmetries. The last
+form is used when it is not important to know the chirality of the supercharges.
+The variation of a field φ(x) is defined by
+δφ(x) = φ′(x) − φ(x).
+(A.21)
+Given an internal symmetry with parameters αa, the Noether current in Lorentzian signature
+is given by:
+Jµ
+a = λ
+∂L
+∂(∂µφ)
+δφ
+δαa ,
+∇µJµ
+a = 0,
+(A.22)
+where L is the Lagrangian which does not include the factor √g for curved spaces and λ is
+some normalization.3 The conserved charges Qa associated to the currents Jµ
+a for a fixed
+spatial slice t = cst are
+Qa = 1
+λ
+�
+Σ
+dD−1x
+√
+h J0
+a,
+(A.23)
+3Including the √g would give the current density √gJµ
+a . The simple derivative of the latter vanishes
+∂µ(√gJµ
+a ) = 0 in view of the identity (B.4).
+268
+
+where Σ is a spatial slice and h is the induced metric. One sets λ = 2π in two dimensions,
+otherwise λ = 1. The variation of a field under a transformation generated by Q is
+δαaφ(x) = iαa[Qa, φ(x)]
+(A.24)
+In Euclidean signature, the current and variation are:
+Jµ
+a = iλ
+∂L
+∂(∂µφ)
+δφ
+δαa ,
+(A.25a)
+δαaφ(x) = −αa[Qa, φ(x)].
+(A.25b)
+Note that the charge is still given by (A.23). The factor of i in (A.25a) can be understood
+as follows.4
+First, the time component J0
+a of the current transforms like time such that
+J0
+a → iJ0
+a, which implies that the charge also gets a factor i, Qa → iQa. This explains the
+minus sign in (A.25b). Then, one needs to make this consistent with the formula (B.9) for
+the charge associated to a general surface. Given a spacelike nµ, the integration measure
+includes the time which transforms with a factor of i: one can interpret it as coming from the
+spatial components of the current, Ji
+a → iJi
+a, while working with a Euclidean region. Another
+way to understand this factor for the spatial vector is by considering the electromagnetic
+case, where J contains a time derivative.
+The term “zero-mode” has two (related) meanings:
+1. given an operator D acting on a space of fields ψ(z), zero-modes ψ0,i(z) of the operator
+are all fields with zero eigenvalue Dψ0,i(z) = 0, i = 1, . . . , dim ker D
+2. the zero-mode of a field expansion ψ = �
+n ψnz−n−h is the mode ψ0 for n = 0: on the
+cylinder, it corresponds to the constant term of the Fourier expansion on the cylinder
+(hence, a zero-mode of ∂z according to the previous definition)
+A prime indicates that the zero-modes are excluded. For example, det′ D is the product of
+non-zero eigenvalues, φ′ is a field without zero-mode and d′φ the corresponding integration
+measure.
+A.4
+Curved space and gravity
+The covariant derivative is defined by
+∇µ = ∂µ + Γµ
+(A.26)
+where Γµ is the connection. For example, one has for a vector field
+∇µAν = ∂µAν + Γ
+ν
+µρ Aρ.
+(A.27)
+The negative-definite Laplacian (or Laplace–Beltrami operator) is defined by
+∆ = gµν∇µ∇ν =
+1
+√g ∇µ
+�√ggµν∇ν).
+(A.28)
+Note that ∇µ does not contain the Christoffel symbol for the index ν because of the identity
+(B.4) (but it contains a connection for any other index of the field). For a scalar field, both
+derivatives become simple derivatives.
+The energy–momentum tensor is defined by
+Tµν = − 2λ
+√g
+δS
+δgµν ,
+(A.29)
+where λ = 2π for D = 2 and λ = 1 otherwise.
+4We stress that these formulas and arguments do not apply to the energy–momentum tensor.
+269
+
+A.5
+List of symbols
+General:
+• D: number of non-compact spacetime dimensions
+• g: loop order for a scattering amplitude
+• n: number of external closed string states
+• xµ: spacetime non-compact coordinates
+• σa = (t, σ): worldsheet coordinates
+• gs: closed string coupling
+• Zg = Ag,0: genus-g vacuum amplitude
+• Ag,n(k1, . . . , kn)α1,...,αn := Ag,n({ki}){αi}: g-loop n-point scattering amplitude for
+states with quantum numbers {ki, αi} (if connected, amputated Green functions for
+n ≥ 3)
+• Gg,n(k1, . . . , kn)α1,...,αn: g-loop n-point Green function for states with quantum num-
+bers {ki, αi}
+• T ⊥
+ab: traceless symmetric tensor or traceless component of the tensor Tab
+• Ψ: generic (set of) matter field(s)
+Hilbert spaces:
+• H: generic Hilbert space (in general, Hilbert space of the matter plus ghost CFT)
+• H± = H ∩ ker b±
+0
+• H0 = H ∩ ker b−
+0 ∩ ker b+
+0
+• H(QB): absolute cohomology of the operator QB inside the space H
+• H−(QB) = H(QB) ∩ H−: semi-relative cohomology of the operator QB inside the
+space H
+• H0(QB) = H(QB) ∩ H0: relative cohomology of the operator QB inside the space H
+• A: Grassmann parity of the operator or state A
+Riemann surfaces:
+• g: Riemann surface genus (number of holes / handles)
+• n: number of bulk punctures / marked points
+• Σg,n: genus-g Riemann surface with n punctures
+• Σg = Σg,0: genus-g Riemann surface
+• Mg,n: moduli space of genus-g Riemann surfaces with n punctures
+• Mg = Mg,0: moduli space of genus-g Riemann surfaces
+• Mg,n = dim Mg,n
+270
+
+• Mc
+g,n = dimC Mg,n
+• Mg = Mg,0 = dim Mg = dim ker P †
+1
+• Mc
+g = Mc
+g,0 = dimC Mg
+• Kg,n: conformal Killing vector group of genus-g Riemann surfaces with n punctures
+• Kg = Kg,0 = ker P1: conformal Killing vector group of genus-g Riemann surfaces
+• Kg,n = dim Kg,n
+• Kc
+g,n = dimC Kg,n = dimC ker P1
+• Kg = Kg,0 = dim ker P1
+• Kc
+gKc
+g,0 = dimC ker P1
+• ψi: real basis of ker P1, CKV
+• φi: real basis of ker P †
+1 , real quadratic differentials
+• (ψK, ¯ψK): complex basis of ker P1, (anti-)holomorphic CKV
+• (φI, ¯φI): complex basis of ker P †
+1 , (anti-)holomorphic quadratic differentials
+• tλ ∈ Mg,n: real moduli of Mg,n
+• mΛ ∈ Mg,n: complex moduli of Mg,n
+• ti ∈ Mg: real moduli of Mg
+• mI ∈ Mg: complex moduli of Mg
+• z: coordinate on the Riemann surface
+• wi: local coordinates around punctures
+• za: local coordinates away from punctures
+• fi(wi): transition functions from wi to z
+• σα: coordinate system on the left of the contour Cα
+• τα: coordinate system on the right of the contour Cα
+CFT:
+• Vα(k; σa) := Vk,α(σa): matter vertex operator with5 momentum k and quantum num-
+bers α inserted at position σa = (z, ¯z)
+• Vα(k; σa): unintegrated vertex operator with momentum k and quantum numbers α
+inserted at position σa
+• Vα(k) =
+�
+d2σ√g Vα(k; σ): integrated vertex operator
+• on-shell (closed bosonic string): Vα(k; σa) = c¯cVα(k; σa) is a (0, 0)-primary, with
+Vα(k; σa) a (1, 1)-primary matter operator
+5When the momentum and/or quantum numbers are not relevant, we remove them or simply index the
+operators by a number.
+271
+
+• �
+O: operator O with zero-modes removed
+• O†: Hermitian adjoint
+• O‡: Euclidean adjoint
+• Ot: BPZ conjugation
+• ⟨O1|O2⟩: BPZ inner-product
+• ⟨O‡
+1|O2⟩: Hermitian inner-product
+• |0⟩: SL(2, C) (conformal) vacuum
+• |Ω⟩: energy vacuum (lowest energy state)
+• :O: : conformal normal ordering (with respect to SL(2, C) vacuum |0⟩)
+•
+⋆
+⋆O
+⋆
+⋆ : energy normal ordering (with respect to energy vacuum |Ω⟩)
+SFT:
+• Ψ: closed string field
+• {φr} = {φα(k)}: basis of H (or some subspace)
+Indices:
+• µ = 0, . . . , D − 1: non-compact spacetime dimensions
+• a = 0, . . . , p: worldvolume coordinates (p = 1: worldsheet)
+• i = 1, . . . , n: external states, local coordinates
+• λ = 1, . . . , Mg,n: real moduli of Mg,n
+• Λ = 1, . . . , Mc
+g,n: complex moduli of Mg,n
+• i = 1, . . . , Mg: real moduli of Mg
+• I = 1, . . . , Mc
+g: complex moduli of Mg
+• i = 1, . . . , Kg: real CKV of Kg
+• K = 1, . . . , Kc
+g: complex CKV of Kg
+• r = (k, α): index for basis state of H (or some subspaces), α: non-momentum indices
+272
+
+Appendix B
+Summary of important formulas
+This appendix summarizes formulas which appear in the book or which are needed but
+assumed to be known to the reader (such as formulas from QFT and general relativity).
+B.1
+Complex analysis
+The Cauchy–Riemann formula is
+�
+Cz
+dw
+2πi
+f(w)
+(w − z)n = f (n��1)(z)
+(n − 1)! ,
+(B.1)
+where f(z) is an holomorphic function.
+One has
+¯∂ 1
+z = 2π δ(2)(z).
+(B.2)
+B.2
+QFT, curved spaces and gravity
+The Green function G of a differential operator D is defined by
+DxG(x, y) = δ(x − y)
+√g
+− P(x, y),
+(B.3)
+where P is the projector on the zero-modes of D.
+The covariant divergence of a vector can be rewritten in terms of a simple derivative:
+∇µvµ =
+1
+√g ∂µ(√gvµ).
+(B.4)
+Under an infinitesimal change of coordinates
+δxµ = ξµ,
+(B.5)
+the metric transforms as
+δgµν = Lξgµν = ∇µξν + ∇νξµ.
