diff --git "a/ItE3T4oBgHgl3EQfugv_/content/tmp_files/2301.04686v1.pdf.txt" "b/ItE3T4oBgHgl3EQfugv_/content/tmp_files/2301.04686v1.pdf.txt"
new file mode 100644--- /dev/null
+++ "b/ItE3T4oBgHgl3EQfugv_/content/tmp_files/2301.04686v1.pdf.txt"
@@ -0,0 +1,5885 @@
+Non-Lorentzian Kaˇc-Moody Algebras
+Arjun Bagchi,a,b Ritankar Chatterjee,a Rishabh Kaushik,a,c Amartya Saha,a and
+Debmalya Sarkar.a,c
+aIndian Institute of Technology Kanpur, Kanpur 208016, India.
+bCentre de Physique Theorique, Ecole Polytechnique de Paris, 91128 Palaiseau Cedex, France.
+cInternational Centre for Theoretical Sciences (ICTS-TIFR), Bengaluru 560089, India.
+E-mail: (abagchi, ritankar, amartyas)@iit.ac.in,
+rishabh.kaushik@icts.res.in, sarkardebmalya01@gmail.com
+Abstract: We investigate two dimensional (2d) quantum field theories which exhibit Non-
+Lorentzian Kaˇc-Moody (NLKM) algebras as their underlying symmetry. Our investigations
+encompass both 2d Galilean (speed of light c → ∞) and Carrollian (c → 0) CFTs with ad-
+ditional number of infinite non-Abelian currents, stemming from an isomorphism between
+the two algebras. We alternate between an intrinsic and a limiting analysis. Our NLKM
+algebra is constructed first through a contraction and then derived from an intrinsically
+Carrollian perspective. We then go on to use the symmetries to derive a Non-Lorentzian
+(NL) Sugawara construction and ultimately write down the NL equivalent of the Knizhnik
+Zamolodchikov equations. All of these are also derived from contractions, thus providing
+a robust cross-check of our analyses.
+arXiv:2301.04686v1  [hep-th]  11 Jan 2023
+
+Contents
+1
+Introduction
+2
+2
+Non-Lorentzian Kaˇc-Moody algebra in 2d
+5
+2.1
+Carrollian and Galilean CFTs
+5
+2.2
+NL Affine Lie algebras
+7
+3
+An intrinsic Carrollian derivation
+10
+3.1
+An Infinity of Conserved Quantities
+10
+3.2
+Current Ward Identities
+11
+3.3
+Current-primary fields
+13
+3.4
+Current-Current OPEs
+15
+3.5
+Global internal symmetry
+18
+3.6
+EM tensor-current OPEs
+19
+3.7
+The Algebra of Modes
+22
+4
+Sugawara Construction
+24
+4.1
+Intrinsic construction from algebra
+24
+4.2
+Consistency with OPEs
+27
+5
+Tensionless String as NLKM
+29
+6
+Non-Lorentzian Knizhnik Zamolodchikov Equations
+31
+7
+NLKM from Contractions
+32
+7.1
+A brief detour to representations of BMS
+32
+7.2
+Contraction of the affine parameters
+33
+7.3
+Contracting the Sugawara construction
+35
+7.4
+NLKZ equations from Contraction
+36
+8
+Conclusions
+38
+8.1
+Summary
+38
+8.2
+Discussions and future directions
+39
+A Carroll Multiplets
+40
+B Calculation of Sugawara Construction Commutators
+42
+C Modified Sugawara Construction
+46
+D Details of OPE calculations
+47
+E K-Z equation in field theory approach
+49
+– 1 –
+
+F NL KZ equation as a limit
+50
+1
+Introduction
+Relativistic conformal field theory (CFT) is one of the most potent tools of modern theo-
+retical physics, with applications ranging from statistical mechanics of phase transitions to
+quantum gravity through holography and string theory. Especially powerful are methods
+of two dimensional (2d) CFTs [1] where symmetries enhance to two copies of the infinite
+dimensional Virasoro algebra. The ideas and methods of 2d CFTs are of particular impor-
+tance to the success of string theory, where this arises as residual symmetry on the string
+worldsheet after the fixing of conformal gauge [2].