+(B.6)
+Stokes’ theorem reads
+�
+V
+dDx ∇µvµ =
+�
+∂V
+dΣµvµ,
+dΣµ := ϵ nµ dD−1Σ,
+(B.7)
+273
+
+where V is a spacetime region, S = ∂V its boundary and dD−1Σ the induced integration
+measure. The vector nµ normal to S points outward and ϵ := nµnµ = 1 (−1) if S is timelike
+(spacelike). If the surface is defined by x0 = cst, then
+dD−1Σ = √g dD−1x,
+nµ = δ0
+µ.
+(B.8)
+We can write a generalization of (A.23) for a charge associated to a general surface S:
+QS = 1
+λ
+�
+S
+dΣµ Jµ
+a .
+(B.9)
+If the current Jµ
+a is conserved, ∇µJµ
+a = 0 (no source), Stokes’ theorem (B.7) shows that the
+charge vanishes QS = 0 if S is a closed surface and that it is conserved QS1 = −QS2 for
+two spacelike surfaces S1 and S2 extending to infinity (if Jµ
+a vanishes at infinity) (see [190,
+chap. 3, 265, sec. 8.4] for more details).
+B.2.1
+Two dimensions
+Stokes’ theorem (B.7) on flat space reads
+�
+d2x ∂µvµ =
+�
+ϵµν dxνvµ =
+�
+(v0dσ − v1dτ),
+(B.10)
+since dΣµ = ϵµνdxν.
+The integral of the curvature is a topological invariant
+χg;b := 1
+4π
+�
+d2σ√g R + 1
+2π
+�
+ds k
+= 2 − 2g − b,
+(B.11)
+called the Euler characteristics and where g is the number of holes and b the number of
+boundaries.
+B.3
+Conformal field theory
+In two dimensions, the energy–momentum tensor is defined by
+Tab = − 4π
+√g
+δS
+δgab .
+(B.12)
+B.3.1
+Complex plane
+Defining the real coordinates (x, y) from the complex coordinate on the complex plane
+z = x + iy,
+z = x − iy,
+(B.13)
+274
+
+we have the formulas:
+ds2 = dx2 + dy2 = dzd¯z,
+gz¯z = 1
+2,
+gzz = g¯z¯z = 0,
+(B.14a)
+ϵz¯z = i
+2,
+ϵz¯z = −2i,
+(B.14b)
+∂ := ∂z = 1
+2 (∂x − i∂y),
+¯∂ := ∂¯z = 1
+2 (∂x + i∂y),
+(B.14c)
+V z = V x + iV y,
+V ¯z = V x − iV y,
+(B.14d)
+d2x = dxdy = 1
+2 d2z,
+d2z = dzd¯z,
+(B.14e)
+δ(z) = 1
+2 δ(2)(x),
+1 =
+�
+d2z δ(2)(z) =
+�
+d2x δ(2)(x),
+(B.14f)
+�
+R
+d2z (∂zvz + ∂¯zv¯z) = −i
+�
+∂R
+�
+dz v¯z − d¯zvz�
+= −2i
+�
+∂R
+(vzdz − v¯zd¯z).
+(B.14g)
+B.3.2
+General properties
+A primary holomorphic field φ(z) of weight h transforms as
+f ◦ φ(z) =
+�df
+dz
+�h
+φ
+�
+f(z)
+�
+(B.15)
+for any local change of coordinates f. A quasi-primary operator transforms like this only
+for f ∈ SL(2, C). Its mode expansion reads
+φ(z) =
+�
+n
+φn
+zn+h ,
+φn =
+�
+C0
+dz
+2πi zn+h−1φ(z),
+(B.16)
+where the integration is counter-clockwise around the origin.
+The SL(2, C) vacuum |0⟩ is defined by
+∀n ≥ −h + 1 :
+φn |0⟩ = 0.
+(B.17)
+Its BPZ conjugate ⟨0| satisfies:
+∀n ≤ h − 1 :
+⟨0| φn = 0.
+(B.18)
+The state–operator correspondence associates a state |φ⟩ to each operator φ(z):
+|φ⟩ := φ(0) |0⟩ = φ−h |0⟩ .
+(B.19)
+The operator corresponding to the vacuum is the identity 1.1
+The Hermitian and BPZ
+conjugated states are
+⟨φ‡| :=⟨0| I ◦ φ†(0) = lim
+z→∞ z2h⟨0| φ†(z),
+⟨φ| :=⟨0| I± ◦ φ(0) = (±1)h lim
+z→∞ z2h⟨0| φ(z).
+(B.20)
+The energy–momentum tensor is a quasi-primary operator of weight h = 2
+T(z) =
+�
+n
+Ln
+zn+2 .
+(B.21)
+1Exceptionally, the state |0⟩ and the operator 1 does not have the same symbol.
+275
+
+The OPE between T and a primary operator h of weight h is
+T(z)φ(w) ∼
+h φ(w)
+(z − w)2 + ∂φ(w)
+z − w .
+(B.22)
+The OPE of T with itself defines the central charge c
+T(z)T(w) ∼
+c/2
+(z − w)4 +
+2T(w)
+(z − w)2 + ∂T(w)
+z − w .
+(B.23)
+B.3.3
+Hermitian and BPZ conjugations
+Both conjugations do not change the ghost number of a state.
+Hermitian
+The Hermitian conjugate of a general state built from n operators Ai and a complex number
+λ is
+(λ A1 · · · An |0⟩)† = λ∗ ⟨0| A†
+n · · · A†
+1.
+(B.24)
+BPZ
+The BPZ conjugate of modes is
+φt
+n = (I± ◦ φ)n = (−1)h(±1)nφ−n,
+(B.25)
+where I±(z) = ±1/z. The plus sign is usually used for the closed string, and the minus sign
+for the open string. Given a general state built from n operators n and a complex number λ,
+the conjugation does not change the order of the operators and does not conjugate complex
+numbers:
+(λ A1 · · · An |0⟩)t = λ ⟨0| (A1)t · · · (An)t.
+(B.26)
+However, it reverses radial ordering such that operators must be (anti-)commuted in radial
+ordered expressions.
+The BPZ product satisfies
+⟨A, B⟩ = (−1)|A||B|⟨B, A⟩.
+(B.27)
+Moreover the inner product is non-degenerate, so
+∀A :
+⟨A|B⟩ = 0
+=⇒
+|B⟩ = 0.
+(B.28)
+Denoting by {|φr⟩} a complete basis of states, then the conjugate basis {⟨φc
+r|} is defined
+by the BPZ product as
+⟨φc
+r|φs⟩ = δrs.
+(B.29)
+We have
+⟨φr|φc
+s⟩ = (−1)|φr|δrs.
+(B.30)
+B.3.4
+Scalar field
+The simplest matter CFT is a set of D scalar field Xµ(z, ¯z) such that the i∂Xµ and i¯∂Xµ
+are of weight h = (1, 0) and h = (0, 1)
+i∂Xµ =
+�
+n
+αµ
+n
+zn+1 ,
+i¯∂Xµ =
+�
+n
+¯αµ
+n
+¯zn+1 .
+(B.31)
+276
+
+The commutation relations between the modes are:
+[αµ
+m, αν
+n] = mδm+n,0ηµν,
+[¯αµ
+m, ¯αν
+n] = mδm+n,0ηµν,
+[αµ
+m, ¯αν
+n] = 0.
+(B.32)
+The zero-modes of both operators are equal and correspond to the (centre-of-mass) mo-
+mentum
+αµ
+0 = ¯αµ
+0 =
+�
+α′
+2 pµ.
+(B.33)
+The conjugate of pµ is the centre-of-mass position xµ:
+[xµ, pν] = ηµν.
+(B.34)
+Vertex operators are defined by
+Vk(z, ¯z) = :eik·X(z,¯z):,
+h = ¯h = α′2k2
+4
+.
+(B.35)
+The scalar vacuum |k⟩ is annihilated by all positive-frequency oscillators and it is char-
+acterized by its eigenvalue for the zero-mode operator
+pµ |k⟩ = kµ |k⟩ ,
+∀n > 0 :
+αµ
+n |k⟩ = 0,
+¯αµ
+n |k⟩ = 0.
+(B.36)
+The vacuum is associated to the vertex operator Vk:
+|k⟩ = Vk(0, 0) |0⟩ = eik·x |0⟩ .
+(B.37)
+The conjugate vacuum is
+⟨k| pµ =⟨k| kµ,
+⟨k| = |k⟩† ,
+⟨−k| = |k⟩t .
+(B.38)
+B.3.5
+Reparametrization ghosts
+The reparametrization ghosts are described by an anti-commuting first-order system with
+the parameters (Chapter 7 and table 7.1):
+ϵ = 1,
+λ = 2,
+cgh = −26,
+qgh = −3,
+agh = −1.
+(B.39)
+We focus on the holomorphic sector.
+The b and c ghosts have weights:
+h(b) = 2,
+h(c) = −1
+(B.40)
+such that the mode expansions are:
+b(z) =
+�
+n∈Z
+bn
+zn+2 ,
+c(z) =
+�
+n∈Z
+cn
+zn−1 ,
+(B.41a)
+bn =
+�
+dz
+2πi zn+1b(z),
+cn =
+�
+dz
+2πi zn−2c(z).
+(B.41b)
+The anti-commutators between the modes bn and cn read:
+{bm, cn} = δm+n,0,
+{bm, bn} = 0,
+{cm, cn} = 0.
+(B.42)
+The energy–momentum tensor and the Virasoro modes are respectively:
+T = −2 :b∂c: − :∂b c:,
+(B.43a)
+Lm =
+�
+n
+�
+n + m
+�
+:bm−ncn: =
+�
+n
+(2m − n) :bncm−n:.
+(B.43b)
+277
+
+The expression of the zero-mode is:
+L0 = −
+�
+n
+n :bnc−n: =
+�
+n
+n :b−ncn:.
+(B.44)
+The commutators between the Ln and the ghost modes are:
+[Lm, bn] =
+�
+m − n
+�
+bm+n,
+[Lm, cn] = −(2m + n)cm+n.
+(B.45)
+In particular, L0 commutes with the zero-modes:
+[L0, b0] = 0,
+[L0, c0] = 0.
+(B.46)
+The anomalous global U(1) symmetry for the ghost number Ngh is generated by the
+ghost current:
+j = −:bc:,
+Ngh,L =
+�
+dz
+2πi j(z),
+(B.47)
+such that
+Ngh(c) = 1,
+Ngh(b) = −1.