+Non-abelian current algebras arise on the string worldsheet when one considers strings
+moving on arbitrary group manifolds [3], generalising the abelian versions which arise
+for strings propagating on flat backgrounds. These Kaˇc-Moody algebras give rise to the
+worldsheet 2d CFT by the Sugawara construction. The construction of strings on arbitrary
+backgrounds is thus intimately linked to these Kaˇc-Moody algebras.
+Kaˇc-Moody (KM) algebras also arise when we think of 2d CFTs augmented by additional
+symmetries [4]. For example, a CFT with additional U(1) global symmetry arises in a
+number of places, including the study of black holes in AdS3 with charged U(1) hair that
+are solutions to Einstein-Chern Simons theory [5].
+Virasoro with U(1) KM symmetry
+forms the chiral algebra of a large number of theories including N = 2 superconformal field
+theories and theories with W1+∞ symmetry.
+In this paper, we will be interested in the construction of Non-Lorentzian (NL) versions
+of Kaˇc-Moody algebras. Specifically, we are concerned with Galilean and Carrollian CFTs
+in 2d with additional symmetry. Galilean and Carrollian CFTs are obtained from their
+relativistic counterparts by a process of contraction where the speed of light is taken to
+infinity (Galilean theory) or zero (Carrollian theory). In two dimensions, the symmetry
+algebras turn out to be isomorphic and it is in this 2d case we will focus our attention. We
+will obtain the algebras of interest by a contraction and then construct various properties
+of these algebras by methods that have no connections with the limiting procedure and can
+be thought of as completely intrinsic analyses. We also show that suitable singular limits
+also reproduce our intrinsic answers.
+Possible applications
+We have a variety of applications in mind for our algebraic explorations in this work.
+Holography of flat spacetimes
+Following closely related observation in [6], it was shown in [7] that d-dimensional Con-
+formal Carrollian algebras are isomorphic to asymptotic symmetry algebras of (d + 1)
+– 2 –
+
+dimensional flat spacetimes discovered first by Bondi, van der Burg, Metzner [8] and Sachs
+[9] and called BMS algebras. The Carroll CFTs can thus act as putative duals to asypm-
+totically flat spacetimes [6, 10, 11]. Some important evidence for this duality has been
+provided in e.g. [12–18]. Of particular importance is the recent result [19] that links 3d
+Carroll CFTs and scattering amplitudes in 4d asymptotically flat spacetimes.
+Our explorations in this paper are of direct importance in the context of 3d flatspace.
+Here, the U(1) version of NLKM symmetries are of interest for the study of Flat Space
+Cosmological solutions [20] with U(1) hair, which are solutions of Einstein-Chern-Simons
+theory, like the AdS case. This was addressed recently in [21, 22].
+The above approach to holography in asymptotically flat spacetimes goes under the name
+of Carrollian holography. There is an alternate formulation called Celestial holography
+which posits that there is a 2d (relativistic) CFT that computes S-matrix elements in 4d
+asymptotically flat spacetimes. This has been instrumental in the uncovering of many new
+results in asymptotic symmetries and scattering amplitudes in 4d. The interested reader is
+pointed to the excellent reviews [23–25]. For connections between Celestial and Carrollian
+holography, we point to [19, 26, 27].
+Interestingly, of late there have been studies of tree-level massless scattering amplitudes
+which suggest that the asymptotic symmetries are far richer than the extended BMS group.
+In [28, 29], it was shown that there is an SL(2) current algebra at level zero underlying the
+symmetries of tree-level graviton amplitudes. More recently, massless scattering amplitudes
+have revealed an infinite dimensional w1+∞ algebra [30].
+If we assume that the field theory duals with these additional symmetries would be related
+to a co-dimension one holographic description of 4d flatspace, and that these theories should
+live on the whole of the null boundary and not only on the celestial sphere, the structure
+emerging from the above discussions should only be a part of the whole symmetries, very
+much like the relation between the two copies of the Virasoro algebra that make up the
+Celestial CFT and the whole (extended) BMS4 algebra. It is very likely that the algebras
+of interest would then be the 3d versions of the Carrollian Kac-Moody algebras that we
+discuss at length in this work.