+(B.48)
+Remember that Ngh = Ngh,L in the left sector, such that we omit the index L. The modes
+of the ghost current are
+jm = −
+�
+n
+:bm−ncn: = −
+�
+n
+:bncm−n:,
+Ngh,L = j0 = −
+�
+n
+:b−ncn:.
+(B.49)
+The commutator of the current modes with itself and with the Virasoro modes are:
+[jm, jn] = m δm+n,0,
+[Lm, jn] = −njm+n − 3
+2 m(m + 1)δm+n,0.
+(B.50)
+Finally, the commutators of the ghost number operator are:
+[Ngh, b(w)] = −b(w),
+[Ngh, c(w)] = c(w).
+(B.51)
+The level operators N b and N c and number operators N b
+n and N c
+n are defined as:
+N b =
+�
+n>0
+n N b
+n,
+N c =
+�
+n>0
+n N c
+n,
+(B.52a)
+N b
+n = :b−ncn:,
+N c
+n = :c−nbn:.
+(B.52b)
+The commutator of the number operators with the modes are:
+[N b
+m, b−n] = b−nδm,n,
+[N c
+m, c−n] = c−nδm,n.
+(B.53)
+The OPE between the ghosts and different currents are:
+c(z)b(w) ∼
+1
+z − w,
+b(z)c(w) ∼
+1
+z − w,
+b(z)b(w) ∼ 0,
+c(z)c(w) ∼ 0,
+(B.54a)
+T(z)b(w) ∼
+2b(w)
+(z − w)2 + ∂b(w)
+z − w ,
+T(z)c(w) ∼
+−c(w)
+(z − w)2 + ∂c(w)
+z − w .
+(B.54b)
+j(z)b(w) ∼ − b(w)
+z − w,
+j(z)c(w) ∼ c(w)
+z − w.
+j(z)O(w) ∼ Ngh(O) O(w)
+z − w,
+(B.54c)
+j(z)j(w) ∼
+1
+(z − w)2 .
+(B.54d)
+T(z)j(w) ∼
+−3
+(z − w)3 +
+j(w)
+(z − w)2 + ∂j(w)
+z − w .
+(B.54e)
+278
+
+any operator O(z) is defined by
+The OPE (B.54e) implies that the ghost number is not conserved on a curved space:
+N c − N b = 3 − 3g,
+(B.55)
+and leads to a shift between the ghost numbers on the plane and on the cylinder:
+Ngh,L = N cyl
+gh,L + 3
+2.
+(B.56)
+The SL(2, C) vacuum |0⟩ is defined by
+∀n > −2 :
+bn |0⟩ = 0,
+∀n > 1 :
+cn |0⟩ = 0.
+(B.57)
+The mode c1 does not annihilate the vacuum and the two degenerate energy vacua are:
+| ↓⟩ := c1 |0⟩ ,
+| ↑⟩ := c0c1 |0⟩ .
+(B.58)
+The zero-point energy of these states is:
+L0 | ↓⟩ = agh | ↓⟩ ,
+L0 | ↑⟩ = agh | ↑⟩ ,
+agh = −1.
+(B.59)
+The energy for the normal ordering of the different currents is:
+Lm =
+�
+n
+�
+n − (1 − λ)m
+� ⋆
+⋆bm−ncn
+⋆
+⋆ + agh δm,0,
+(B.60a)
+jm =
+�
+n
+⋆
+⋆bm−ncn
+⋆
+⋆ + δm,0.
+(B.60b)
+The energy–momentum and ghost current zero-modes are explicitly:
+L0 =
+�
+n
+n
+⋆
+⋆b−ncn
+⋆
+⋆ + agh = �L0 − 1,
+(B.61a)
+Ngh,L = j0 =
+�
+n
+⋆
+⋆b−ncn
+⋆
+⋆ + 1 = �
+Ngh,L + 1
+2
+�
+N c
+0 − N b
+0
+�
+− 3
+2,
+(B.61b)
+�L0 = N b + N c,
+�
+Ngh,L :=
+�
+n>0
+�
+N c
+n − N b
+n
+�
+.
+(B.61c)
+Then, one can straightforwardly compute the ghost number of the vacua:
+Ngh |0⟩ = 0,
+Ngh | ↓⟩ = | ↓⟩ ,
+Ngh | ↑⟩ = 2 | ↑⟩ .
+(B.62)
+Using (B.56) allows to write the ghost numbers on the cylinder:
+N cyl
+gh | ↓⟩ = −1
+2 | ↓⟩ ,
+N cyl
+gh | ↑⟩ = 1
+2 | ↑⟩ .
+(B.63)
+The bn and cn are Hermitian:
+b†
+n = b−n,
+c†
+n = c−n.
+(B.64)
+The BPZ conjugates of the modes are:
+bt
+n = (±1)nb−n,
+ct
+n = −(±1)nc−n,
+(B.65)
+using I±(z) with (6.111).
+279
+
+The conjugates of the vacuum read:
+| ↓⟩‡ =⟨0| c−1,
+| ↑⟩‡ =⟨0| c−1c0.
+(B.66)
+The BPZ conjugates of the vacua are:
+⟨↓ | := | ↓⟩t = ∓⟨0| c−1,
+⟨↑ | := | ↑⟩t = ±⟨0| c0c−1.
+(B.67)
+We have the following relations:
+⟨↓ | = ∓ | ↓⟩‡ ,
+⟨↑ | = ∓ | ↑⟩‡ .
+(B.68)
+The ghost are normalized with
+⟨↑ | ↓⟩ =⟨↓ | c0 | ↓⟩ =⟨0| c−1c0c1 |0⟩ = 1,
+(B.69)
+which selects the minus sign in the BPZ conjugation. The conjugate of the ghost vacuum is
+⟨0c| =⟨0| c−1c0c1.
+(B.70)
+Considering both the holomorphic and anti-holomorphic sectors, we introduce the com-
+binations:
+b±
+n = bn ± ¯bn,
+c±
+n = 1
+2 (cn ± ¯cn).
+(B.71)
+The normalization of b±
+m is chosen to match the one of L±
+m (B.75), and the one of c±
+m such
+that
+{b+
+m, c+
+n } = δm+n,
+{b−
+m, c−
+n } = δm+n.
+(B.72)
+We have the following useful identities:
+b−
+n b+
+n = 2bn¯bn,
+c−
+n c+
+n = 1
+2 cn¯cn.
+(B.73)
+B.4
+Bosonic string
+The BPZ conjugates of the scalar and ghost modes are
+(αn)t = −(±1)n α−n,
+(bn)t = (±1)n b−n,
+(cn)t = −(±1)n c−n.
+(B.74)
+Combinations of holomorphic and anti-holomorphic modes:
+L±
+n = Ln ± ¯Ln,
+b±
+n = bn ± ¯bn,
+c±
+n = 1
+2 (cn ± ¯cn).
+(B.75)
+The closed string inner product is defined from the BPZ product by an additional inser-
+tion of c−
+0
+⟨A, B⟩ =⟨A| c−
+0 |B⟩ ,
+(B.76)
+while the open string inner product is equal to the BPZ product
+⟨A, B⟩ = ⟨A|B⟩ .
+(B.77)
+The vacuum for the matter and ghosts is
+|k, 0⟩ := |k⟩ ⊗ |0⟩ ,
+|k, ↓⟩ := |k⟩ ⊗ | ↓⟩ .
+(B.78)
+The vacuum is normalized as
+open:
+⟨k, ↓ | c0 |k, ↓⟩ =⟨k′, 0| c−1c0c1 |k, 0⟩ = (2π)Dδ(D)(k + k′),
+(B.79a)
+closed:
+⟨k, ↓↓ | c0¯c0 |k, ↓↓⟩ =⟨k′, 0| c−1¯c−1c0¯c0c1¯c1 |k, 0⟩ = (2π)Dδ(D)(k + k′),
+(B.79b)
+280
+
+Appendix C
+Quantum field theory
+In this appendix, we gather useful information on quantum field theories. The first section
+describes how to compute with path integral with non-trivial measures, generalizing tech-
+niques from finite-dimensional integrals. Then, we summarize the important concepts from
+the BRST and BV formalisms.
+C.1
+Path integrals
+In this section, we explain how analysis, algebra and differential geometry are generalized
+to infinite-dimensional vector spaces (fields).
+C.1.1
+Integration measure
+In order to construct a path integral for the field Φ, one needs to define a notion of distance
+on the space of fields. The distance between a field Φ and a neighbouring field Φ + δΦ is
+|δΦ|2 = G(Φ)(δΦ, δΦ),
+(C.1)
+where G is the (field-dependent) metric on the field tangent space (the field dependence will
+be omitted when no confusion is possible). This induces a metric on the field space itself
+|Φ|2 = G(Φ)(Φ, Φ),
+(C.2)
+from which the integration measure over the field space can be defined as
+dΦ
+�
+det G(Φ).
+(C.3)
+Moreover, the field metric also defines an inner-product between two different elements of
+the tangent space or field space:
+(δΦ1, δΦ2) = G(Φ)(δΦ1, δΦ2),
+(Φ1, Φ2) = G(Φ)(Φ1, Φ2).
+(C.4)
+Remark C.1 (Metric in component form) If one has a set of spacetime fields Φa(x),
+then a local norm is defined by
+|δΦa|2 =
+�
+dx ρ(x)γab
+�
+Φ(x)
+�
+δΦa(x)δΦb(x),
+(C.5)
+which means that the metric in component form is
+Gab(x, y)(Φ) = δ(x − y)ρ(x)γab
+�
+Φ(x)
+�
+.
+(C.6)
+281
+
+Locality means that all fields are evaluated at the same point.
+On a curved space, it is
+natural to write γ only in terms of the metric g and to set ρ(x) =
+�
+det g(x), such that the
+inner-product is diffeomorphism invariant.
+Since a Gaussian integral is proportional to the squareroot of the operator determinant,
+the integration measure can be determined by considering the Gaussian integral over the
+tangent space:
+�
+dδΦ e−G(Φ)(δΦ,δΦ) =
+1
+�
+det G(Φ)
+.
+(C.7)
+Note that one needs to work on the tangent space because G(Φ) can depend on the field,
+which means that the integral
+�
+dΦ e−G(Φ)(Φ,Φ).