+Tensionless strings
+The tensionless limit of strings, which is analogous to the massless limit of point particles,
+is an important sector of string theory that remains relatively less explored. In this limit,
+the string worldsheet becomes null [31, 32] and the 2d relativistic conformal symmetry that
+arises on the tensile worldsheet is replaced by 2d Carrollian Conformal symmetry in the
+tensionless theory [33–35]. The study of tensionless strings on arbitrary group manifolds
+would naturally incorporate the NLKM algebras we study in this paper. We would, in
+future, attempt a construction of a Wess-Zumino-Witten model with the NLKM algebras
+we discuss in this work.
+It is of interest to mention that in [36], it was argued that tensionless strings appeared when
+the level of the affine algebra corresponding to the WZW model of the strings propagating
+in a group manifold. We believe that the intrinsic formulation of such strings should involve
+– 3 –
+
+the NLKM algebras we are studying here. A direction of future work is the connection of
+these two ideas and the aim would be to show that the NLKM algebras appear when the
+level of the relativistic affine algebras are dialled to their critical value.
+Other applications
+There are, of course, other natural applications. The Galilean version of our story is of
+relevance for understanding 2d Galilean CFTs with additional symmetry which may appear
+in real life non-relativistic systems. These would be of interest also in understanding non-
+relativistic strings in curved Newton-Cartan backgrounds [37].
+Interesting enough, Carrollian structures also arise condensed matter systems, e.g. in the
+physics of flat bands that is of relevance in the context of “magic” superconductivity in
+bi-layer graphene and also in fractional quantum Hall systems [38]. It is easy to envision
+condensed matter systems with additional symmetry and hence our methods in this paper
+which lay the foundation for systems with Carrollian (and Galilean) affine Lie algebras,
+should have applicability in a wide number of condensed matter systems. Finally, a U(1)
+affine algebra was also found to emerge in studies of the BMS scalar field theory in 2d [39].
+It would be of interest to figure out if this is a feature of all free BMS field theories. Our
+methods outlined in this paper should then be useful even for free BMS field theories.
+Outline of the paper
+In section 2, we will start with a brief review of Carrollian and Galilean CFTs in gen-
+eral dimensions, subsequently specializing the discussion to two dimensions followed by
+a brief introduction to Non Lorentzian Kac Moody (NLKM) algebras via contraction of
+relativistic Affine Lie Algebras. In section 3, we present an intrinsic carrollian derivation
+to the NLKM Algebra. Then, we formulate the non-Lorentzian version of the Sugawara
+construction for these algebras in section 4 and verify its validity by showing its consis-
+tency with the OPEs we obtained in section 3. In section 5, we work out the example of
+tensionless strings on a flat background geometry which exhibits the U(1) NLKM algebra
+as the current algebra. Section 6 contains the derivation of the Non-Lorentzian analog
+of Knizhnik Zamolodchikov(KZ) equations using the OPE definition of the primaries in
+section 3. And finally in section 7, we derive NLKM Algebra, Sugawara Construction and
+the Non Lorentzian KZ equations through contractions in detail. There are six appendices
+which collect the details of various calculations that have been skipped in the main text
+for the ease of readability.
+– 4 –
+
+2
+Non-Lorentzian Kaˇc-Moody algebra in 2d
+In this section, we will begin our analysis by building the algebra we wish to study in the
+remainder of the paper. The NLKM algebra will be defined as a limit from the relativistic
+Virasoro KM algebra. We will see later how the current part of the algebra can be used
+to generate the entire algebra by a NL version of the Suwagara construction. We begin by
+reminding the reader of the Galilean (c → ∞) and the Carrollian (c → 0) contractions of
+relativistic CFTs in generic dimensions and the fact that this leads to isomorphic algebras
+in d = 2
+2.1
+Carrollian and Galilean CFTs
+The Galilean (c → ∞) and Carrollian (c → 0) limits of the Poincare algebra leads to two
+different contractions of the parent relativistic algebra and two different algebras in the
+limit in general dimensions. The Poincare algebra in D dimension ISO(D − 1, 1) is given
+by
+[Pµ, Pν] = 0,
+[Mµν, Pρ] = −2ηρ[µPν],
+[Mµν, Mρσ] = 4η[µρMσ]ν],
+(2.1)
+where µ = 0, 1, 2..., D − 1, Pµ = −∂µ are translation generators and Mµν = xµ∂ν − xν∂µ
+are Lorentz generators. Galilean limit is achieved by taking c → ∞ limit, alternatively by
+taking the contraction t → t, xi → ϵxi, ϵ → 0 limit where i ∈ {1, 2, ..., D − 1} [40, 41].