+(C.8)
+is not Gaussian.
+Having constructed the Gaussian measure with respect to the metric G(Φ), it is now
+possible to consider the path integral of general functional F of the fields:
+�
+dΦ
+�
+det G(Φ) F(Φ).
+(C.9)
+The (effective) action S(Φ) provides a natural metric on the field space by defining
+√
+det G =
+e−S, or
+S = −1
+2 tr ln G(Φ).
+(C.10)
+However, it can be simpler to work with a Gaussian measure by considering only the quad-
+ratic terms in S, and expanding the rest in a power series.
+In particular, the partition
+function is defined from the classical action Scl by
+Z =
+�
+dΦ e−Scl(Φ).
+(C.11)
+Given an operator D, its adjoint D† is defined with respect to the metric as
+G(δΦ, DδΦ) = G(D†δΦ, δΦ).
+(C.12)
+The free-field measure is such that the metric on the field space is independent from the
+field itself: G(X) = G0. In particular, this implies that the metric is flat and its determinant
+can be absorbed in the measure, setting det G0 = 1. In this case, the measure is invariant
+under shift of the field:
+Φ → Φ + ε
+(C.13)
+such that
+�
+dΦ e− 1
+2 |Φ+ε|2 =
+�
+dΦ e− 1
+2 |Φ|2.
+(C.14)
+This property allows to complete squares and shift integration variables (for example to
+generate a perturbative expansion and to derive the propagator).
+Computation – Equation (C.14)
+�
+dΦ e− 1
+2 |Φ+ε|2 =
+�
+d�Φ det δΦ
+δ�Φ
+e− 1
+2 |�
+Φ|
+2
+=
+�
+d�Φ e− 1
+2 |�
+Φ|
+2
+(C.15)
+The first equality follows by setting �Φ = Φ + ε, and the result (C.14) follows by the
+redefinition �Φ = Φ.
+282
+
+C.1.2
+Field redefinitions
+Under a field redefinition Φ → Φ′, the norm and the measure are invariant:
+dΦ
+�
+det G(Φ) = d�Φ
+�
+det �G(�Φ),
+G(Φ)(δΦ, δΦ) = �G(�Φ)(δ�Φ, δ�Φ).
+(C.16)
+Conversely, one can find the Jacobian J(Φ, �Φ) between two coordinate systems by writing
+dΦ = J(Φ, �Φ)d�Φ,
+J(Φ, �Φ) =
+����det ∂Φ
+∂�Φ
+���� =
+�
+det �G(�Φ)
+det G(Φ).
+(C.17)
+If the measure of the initial field coordinate is normalized such that det G = 1, or equivalently
+�
+dδΦ e−|δΦ|2 = 1,
+(C.18)
+one can determine the Jacobian by performing explicitly the integral
+J(�Φ)−1 =
+�
+dδ�Φ e−�
+G(δ�
+Φ,δ�
+Φ).
+(C.19)
+Remark C.2 (Identity of the Jacobian for Φ and δΦ) The Jacobian agrees on the space
+of fields and on its tangent space. This is most simply seen by using a finite-dimensional
+notation: considering the coordinates xµ and a vector v = vµ∂µ, the Jacobian for changing
+the coordinates to ˜xµ is equivalently
+J = det ∂˜xµ
+∂xµ = det ∂˜vµ
+∂vµ
+(C.20)
+since the vector transforms as
+˜vµ = vν ∂˜xµ
+∂xν .
+(C.21)
+C.1.3
+Zero-modes
+A zero-mode Φ0 of an operator D is a field such that
+DΦ0 = 0.
+(C.22)
+In the definition of the path integral over the space of fields Φ, the measure is defined
+over the complete space. However, this will lead respectively to a divergent or vanishing
+integral if the field is bosonic or fermionic, because the integration over the zero-modes can
+be factorized from the rest of the integral. Writing the field as
+Φ = Φ0 + Φ′,
+(Φ0, Φ′) = 0,
+(C.23)
+where Φ′ is orthogonal to the zero-mode Φ0, a Gaussian integral of an operator D reads:
+Z[D] =
+�
+dΦ
+√
+det G e− 1
+2 (Φ,DΦ) =
+��
+dΦ0
+� �
+dΦ′ e− 1
+2 (Φ′,DΦ′)
+(C.24)
+A first solution could be to simply strip the first factor (for example, by absorbing it in the
+normalization), but this is not satisfactory. In particular, the partition function with source
+Z[D, J] =
+�
+dΦ
+√
+det G e− 1
+2 (Φ,DΦ)−(J,Φ)
+(C.25)
+283
+
+will depend on the zero-modes through the sources.
+But, since the zero-modes are still
+singled out, it is interesting to factorize the integration
+Z[D, J] =
+�
+dΦ0 e−(J,Φ0)
+�
+dΦ′ e− 1
+2 (Φ′,DΦ′)−(J,Φ′)
+(C.26)
+and to understand what makes it finite. Ensuring that zero-modes are correctly inserted
+is an important consistency and leads to powerful arguments. Especially, this can help to
+guess an expression when it cannot be derived easily from first principles.
+To exemplify the problem, consider the cases where there is a single constant zero-mode
+denoted as x (bosonic) or θ (fermionic). The integral over x is infinite:
+�
+dx = ∞.
+(C.27)
+Oppositely, the integral of a Grassmann variable θ vanishes:
+�
+dθ = 0.
+(C.28)
+A Grassmann integral satisfies also
+�
+dθ θ =
+�
+dθ δ(θ) = 1,
+(C.29)
+such that an integral over a zero-mode does not vanish if there one zero-mode in the integrand
+(due to the Grassmann nature of θ, the integrand can be at most linear). By analogy with
+the fermionic case, a possibility for getting a finite bosonic integral is to insert a delta
+function:
+�
+dx δ(x) = 1.
+(C.30)
+We will see that this is exactly what happens for the ghosts and super-ghosts in (super)string
+theories.
+Since ker D is generally finite-dimensional, it is interesting to decompose the zero-mode
+on a basis and to integrate over the coefficients in order to obtain a finite-dimensional
+integral. Writing the zero-mode as
+θ0(x) = θ0iψi(x),
+ker D = Span{ψi}
+(C.31)
+where the coefficients θ0i are constant Grassmann numbers, the change of variables θ →
+(θ0i, θ′) implies:
+dθ =
+1
+�
+det(ψi, ψj)
+dθ′
+n
+�
+i=1
+dθ0i,
+(C.32)
+where n = dim ker D.
+Next, according to the discussion above, one can ask if it is possible to rewrite an integ-
+ration over dθ′ in terms of an integration over dθ together with zero-mode insertions. This
+is indeed possible and one finds:
+dθ
+n
+�
+i=1
+θ(xi) =
+det ψi(xj)
+�
+det(ψi, ψj)
+dθ′.
+(C.33)
+284
+
+Computation – Equation (C.32)
+1 =
+�
+dθ e−|θ|2 =
+�
+dθ′dθ0 e−|θ|2−|θ0|2
+= J
+�
+dθ′ �
+i
+dθ0i e−|θ′|2−|θ0iψi|2 = J
+�
+det(ψi, ψj)
+Computation – Equation (C.33)
+The simplest approach is to start with the LHS. This formula is motivated from the
+previous discussion: if the integration measure contains n zero-modes, it will vanish
+unless there are n zero-mode insertions. Moreover, one can replace each of them by the
+complete field since only the zero-mode part can contribute:
+�
+dθ0
+n
+�
+j=1
+θ(xj) =
+�
+dθ0
+n
+�
+j=1
+θ0(xj) =
+1
+�
+det(ψi, ψj)
+�
+dnθ0i
+n
+�
+j=1
+�
+θ0iψi(xj)
+�
+=
+det ψi(xj)
+�
+det(ψi, ψj)
+� �
+i
+dθ0i θ0i =
+det ψi(xj)
+�
+det(ψi, ψj)
+.
+The third equality follows by developing the product and ordering the θ0i: minus signs
+result from anticommuting the θ0i such that one gets the determinant of the basis
+elements.
+C.2
+BRST quantization
+Consider an action Sm[φi] which depends on some fields φi subject to a gauge symmetry:
+δφi = ϵaδaφi = ϵaRi
+a(φ),
+(C.34)
+where ϵa are the (local) bosonic parameters, such that the action is invariant
+ϵaδaSm = 0.
+(C.35)
+The gauge transformations form a Lie algebra with structure coefficients f c
+ab
+[δa, δb] = f c
+abδc.
+(C.36)
+It is important 1) that the algebra closes off-shell (without using the equations of motion),
+2) that the structure coefficients are field independent and 3) that the gauge symmetry is
+irreducible (each gauge parameter is independent).
+Remark C.3 (Interpretation of the Ri
+a matrices) If the φi transforms in a represent-
+ation R of the gauge group, then the transformation is linear in the field
+Ri
+a(φ) = (T R
+a )i
+jφj,
+(C.37)
+with T R
+a the generators in the representation R. But, in full generality, this is not the case:
+for example the gauge fields Aa
+µ do not transform in the adjoint representation even if they
+carry an adjoint index (only the field strength does), and in this case
+Rb
+aµ = δb
+a∂µ + f c
+abAb
+µ.
+(C.38)
+When the fields φi form a non-linear sigma models, the Ri
+a(φ) correspond to Killing
+vectors of the target manifold.
+285
+
+In order to fix the gauge symmetry in the path integral
+Z = Ω−1
+gauge
+�
+dφi e−Sm,
+(C.39)
+gauge fixing conditions must be imposed:
+F A(φi) = 0.
+(C.40)
+Indeed, without gauge fixing, the integration is performed over multiple identical configura-
+tions and the result diverges. The index A is different from the gauge index a because they
+can refer to different representations, but for the gauge fixing to be possible they should run
+over as many values.
+Next, ghost fields ca (fermionic) are introduced for every gauge parameter, anti-ghosts
+bA (fermionic) and auxiliary (Nakanishi–Laudrup) fields BA (bosonic) for every gauge con-
+dition. The gauge-fixing and ghost actions are then defined by
+Sgh = bAca δaF A(φi),
+(C.41a)
+Sgf = −i BAF A(φi)
+(C.41b)
+such that the original partition function is equivalent to
+Z =
+�
+dφi dbA dca dBA e−Stot
+(C.42)
+where
+Stot = Sm + Sgf + Sgh.
+(C.43)
+The total action is invariant
+δϵStot = 0.
+(C.44)
+under the (global) BRST transformations
+δϵφi = iϵ caδaφi,
+δϵca = − i
+2 ϵ f a
+bccbcc,
+δϵbA = ϵ BA,
+δϵBA = 0,
+(C.45)
+where ϵ is an anti-commuting constant parameter.
+Note that the original action Sm is
+invariant by itself since the transformation acts like a gauge transformation with parameter
+ϵca. The transformation of ca follows because it transforms in the adjoint representation of
+the gauge group. Direct computations show that this transformation is nilpotent
+δϵδϵ′ = 0.
+(C.46)
+These transformations are generated by a (fermionic) charge QB called the BRST charge
+δϵφi = i [ϵQB, φi]
+(C.47)
+and similarly for the other fields (stripping the ϵ outside the commutator turns it to an
+anticommutator if the field is fermionic). Taking the ghosts to be Hermitian leads to an
+Hermitian charge.
+An important consequence is that the two additional terms of the action can be rewritten
+as a BRST exact terms
+Sgf + Sgh = {QB, bAF A}.
+(C.48)
+A small change in the gauge-fixing condition δF leads to a variation of the action
+δS = {QB, bAδF A}.
+(C.49)
+286
+
+The BRST charge should commute with the Hamiltonian in order to be conserved: this
+should hold in particular when changing the gauge fixing condition
+[QB, {QB, bAδF A}] = 0
+=⇒
+Q2
+B = 0.
+(C.50)
+Some vocabulary is needed before proceeding further. A state |ψ⟩ is said to be BRST
+closed if it is annihilated by the BRST charge
+|ψ⟩ closed
+⇐⇒
+|ψ⟩ ∈ ker QB
+⇐⇒
+QB |ψ⟩ = 0.
+(C.51)
+States which are in the image of QB (i.e. they can be written as QB applied on some other
+states) are said to be exact
+|ψ⟩ exact
+⇐⇒
+|ψ⟩ ∈ Im QB
+⇐⇒
+∃ |χ⟩ : |ψ⟩ = QB |χ⟩ .
+(C.52)
+The cohomology H(QB) of QB is the set of closed states which are not exact
+|ψ⟩ ∈ H(QB)
+⇐⇒
+|ψ⟩ ∈ ker QB,
+∄ |χ⟩ : |ψ⟩ = QB |χ⟩ .
+(C.53)
+Hence the cohomology corresponds to
+H(QB) = ker QB
+Im QB
+.
+(C.54)
+Two elements of the cohomology differing by an exact state are in the same equivalence class
+|ψ⟩ ≃ |ψ⟩ + QB |χ⟩ .
+(C.55)
+Considering the S-matrix ⟨ψf|ψi⟩ between a set of physical initial states ψi and final
+states ψf, a small change in the gauge-fixing condition leads to
+δF ⟨ψf|ψi⟩ =⟨ψf| {QB, bAδF A} |ψi⟩
+(C.56)
+after expanding the exponential to first order. Since the S-matrix should not depend on the
+gauge this implies that a physical state ψ must be BRST closed (i.e. invariant)
+QB |ψ⟩ = 0.
+(C.57)
+Conversely, this implies that any state of the form QB |χ⟩ cannot be physical because it is
+orthogonal to every physical state |ψ⟩
+⟨ψ| QB |χ⟩ = 0.
+(C.58)
+This implies in particular that the amplitudes involving |ψ⟩ and |ψ⟩ + QB |χ⟩ are identical,
+and any amplitude for which an external state is exact vanishes. As a conclusion, physical
+states are in the BRST cohomology
+|ψ⟩ physical
+⇐⇒
+|ψ⟩ ∈ H(QB).
+(C.59)
+If there is a gauge where the ghosts decouple from the matter field, then the invariance
+of the action and of the S-matrix under changes of the gauge fixing ensures that this state-
+ment holds in any gauge (but, one still need to check that the gauge preserves the other
+symmetries). If such a gauge does not exist, then one needs to employ other methods to
+show the desired result.
+Note that BA can be integrated out by using its equations of motion
+δF A
+δφi BA = −δSm
+δφi ,
+(C.60)
+287
+
+and this modifies the BRST transformation of the anti-ghost to
+δϵbA = −ϵ
+�δF A
+δφi
+�−1 δSm
+δφi .
+(C.61)
+It is also possible to introduce a term
+{QB, bABBM AB} = i BAM ABBB
+(C.62)
+for any constant matrix M AB. Since this is also a BRST exact term, the amplitudes are
+not affected. Integrating over BA produces a Gaussian average instead of a delta function
+to fix the gauge.
+In the previous discussion, the BRST symmetry was assumed to originate from the
+Faddeev–Popov gauge fixing. But, in fact, it is possible to start directly with an action of
+the form
+S[φ, b, c, B] = S0[φ] + QBΨ[φ, b, c, B]
+(C.63)
+where Ψ has ghost number −1. It can be proven that this is the most general action invariant
+under the BRST transformations (C.45). This can describe gauge fixed action which cannot
+be described by the Faddeev–Popov procedure: in particular, the latter yields actions which
+are quadratic in the ghost fields (by definition of the Gaussian integral representation of the
+determinant), but this does not exhaust all the possibilities. For example, the background
+field method applied to Yang–Mills theory requires using an action quartic in the ghosts.
+In this section, several hypothesis have been implicit (off-shell closure, irreducibility and
+constant structure coefficients). If one of them breaks, then it is necessary to employ the
+more general BV formalism.
+C.3
+BV formalism
+The Batalin–Vilkovisky (BV, or also field–antifield) formalism is the most general framework
+to quantize theories with a gauge symmetry. While the BRST formalism (Appendix C.2)
+is sufficient to describe simple systems, it breaks down when the structure of the gauge
+symmetry is more complicated, for example in systems implying gravity. The BV formalism
+is required in the three following cases (which can occur simultaneously):
+1. the gauge algebra is open (on-shell closure);
+2. the structure coefficients depend on the fields;
+3. the gauge symmetry is reducible (not all transformations are independent).
+The BV formalism is also useful for standard gauge symmetries to demonstrate renormaliz-
+ability and to deal with anomalies.
+As explained in the previous section, the ghosts and the BRST symmetry are crucial to
+ensure the consistency of the gauge theory. The idea of the BV formalism is to put on an
+equal footing the physical fields and all the required auxiliary and ghost fields (before gauge
+fixing). The introduction of antifields – one for each of the fields – and the description of the
+full quantum dynamics in terms of a quantum action (constrained by the quantum master
+equation) ensure the consistency of the system. Additional benefits are the presence of a
+(generalized) BRST symmetry, the existence of a Poisson structure (which allows to bring
+concepts from the Hamiltonian formalism), the covariance of the formalism and the simple
+interpretation of counter-terms as corrections to the classical action.
+For giving a short intuition, the BV formalism can be interpreted as providing a (anti)ca-
+nonical structure in the Lagrangian formalism, the role of the Hamiltonian being played by
+the action.
+288
+
+C.3.1
+Properties of gauge algebra
+Before explaining the BV formalism, we review the situations listed above. The classical
+action for the physical fields φi is denoted by S0[φ] and the associated equations of motion
+by
+Fi(φ) = ∂S0
+∂φi .
+(C.64)
+Then, a gauge algebra is open and has field-dependent structure coefficients F c
+ab(φ) if:
+[Ta, Tb] = F c
+ab(φ)Tc + λi
+abFi(φ).
+(C.65)
+On-shell, Fi = 0 and the second term is absent, such that the algebra closes. The fields
+themselves are constants from the point of view of the gauge algebra, but their presences in
+the structure coefficients complicate the analysis of the theory. Moreover, the path integral
+is off-shell and for this reason one needs to take into account the last term.
+Finally, the gauge algebra can be reducible: in brief, it means that there are gauge
+invariances associated to gauge parameters – and correspondingly ghosts for ghosts –, and
+this recursively. Since there is one independent ghost for each generator, there are too many
+ghosts if the generators are not all independent, and there is a remnant gauge symmetry for
+the ghost fields (in the standard Faddeev–Popov formalism, the ghosts are not subject to
+any gauge invariance). This originates from relations between the generators Ri
+a: denoting
+by m0 the number of level-0 gauge transformations, the number of independent generators
+is rank Ri
+a. Then, the
+m1 = m0 − rank Ri
+a
+(C.66)
+relations between the generators translate into a level-1 gauge invariance of the ghosts. This
+symmetry can be gauge fixed by performing a second time the Faddeev–Popov procedure,
+yielding commuting ghosts. This symmetry can also be reducible, and the procedure can
+continue without end. If one finds that the gauge invariance at level n = ℓ is irreducible, one
+says that the gauge invariance is ℓ-reducible. If this does not happen, one defines ℓ = ∞.
+The number of generators at level n is denoted by mn.
+Example C.1 – p-form gauge theory
+A p-form gauge theory is written in terms of a gauge field Ap with a a gauge invariance
+δAp = dλp−1.
+(C.67)
+But, due to the nilpotency of the derivative, deformations of the gauge parameter
+satisfying
+δλp−1 = dλp−2
+(C.68)
+does not translate into a gauge invariance of Ap. Similarly from this should be excluded
+the transformation
+δλp−2 = dλp−3,
+(C.69)
+and so on until one reaches the case p = 0. Hence, a p-form field has a p-reducible
+gauge invariance.
+C.3.2
+Classical BV
+Denoting the fields collectively as
+ψr = {φi, BA, bA, ca},
+(C.70)
+289
+
+the simplest BV action reads
+S[ψr, ψ∗
+r] = S0[φ] + QBψr ψ∗
+r
+(C.71)
+with the antifields
+ψ∗
+r = {φ∗
+i , BA∗, bA∗, c∗
+a}.
+(C.72)
+The action (C.63) is recovered by writing
+ψ∗
+r = ∂Ψ
+∂ψr .
+(C.73)
+This indicates that the general BRST formalism could be rephrased in the BV language.
+But, in the same way that the BRST formalism generalizes the Faddeev–Popov formalism,
+it is in turn generalized by the BV formalism. Indeed, the above action is linear in the
+antifields: this constraint is not required and one can write more general actions. In the rest
+of this section, we explain how this works at the level of the action (classical level) and how
+the sets of fields and antifields are defined.