+Under this limit we see that
+M0i = t∂i + xi∂t → 1
+ϵ t∂i + ϵxi∂t
+=⇒ Bi = lim
+ϵ→0 ϵM0i = t∂i.
+(2.2)
+The spatial rotation generators Mij remains same under this contraction.
+Doing this
+contraction, we end up with the Galilei algebra, where all the non-zero commutators are
+given by
+[Mij, Mkl] = 4δi[kMl]j], [Mij, Pk] = −2δk[iPj], [Mij, Bk] = 2δk[iBj], [Bi, H] = −Pi.
+(2.3)
+Carrollian limit is achieved by taking c → 0 limit, alternatively by taking the contraction
+t → ϵt, xi → xi, ϵ → 0. Under this limit, we have
+M0i = ϵt∂i + xi∂t → ϵt∂i + 1
+ϵ xi∂t
+=⇒ Bi = lim
+ϵ→0 ϵM0i = xi∂t,
+(2.4)
+with Mij intact again. This gives us the Carroll algebra where the non-zero commutators
+are given by
+[Mij, Mkl] = 4δi[kMl]j], [Mij, Pk] = −2δk[iPj], [Mij, Bk] = −2δk[iBj] [Pi, Bj] = δijH, (2.5)
+where H = −∂t is the Hamiltonian which has now become a central element.
+The tale of the two contractions is also true for the relativistic conformal algebra.
+In
+relativistic Conformal symmetry group there are two additional generators
+D = −xµ∂µ
+Kµ = −(2xµxν∂ν − x2∂µ),
+(2.6)
+– 5 –
+
+giving us the following additional commutators along with (2.1)
+[D, Pµ] = Pµ, [D, Kµ] = −Kµ, [Kµ, Pν] = 2(ηµνD − Mµν), [Kρ, Lµν] = 2ηρ[µKν].
+(2.7)
+Taking the non-relativistic limit this time, we will end up with the following additional
+generators along with the generators of Galilei algebra
+D = −(xi∂i + t∂t)
+K = K0 = −(2txi∂i + t2∂t)
+Ki = t2∂i.
+(2.8)
+These generators, along with the generators of the Galilei algebra gives us the Galilean
+Conformal Algebra (GCA) in D dimensions [40] where non-zero commutators apart from
+(2.3) are given by
+[K, Bi] = Ki,
+[K, Pi] = 2Bi
+[Mij, Kr] = −2K[iδj]r,
+[H, Ki] = −2Bi
+[D, Ki] = −2Ki,
+[D, Pi] = Pi
+[D, H] = H,
+[H, K] = −2D,
+[D, K] = −K.
+(2.9)
+When we take the Carrollian limit of the relativistic Conformal algebra, we get the following
+new generators apart form the Carroll generators previously encountered:
+D = −(xi∂i + t∂t)
+K = K0 = −xixi∂t
+Kj = −2xj(xi∂i + t∂t) + xixi∂j.
+(2.10)
+These additional generators give us the following non-vanishing commutators along with
+those given in (2.5)
+[Mij, Kk] = δk[jKi]
+[Bi, Kj] = δijK,
+[D, K] = −K
+[K, Pi] = −2Bi
+[Ki, Pj] = −2Dδij − 2Mij,
+[H, Ki] = 2Bi
+[D, H] = −H,
+[D, Pi] = −Pi
+[D, Ki] = Ki.
+(2.11)
+(2.5) and (2.11) together form Carrollian Conformal Algebra (CCA). We see that in general
+dimensions the two different contractions give us different algebras that are not isomorphic
+to each other.
+Let us now move to the interesting case of d = 2. In 2d, the relativistic conformal algebra
+becomes infinite dimensional and is given by two copies of the Virasoro algebra:
+[Lm, Ln] = (m − n)Lm+n + c
+12(m3 − m)δm+n,0,
+[ ¯Lm, ¯Ln] = (m − n) ¯Lm+n + ¯c
+12(m3 − m)δm+n,0,
+(2.12)
+[Lm, ¯Ln] = 0.