+Consider a set of physical fields φi with the gauge invariance
+δφi = ϵa0
+0 Ri
+a0(φi).
+(C.74)
+Then, associate a ghost field ca0 to each of the gauge parameters ϵa0. If the gauge symmetry
+is reducible, a new gauge invariance is associated to the ghosts
+δca0
+0 = ϵa1
+1 Ra0
+a1(φi, ca0).
+(C.75)
+This structure is recurring and the ghosts of the level-n gauge invariance are denoted by can
+and they satisfy
+δcan
+n = ϵan+1
+n+1 Ran
+an+1(φi, ca0
+0 , . . . , can
+n ).
+(C.76)
+Thus, the set of fields is
+ψr = {can
+n }n=−1,...,ℓ,
+c−1 := φ.
+(C.77)
+A ghost number is introduced
+Ngh(φi) = 0,
+Ngh(can
+n ) = n + 1,
+(C.78)
+and the Grassmann parity of the ghosts is defined to be opposite (resp. identical) of the
+parity of the associated gauge parameter for even (resp. odd) n
+|cn| = |ϵan
+n | + n + 1.
+(C.79)
+To each of these fields is associated an antifield ψ∗
+r of opposite parity as ψr and such that
+their ghost numbers sum to −1
+Ngh(ψ∗
+r) = −1 − Ngh(ψr),
+|ψ∗
+r| = −|ψr|.
+(C.80)
+The fields and antifields together are taken to define a graded symplectic structure
+ω =
+�
+r
+dψr ∧ dψ∗
+r
+(C.81)
+with respect to which they are conjugated to each other
+(ψr, ψ∗
+s) = δrs,
+(ψr, ψs) = 0,
+(ψ∗
+r, ψ∗
+s) = 0.
+(C.82)
+290
+
+The antibracket (graded Poisson bracket) (·, ·) reads
+(A, B) = ∂RA
+∂ψr
+∂LB
+∂ψ∗r
+− ∂RA
+∂ψ∗r
+∂LB
+∂ψr ,
+(C.83)
+where the L and R indices indicate left and right derivatives. It is graded symmetric, which
+means
+(A, B) = −(−1)(|A|+1)(|B|+1)(B, A).
+(C.84)
+It also satisfies a graded Jacobi identity and the property
+Ngh((A, B)) = Ngh(A) + Ngh(B) + 1,
+|(A, B)| = |A| + |B| + 1
+mod 2.
+(C.85)
+Moreover, the antibracket acts as a derivative
+(A, BC) = (A, B)C + (−1)|B|C(A, C)B.
+(C.86)
+The dynamics of the theory is described by the (classical) master action S[ψr, ψ∗
+r] which
+satisfies
+Ngh(S) = 0,
+|S| = 0.
+(C.87)
+In order to reproduce correctly the dynamics of the classical system without ghosts, this
+action is required to satisfy the boundary condition
+S[ψr, ψ∗
+r = 0] = S0[φi],
+∂L∂RS
+∂c∗
+n−1,an−1∂can
+n
+����
+ψ∗=0
+= Ran−1
+an
+.
+(C.88)
+Indeed, if the antifields are set to zero, the ghost fields cannot appear because they all have
+positive ghost numbers and it is not possible to build terms with vanishing ghost numbers
+from them.
+In analogy with the Hamiltonian formalism, the master action can be used as the gen-
+erator of a global fermionic symmetry, and inspection will show that it corresponds to a
+generalization of the BRST symmetry. Writing the generalized and classical BRST operator
+as s, the transformations of the fields and antifields read
+δθψr = θ sψr = −θ (S, ψr) = θ ∂RS
+∂ψ∗r
+,
+(C.89a)
+δθψ∗
+r = θ sψ∗
+r = −θ (S, ψ∗
+r) = −θ ∂RS
+∂ψr ,
+(C.89b)
+where θ is a constant Grassmann parameter. The variation of a generic functional F[ψr, ψ∗
+r]
+is
+δθF = θ sF = −θ (S, F).
+(C.90)
+For the BRST transformation to be a symmetry of the action, the action must satisfy the
+classical master equation
+(S, S) = 0.
+(C.91)
+This equation can easily be solved by expanding S in the ghosts: the various terms can be
+interpreted in terms of properties of the gauge algebra. Then, the Jacobi identity used with
+two S and an arbitrary functional gives
+(S, (S, F)) = 0
+(C.92)
+and this implies that the transformation is nilpotent
+s2 = 0.
+(C.93)
+291
+
+A classical observable O satisfies
+sO = 0.
+(C.94)
+Due to the BRST symmetry, the action is not uniquely defined and the action
+S′ = S + (S, δF)
+(C.95)
+also satisfies the master equation, where δF is arbitrary up to the condition Ngh(δF) = −1.
+This can be interpreted as the action S in a new coordinate system (ψ′r, ψ′∗
+r ) with
+ψ′ = ψ − δF
+δψ∗ ,
+ψ′∗ = ψ∗ + δF
+δψ
+(C.96)
+such that
+S′[ψ, ψ∗] = S
+�
+ψ − δF
+δψ∗ , ψ∗ + δF
+δψ
+�
+.
+(C.97)
+Indeed, for F = F[ψ, ψ∗], one has
+S′[ψ, ψ∗] = S[ψ, ψ∗] + (S, ψ)δF
+δψ + (S, ψ∗) δF
+δψ∗ = S[ψ, ψ∗] − ∂RS
+∂ψ∗
+δF
+δψ + ∂RS
+∂ψ
+δF
+δψ∗ .
+(C.98)
+It can be shown that this transformation preserves the antibracket and the master equation
+(ψ′r, ψ′∗
+s ) = δrs,
+(S′, S′) = 0.
+(C.99)
+More generally, any transformation preserving the antibracket is called an (anti)canonical
+transformation. One can also consider generating functions depending on both the old and
+new coordinates, as is standard in the Hamiltonian formalism. Under a transformation, any
+object depending on the coordinates changes as
+G′ = G + (δF, G).
+(C.100)
+One can consider finite transformation without problems.
+In order to perform the gauge fixing, one needs to eliminate the antifields. A convenient
+condition is
+SΨ[ψr] = S
+�
+ψr, ∂Ψ
+∂ψr
+�
+,
+ψ∗
+r = ∂Ψ
+∂ψr ,
+(C.101)
+where Ψ[ψr] is called the gauge fixing fermion and satisfies
+Ngh(Ψ) = −1,
+|Ψ| = 1.
+(C.102)
+From the discussion on coordinate transformations this amounts to work in new coordinates
+where ψ′∗
+r = 0. But such a function Ψ cannot be built from the fields because they all have
+positive ghost numbers. One needs to introduce trivial pairs of fields.
+A trivial pair (B, ¯c) is defined by the properties
+|B| = −|¯c|,
+Ngh(B) = Ngh(¯c) + 1,
+(C.103a)
+s¯c = B,
+sB = 0
+(C.103b)
+and the new action reads
+¯S = S[ψr, ψ∗
+r] − B¯c∗
+(C.104)
+(the position dependence is kept implicit). In this context ψr and ψ∗
+r are sometimes called
+minimal variables. From this, one learns that
+( ¯S, ¯S) = (S, S) = 0.
+(C.105)
+292
+
+At level-0, one introduces the pair
+(B0a0, ¯c0a0) := (B0
+0a0, ¯c0
+0a0)
+(C.106)
+and the associated antifields. The field ¯c0 := b is the Faddeev–Popov anti-ghost associated
+to c0 and the trivial pair satisfies
+|B0| = |ϵ0|,
+|¯c0| = −|ϵ0|,
+Ngh(B0) = 0,
+Ngh(c0) = −1.
+(C.107)
+For the level 1, two additional pairs are introduced:
+(B0
+1a1, ¯c0
+1a1),
+( ¯B1a1
+1
+, c1a1
+1
+)
+(C.108)
+and the corresponding antifields. The motivation for adding an additional pair is that the
+level-0 pair only fixes m0 − m1 of the generators: the additional m1 extra-ghosts c1a1
+1
+can
+be fixed by the residual level-0 symmetry. The first level-1 pair fixes the level-1 symmetry.
+Then, the gauge fixed action enjoys a BRST symmetry acting only on the fields
+δθψr = θ sψr = θ ∂RS
+∂ψ∗r
+����
+ψ∗
+r =∂rΨ
+.
+(C.109)
+Note that this BRST operator is generically nilpotent only on-shell
+s2 ∝ eom.
+(C.110)
+C.3.3
+Quantum BV
+At the quantum level, one considers the path integral
+Z =
+�
+dψrdψ∗
+r e−W [ψr,ψ∗
+r ]/ℏ
+(C.111)
+where W is called the quantum master action. The reason for distinguishing it from the
+classical master action S is that the measure is not necessarily invariant by itself under
+the generalized BRST transformation – this translates into a non-gauge invariance of the
+measure of the physical fields, i.e. a gauge anomaly.
+Quantum BRST transformation are generated by the quantum BRST operator σ
+δθF = θ σF = (W, F) − ℏ ∆F,
+(C.112)
+where
+∆ = ∂R
+∂ψ∗r
+∂L
+∂ψr .
+(C.113)
+Then, the path integral is invariant if W satisfies the quantum master equation
+(W, W) − 2ℏ∆W = 0,
+(C.114)
+which can also be written as
+∆e−W/ℏ = 0.
+(C.115)
+This can be interpreted as the invariance of Z under changes of coordinates: indeed one
+finds that
+δW = 1
+2(W, W),
+(C.116)
+and the integration measure picks a Jacobian
+sdet J ∼ 1 + ∆W.
+(C.117)
+293
+
+In the limit ℏ → 0, one recovers the classical master equation. More generally, the action
+can be expanded in powers of ℏ
+W = S +
+�
+p≥1
+ℏpWp.
+(C.118)
+Observables are given by operators O[ψ, ψ∗] invariant under σ:
+σO = 0,
+(C.119)
+which ensures that the expectation value is invariant under changes of Ψ
+δ⟨O⟩ = 0.
+(C.120)
+Note that if O depends just on ψ the condition reduces to sO = 0, but generically there is
+no such operators (except constants) satisfying this condition for open algebra.
+Consider the gauge fixed integral
+Z =
+�
+dψr e−WΨ[ψr],
+WΨ[ψr] = W
+�
+ψr, ∂Ψ
+∂ψr
+�
+.
+(C.121)
+Varying the gauge fixing fermion by δΨ gives
+Z =
+�
+dψr e−WΨ[ψr]
+�∂RS
+∂ψ∗r
+�
+ψ∗=∂ψΨ
+∂(δΨ)
+∂ψr .
+(C.122)
+Integrating by part gives the quantum master equation.
+C.4
+Suggested readings
+• Manipulations of functional integral are given in [100, sec. 15.1, 22.1, 172, chap. 14,
+191, 53].
+• Zero-modes are discussed in [23].
+• A general summary of path integrals for bosonic and fermionic fields can be found
+in [193, app. A].
+• BRST formalism: most QFT books contain an introduction, more complete references
+are [251, chap. 15, 247, 105];
+• BV formalism [251, chap. 15, 88, 93, 247, 105, 49] (several explicit examples are given
+in [93, sec. 3], see [12, 231, 254] for more specific details).
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+309
+
+Index
+#, 187, 190
+Σ0, see Riemann sphere
+Σg, Σg,n, see Riemann surface
+χg, χg,n, see Euler characteristics
+| ↓⟩, see first-order system, vacuum
+| ↑⟩, see first-order system, vacuum
+⟨·⟩, 100
+ωg,n
+p
+, see Pg,n space, p-form
+∼, 95
+†, see Hermitian adjoint
+‡, see Euclidean adjoint
+c, see conjugate state
+t, see BPZ conjugation
+[· · · ]g, see string product
+{· · · }g, see fundamental vertex
+1PI vertex, 222, 236
+region, 192
+1PR region, 192
+2d gravity, 51, 135
+adjoint, see Euclidean, Hermitian adjoint
+AdS/CFT, 239
+Ag,n, see (off-shell) string amplitude
+αn, see scalar field CFT, mode expansion
+background independence, 238
+background metric, 32
+Batalin–Vilkovisky formalism, 23, 234, 288
+antibracket, 291
+BRST transformation, 291
+classical master equation, 291
+field redefinition, 292
+fields and antifields, 289
+gauge fixing, 292
+quantum BRST transformation, 293
+quantum master equation, 293
+bc CFT, see first-order CFT, reparametriza-
+tion bc ghosts
+Beltrami differential, 41
+Berkovits’ super-SFT, 256
+βγ CFT, see first-order CFT, superconformal
+βγ ghosts
+bosonic string CFT, 138
+L0, 138, 142
+complex parametrization, 141
+Hilbert space, 138
+level operator, 138, 142
+light-cone parametrization, 141
+boundary condition, 97
+Neveu–Schwarz (NS), 98
+Ramond (R), 98
+twisted, 98
+untwisted, 98
+BPZ conjugation, 97
+mode, 98
+state, 101
+BRST cohomology, see also string states, 69,
+134
+absolute, 69, 139, 146
+relative, 70, 141, 144, 145, 147
+semi-relative, 70, 147
+two flat directions, 138–148
+BRST current, 135
+OPE, 135–136
+BRST operator, 135
+commutator, 137
+full
+zero-mode decomposition, 147
+mode expansion, 136
+zero-mode decomposition, 136
+BRST quantization
+change of gauge fixing condition, 287
+charge nilpotency, 287
+cohomology, 287
+Nakanishi–Lautrup auxiliary field, 286
+physical states, 287
+transformations, 286
+BRST SFT, see covariant SFT
+BV formalism, see Batalin–Vilkovisky form-
+alism
+central charge, 33, 90, 95
+CFT, see conformal field theory
+310
+
+CISO, see conformal isometry group
+CKV, see conformal Killing vector
+classical solution
+marginal deformation, 242
+closed string amplitude
+Feynman diagram decomposition, 215
+on punctured moduli space, 178
+tree-level
+3-point, 171
+4-point, 173, 200
+closed string fundamental vertex, 214
+g-loop
+0-point, 221
+1-loop
+0-point, 221
+1-point, 218
+properties, 223
+recursive construction, 217
+special vertices, 221
+tree-level
+0-point, 221
+1-point, 221
+2-point, 221, 232
+3-point, 172
+4-point, 175
+closed string product, 223
+g = 0, n = 1, 230, 233
+ghost number, 223
+closed string states
+dilaton Φ, 17
+Kalb–Ramond Bµν, 17
+level-matching, see level-matching
+massless field, 17
+metric Gµν, 17
+tachyon T, 17
+physical, 148
+zero-mode decomposition, 176
+closed string vertex
+fundamental identity, 223
+commutator
+[Lm, Ln], 90
+[Lm, φn], 99
+CFT, 92–93
+complex coordinates, 72, 82
+cylinder, 83
+conformal
+anomaly, see Weyl anomaly
+dimension, 88
+factor, see Liouville field
+spin, 88
+vacuum, see SL(2, C) vacuum
+weight, 88
+conformal algebra, 79
+2d, see Witt algebra, Virasoro algebra
+conformal field theory
+classical, 84
+cylinder, 105
+definition, 88
+finite transformation, 88
+conformal isometry group, see also conformal
+Killing vector, 78
+sphere, see SL(2, C)
+conformal Killing
+equation, 44, 79
+group volume, 46
+vector, 44, 62, 79
+conformal structure, 31
+conjugate state, 102
+conserved charge, 99, 268, 274
+CFT, 93
+conserved current, 268
+CFT, 93
+mode expansion, 99
+contracting homotopy operator, 139, 144
+contraction, 95
+conventions
+complex coordinates, 268
+ghost number, 129
+spacetime momentum current, 109
+correlation function, 90
+quasi-primary operator, 90
+sphere 1-point (-), 91
+sphere 2-point (-), 91
+sphere 3-point (-), 91
+covariant SFT, see also classical (-), free (-),
+quantum (-)
+background independence, 239
+closed bosonic
+1PI action, 236
+1PI gauge symmetry, 237
+classical action, 232
+classical equation of motion, 233
+classical gauge algebra, 233
+classical gauge transformation, 233
+fields and antifields, 235
+gauge fixed action, 230, 231
+inner product, 168
+normalization, 231
+quantum action, 235
+quantum BV master equation, 235
+open bosonic
+inner product, 156
+parameters, 231
+renormalization, 231
+311
+
+critical dimension, 14, 19, 48, 77, 135, 137
+superstring, 245
+cross-ratio, 91
+degeneration limit, 191
+diffeomorphism, 30
+group volume, 39, 44, 46
+infinitesimal (-), 30
+large (-), 31
+dual state, see conjugate state
+ϵ (scalar action sign), 107
+ηξ ghosts, 246
+energy vacuum, 100
+energy–momentum tensor
+finite transformation, 95
+mode expansion, 99
+Euclidean adjoint, 96–97
+mode, 98
+state, 101
+Euler characteristics, 30, 59
+extended complex plane ¯C, 81
+f◦, see CFT, finite transformation
+factorization, see string amplitude
+Faddeev–Popov
+determinant, 44
+gauge fixing, see path integral
+Fg,n, see propagator region
+F1PR
+g,n , see 1PR region
+field space, see also path integral
+DeWitt metric, 35
+inner-product, 34, 281
+norm, 34, 281
+field theory space, 239
+connection, 240
+Hilbert space bundle, 240
+first-order CFT, 119
+L0, 124, 128
+U(1) ghost current, 120, 129
+mode expansion, 124
+transformation law, 122
+U(1) symmetry, 119, 120
+action, 119
+boundary condition, 123
+BPZ conjugate
+modes, 132
+vacuum, 132
+central charge, 121
+commutator, 125
+complex components, 119
+cylinder, 122
+energy normal ordering, 128
+energy–momentum tensor, 120
+mode expansion, 124
+equation of motion, 119
+Euclidean adjoint
+modes, 132
+vacuum, 132
+Fock space, see Hilbert space, 131
+ghost charge, 121
+ghost number, 120, 129
+cylinder, 122
+Hilbert space
+full, 131
+holomorphic, 131
+inner product, 133
+level operator, 124
+mode expansion, 123
+number operator, 124
+OPE, 121–123
+propagator, 120
+summary, 133
+vacuum
+SL(2, C), 126
+conjugate, 133
+energy, 127–128
+Virasoro operators, 124, 129
+weight, 120
+zero-mode decomposition, 131
+zero-point energy, 127
+free covariant SFT
+closed bosonic, 168
+action, 232
+classical action, 168
+equation of motion, 167
+gauge fixed action, 169, 228
+gauge fixed equation of motion, 159,
+169, 231
+gauge transformation, 169
+open bosonic
+BV action, 164
+classical action, 156, 157
+equation of motion, 155
+gauge fixed action, 159
+gauge transformation, 157, 165
+zero-mode decomposition, 156
+path integral, see string field path integ-
+ral
+free SFT, 140
+free super-SFT
+action, 257, 258
+gauge transformation, 257, 258
+fundamental vertex, see also closed string (-)
+312
+
+interpretation, 231
+region, 191
+G, see spacetime ghost number
+ˆgab, see background metric
+gab, see worldsheet metric
+ghost number, 52
+anomaly, 204, 250
+ghosts, see first-order CFT, reparametriza-
+tion bc ghosts, superconformal βγ
+ghosts
+gluing compatibility, 190
+Green function, 59–61, 215
+tree-level 2-point, 221
+group
+fundamental domain, 39
+volume, 38
+GSO symmetry, 247
+Hermitian adjoint, 96
+higher-genus Riemann surface
+conformal group, 89
+Hilbert space
+CFT, 100
+holomorphic factorization, 77, 251
+holomorphic/anti-holomorphic sectors, 83, 267
+I(z), I±(z), see inversion
+iε-prescription, 262
+index
+amplitude, 214
+Riemann surface, 194, 214
+inversion map, 87, 97
+ISO, see isometry group
+isometry group, 79
+ker P1, see conformal Killing vector
+ker P †
+1 , see quadratic differential
+Kg, see conformal Killing vector
+Killing vector, 79
+L±, 99
+L∞ algebra, 234, 255
+left/right-moving sectors, 83, 267
+level-matching condition, 16, 70, 147, 168,
+176, 184, 203, 205, 216, 228, 257
+ℓg,n, see string product
+light-cone coordinates, 83
+Liouville
+action, 47
+central charge, 48
+field, 32, 47
+free-field measure, 40
+theory, 88, 90, 264
+local coordinates, 23, 171, 179