+Here c, ¯c are central charges of the Virasoro algebra (not to be confused with the speed
+of light which we also had called c earlier). The Galilean contraction [42] of the above is
+given by
+Ln = Ln + ¯Ln,
+Mn = ϵ( ¯Ln − Ln),
+(2.13)
+– 6 –
+
+This contraction of the Virasoro algebra leads to
+[Lm, Ln] = (m − n)Lm+n + cL
+12(m3 − m)δn+m,0,
+(2.14a)
+[Lm, Mn] = (m − n)Mm+n + cM
+12 (m3 − m)δn+m,0,
+(2.14b)
+[Mm, Mn] = 0.
+(2.14c)
+The way to see that this combination yields the non-relativistic limit, it is instructive to
+write the generators of the Virasoro algebra in cylindrical cooridinates
+Ln = einω∂ω,
+¯Ln = ein¯ω∂¯ω,
+ω, ¯ω = τ ± σ,
+(2.15)
+The limit (2.13) then translates to
+σ → ϵσ, τ → τ, ϵ → 0.
+(2.16)
+which essentially means scaling velocities to be very small compared to 1 and since we are
+doing all of this in units of speed of light c = 1, this is indeed the non-relativistic limit
+c → ∞.
+On the other hand, the Carrollian contraction of the Virasoro algebra is given by
+Ln = Ln − ¯L−n,
+Mn = ϵ(Ln − ¯L−n).
+(2.17)
+Again if we go back to the cylindrical coordinates, this limit translates to
+σ → σ, τ → ϵτ, ϵ → 0.
+(2.18)
+The velocities are very large compared to 1 now and this means v/c → ∞, which equiv-
+alently translates to c → 0. This is thus the Carrollian limit. The surprising thing is
+that even this contraction yields the same algebra (2.14). In order to avoid confusion with
+Galilean or Carrollian notions, we will exclusively call this the BMS3 algebra. Galilei and
+Carroll contractions in d = 2 yield isomorphic algebras and this is down to the fact that
+there is only one contracted and one uncontracted direction in each case. The algebra does
+not differentiate between a contracted spatial and a contracted temporal direction.
+2.2
+NL Affine Lie algebras
+We start with the two copies of Virasoro Kac-Moody algebra whose holomorphic part is,
+[Lm, Ln] = (m − n)Lm+n + c
+12(m3 − m)δm+n,0,
+[Lm, ja
+n] = −nja
+m+n
+[ja
+m, jb
+n] = i
+dim(g)
+�
+c=1
+fabcjc
+m+n + mkδm+n,0δab.
+(2.19)
+There is an equivalent anti-holomorphic part with ¯fabc, ¯c and ¯k in place of fabc, c and k
+respectively which are not necessarily equal to their holomorphic counterparts. We are
+take fabc ̸= ¯fabc for generality, but the dimensions of the two Lie groups are the same.
+– 7 –
+
+We will take a contraction of the algebra as follows. We will work with the following linear
+combinations of the relativistic KM generators:
+Ln = Ln + ¯Ln,
+Mn = ϵ( ¯Ln − Ln),
+Ja
+m = ja
+m + ¯ja
+m,
+Ka
+m = ϵ(¯ja
+m − ja
+m).
+(2.20)
+We will the consider the limit ϵ → 0. The contracted algebra is given by:
+[Lm, Ln] = (m − n)Lm+n + cL
+12(m3 − m)δn+m,0,
+(2.21a)
+[Lm, Mn] = (m − n)Mm+n + cM
+12 (m3 − m)δn+m,0,
+(2.21b)
+[Lm, Ja
+n] = −nJa
+m+n , [Lm, Ka
+n] = −nKa
+m+n , [Mm, Ja
+n] = −nKa
+m+n
+(2.21c)
+[Ja
+m, Jb
+n] = i
+dim(g)
+�
+c=1
+F abcJc
+m+n + i
+dim(g)
+�
+c=1
+GabcKc
+m+n + mk1δabδm+n,0,
+(2.21d)
+[Ja
+m, Kb
+n] = i
+dim(g)
+�
+c=1
+F abcKc
+m+n + mk2δabδm+n,0,
+(2.21e)
+with rest of the commutators vanishing. We recognise the first two lines as the familiar
+BMS3 algebra, equivalently the 2d Galilean or 2d Carrollian Conformal Algebra. Through-
+out the paper, we will call this sub-algebra the BMS algebra. In the above, the structure
+constants are related to their relativistic counterparts by
+F abc = 1
+2
+�
+fabc + ¯fabc�
+, Gabc = 1
+2ϵ
+�
+¯fabc − fabc�
+.