+constraints, 184, 190
+global phase, 184, 202, 204
+global rescaling, 195
+reparametrization, 184, 202, 203
+transition function, 184, 185
+logarithmic CFT, 91
+mapping class group (MCG), see modular
+group
+marginal deformation
+action, 239, 240
+correlation function, 240
+marked Riemann surface, see punctured Riemann
+surface
+metric
+gauge
+conformal (-), 32
+conformally flat (-), 32, 72
+flat (-), 32
+gauge decomposition, 39, 41, 43, 50
+gauge fixing, 31, 37
+uniformization gauge, 32
+local rescaling, see Weyl transformation
+Mg, see moduli space
+Möbius group, see SL(2, C)
+mode expansion, 97
+Hermiticity, 98
+mode range, 97
+modular group, 31, 38
+moduli space, 21, 37
+complex coordinates, 76
+plumbing fixture decomposition, 190–194
+with punctures, 178
+momentum-space SFT
+action, 261
+consistency, 264
+Feynman rules, 261
+finiteness
+infinite number of states, 222, 261
+UV divergence, 222, 261
+Green function, 262
+interaction vertex, 261
+properties, 260
+string field expansion, 260
+Ngh, see ghost number
+non-locality, 12, 260
+normal ordering, 103
+conformal (-), 103
+energy (-), 103
+313
+
+mode relation, 105
+off-shell closed string amplitude, 179, 199
+contribution from subspace, 199
+tree-level
+3-point, 172
+4-point, 218
+off-shell string amplitude, 23, 177
+conformal invariance, 172
+off-shell superstring amplitude, 250
+consistency, 250, 252
+factorization, 251–252
+PCO insertions, 250
+supermoduli space, 251
+old covariant quantization (OCQ), 146
+on-shell condition, 16, 60, 70, 92, 140, 144,
+146, 147
+OPE, see operator product expansion
+open string states
+gauge field Aµ, 17
+gauge invariance, 17
+momentum expansion, 17
+tachyon T, 17
+operator product expansion, 94
+GG, 246
+T-primary, 95
+TG, 246
+TT, 95
+identity-primary, 94
+out-of-Siegel gauge constraint, 159
+p-brane, 11
+P1, 41, 42
+path integral
+Faddeev–Popov gauge fixing, 36
+field redefinition, 283
+measure, 34, 282
+free-field, 34, 282
+ultralocality, 34, 49
+PCO, see picture changing operator
+ˆPg,n, 184
+Pg,n space, 179
+p-form, 197, 198
+BRST identity, 203
+properties, 201–204
+0-form, 197
+1-form, 198
+coordinates, 185
+section, 179, 214
+vector, 185
+�Pg,m,n space, 250
+p-form, 250
+1-form, 250
+picture changing operator, 256
+picture number, 247
+anomaly, 247, 250
+plumbing fixture, 187, 208, 251
+p-form, 209
+ghost 1-form, 209
+moduli, 187
+non-separating, 190, 212
+separating, 187–190, 208, 222
+vector field, 209
+Polyakov path integral, 29
+complex representation, 75
+Faddeev–Popov gauge fixing, 36, 61
+primary operator/state, 88
+finite transformation, 88
+weight-0, 91
+projector
+on-shell, ker L0, 140
+propagator, 139
+closed bosonic, 212, 215–217
+with stub, 222
+closed string, 174
+NS sector, 251
+open bosonic, 158
+R sector, 252
+Schwinger parametrization, 22, 216
+superstring, 256
+propagator region, 191
+puncture, see Riemann surface
+punctured Riemann surface, 178
+Euler characteristics, 178
+parametrization, 182–183
+quadratic differential, 42
+quasi-primary operator/state, 88
+r(Σg,n), see index
+radial quantization, 92
+reparametrization bc ghost
+zero-mode, 69
+reparametrization bc ghosts, see also first-
+order CFT, 51
+action, 51, 76
+central charge, 47
+equation of motion, 51
+zero-mode, 54
+Rg,n, see off-shell string amplitude contribu-
+tion
+Riemann sphere S2, 81
+complex plane map, 81
+cylinder map, 83
+314
+
+Riemann surface
+degeneration, 23
+g = 0, see Riemann sphere
+genus, 29
+puncture, 20, 58
+scalar field CFT
+L0, 115–116
+U(1) current, 108, 109
+U(1) symmetry, 108
+action, 107, 109
+boundary condition
+periodic, 115
+BPZ conjugate
+modes, 118
+vacuum, 118
+central charge, 111
+commutator, 116
+complex components, 109
+complex plane, 109
+cylinder, 110
+dual position, 114
+energy–momentum tensor, 107, 110
+equation of motion, 107, 109
+Euclidean adjoint
+modes, 118
+vacuum, 118
+Fock space, 117
+Hilbert space, 117
+inner product, 118
+level operator, 115
+mode expansion, 16, 113–115
+momentum, 108, 110, 114, 115
+normal ordering, 113
+number operator, 115
+OPE, 111–113
+periodic boundary condition, 107
+position (center of mass), 114
+propagator, 107
+topological current, 108, 110
+vacuum, 117
+conjugate, 118
+vertex operator, 110
+Virasoro operators, 115
+winding number, 109, 110, 114, 115
+zero-mode, 114
+scattering amplitude, 59–61
+tree-level 2-point (-), 60
+Schwarzian derivative, 95
+Schwinger parameter, see propagator
+section of Pg,n
+generalized, 224
+overlap, 224
+SFT, see string field theory
+Siegel gauge, 141, 158, 165, 169
+SL(2, C) group, 86
+SL(2, C) vacuum, 100
+S-matrix, see scattering amplitude
+spacetime ghost number
+closed string, 227
+open string, 161
+spacetime level-truncated action
+open bosonic, 165
+gauge transformation, 166
+spacetime momentum, see also scalar field
+CFT, 109
+spurious pole, 250, 252–255
+state (CFT)
+bra, 101
+ket, 100
+state–operator correspondence, 100
+Stokes’ theorem, 75
+string
+gauge group, 19
+interactions, 12, 20
+orientation, 19
+parameters, 21
+properties, 14
+string amplitude, 20
+CKV gauge fixing, 62
+closed string, see closed string amplitude
+conformal invariance, 171
+divergence, 22
+factorization, 174, 208, 214
+ghost number, 204
+g-loop vacuum (-), 34, 49, 52, 54, 55, 76
+matching QFT, 59–60, 64
+normalization, 55, 58
+properties, 204–207
+pure gauge states decoupling, 206
+section independence, 206
+tree-level 2-point (-), 64–67
+string amplitudegn
+g-loop n-point (-), 59, 63, 64
+string coupling constant, 21
+string Feynman diagram, see also momentum-
+space SFT, 172
+1PR diagram, 211, 222
+change of stub parameter, 222
+Feynman rules, 213
+intermediate states
+ghost number, 212, 213
+momentum, 212, 213
+IR divergence, 231
+315
+
+loop diagram, 213
+string field, see also string states
+closed bosonic, 168
+classical, 232
+expansion, 226, 235
+quantum, 235
+expansion, 152
+functional, 150
+gauge fixing, see Siegel gauge
+ket representation, 151
+momentum expansion, 25, 151, 260
+open bosonic, 155
+classical, 156
+expansion, 160
+Nakanishi–Lautrup auxiliary field, 165
+parity, 156
+quantum, 161
+reality condition, 156
+position representation, 151
+string field path integral
+Faddeev–Popov gauge fixing, 162
+free covariant open bosonic string, 162
+string field theory, 25
+construction, 23
+string states, see also closed, open (-), 14,
+175–177
+dual, 176
+off-shell, 175
+on-shell, see on-shell condition
+physical, see BRST cohomology
+resolution of identity, 176
+string theory
+background independence, 238
+consistency, 64, 264
+motivations, 11–13
+structure constant (CFT), 91
+stub, 194–195
+Feynman diagram, 222
+parameter, 194
+stub parameter, 232, 261
+super-SFT, 255
+superconformal βγ ghosts, see also first-order
+CFT, 246
+bosonization, 246
+conformal weights, 246
+energy–momentum tensor, 246
+OPE, 246
+superstring, 18, 245
+BRST current, 247
+heterotic (-), 19, 245
+Hilbert space, 248
+large, 248
+picture number, 248
+small, 248
+motivations, 18
+normalization, 248
+type I (-), 19
+type II (-), 19
+superstring field
+auxiliary field, 257
+constrained Ramond field, 256
+large Hilbert space, 258
+Ramond field, 255
+small Hilbert space, 256
+supersymmetry, 18
+surface state, 200, 209
+T-duality, 114
+tachyon, 17
+instability, 18
+Teichmüller deformation, 39, 41, 42
+Teichmüller space, 37
+ultralocality, see path integral measure
+Verma module, 102
+vertex operator, see also scalar field CFT, see
+also string states
+integrated, 58
+unintegrated, 64
+Weyl invariance, 61
+vertex state, 223
+vertical integration, 253
+Vg,n, see fundamental vertex
+V1PI
+g,n , see 1PI vertex
+Virasoro algebra, 90
+Virasoro operators, 90, 99
+Hermiticity, 99
+Weyl
+anomaly, 35, 47, 49
+ghost, 52, 68
+group volume, 39
+symmetry, 31, 52
+Wick rotation, 262, 266
+generalized, 263
+Wick theorem, 103
+winding number, see also scalar field CFT
+Witt algebra, 85
+worldsheet
+action
+Einstein–Hilbert (-), 56
+gauge-fixing (-), 68
+Nambu–Goto (-), 29
+316
+
+Polyakov (-), 29
+sigma model (-), 30
+boundary conditions, 14
+CFT, 14, 18
+classification, 14
+cosmological constant, 48
+cylinder, 83
+energy–momentum tensor, 33
+trace, 33
+ghosts, see reparametrization ghosts
+Nakanishi–Lautrup auxiliary field, 68
+path integral, see Polyakov path integral
+Riemann surface, 20, 29
+symmetry, 30
+background diffeomorphisms, 50
+background Weyl, 50
+BRST, 68
+diffeomorphisms, 30
+Weyl, 30
+worldsheet metric, 29
+worldvolume description, 12
+X, see picture changing operator
+Zamolodchikov metric, 91
+zero-mode, 269, 283
+zero-point energy, 101
+317
+