+(2.22)
+while the central terms are given by
+cL = c + ¯c,
+cM = ϵ(¯c − c), k1 = ¯k + k, k2 = ϵ(¯k − k).
+(2.23)
+We will call the algebra (2.21) the 2d Non-Lorentzian Kaˇc-Moody algebra. We can also carry
+out the contraction ultrarelativistically for which we take the following linear combinations
+of generators,
+Ln = Ln − ¯L−n,
+Mn = ϵ(Ln + ¯L−n),
+Ja
+n = (ja
+n + ¯ja
+−n),
+Ka
+n = ϵ(ja
+n − ¯ja
+−n).
+(2.24)
+Contracted algebra will be same as (2.21) with relations analogous to (2.22) and(2.23)
+taking the following form:
+F abc = 1
+2
+�
+fabc + ¯fabc
+�
+, Gabc = 1
+2ϵ
+�
+fabc − ¯fabc
+�
+(2.25)
+cL = c − ¯c,
+cM = ϵ(c + ¯c),
+k1 = k − ¯k,
+k2 = ϵ(k + ¯k).
+(2.26)
+This (2.21) will be the algebra of interest for the rest of our paper.
+Note that if we start with 2 identical Kac-Moody algebras for the holomorphic and anti-
+holomorphic sections, then after contraction we get
+fabc = ¯fabc ⇒ F abc = fabc, Gabc = 0.
+(2.27)
+– 8 –
+
+For most of our analysis, we will use this simplified algebra, but the results can be easily
+generalised for the general algebra.
+We should also clarify something about the new Lie algebra structure and our notation.
+We started with (the affine version of) two copies of a Lie algebra g of dimension
+n = dim(g),
+constructed from the generators {ja, a ∈ {1, 2, . . . , n}} and {¯ja, a ∈ {1, 2, . . . , n}}. Via
+contraction we obtain a (non-semisimple) Lie algebra ˜g of dimension
+˜n = dim(˜g) = 2 dim(g)
+consisting of generators {Ja, Ka, a ∈ {1, 2, . . . , n}}.
+So in all our notation, the indices
+a, b, c run from 1 to n = dim(g) (not ˜n), and whenever we write dim(g) (like in expressions
+of central charge cL later in the paper), we mean the dimension of the parent algebra g,
+which is actually half of the dimension of the new algebra ˜g. This is done for the tidiness
+of expressions.
+Before we conclude this section, we would like to comment on the choice of contraction of
+the currents to get to the NLKM algebra. Note that the chosen linear combinations of the
+Kac Moody generators are motivated by the Galilean/Carrollian contractions analogous to
+(2.16) and (2.18). The generators in cylindrical coordinates look like,
+ja
+n = ja ⊗ einω,
+¯ja
+n = ¯ja ⊗ ein¯ω,
+ω, ¯ω = τ ± σ
+(2.28)
+Galilean limit will correspond to (2.16) and Carrollian limit corresponds to (2.18).
+In
+Galilean case, we have (upto linear order in ϵ),
+ja
+n + ¯ja
+n = (ja + ¯ja) ⊗ einτ + inσϵ(ja − ¯ja) ⊗ einτ,
+(2.29a)
+ja
+n − ¯ja
+n = (ja − ¯ja) ⊗ einτ + inσϵ(ja + ¯ja) ⊗ einτ.
+(2.29b)
+We now have two choices, keeping the BMS contraction the same:
+• Ja = ja + ¯ja, Ka = ϵ(¯ja − ja) ⇒ (2.20).
+• Ja = ¯ja − ja, Ka = ϵ(¯ja + ja) ⇒ Ja
+m = ¯ja
+m − ja
+m, Ka
+m = ϵ(¯ja
+m + ja
+m).
+Both of these choices lead to relations (2.23) but (2.22) need to be altered for the second
+choice to:
+F abc = 1
+2( ¯fabc − fabc), Gabc = 1
+2ϵ(fabc + ¯fabc).
+(2.30)
+Similarly, in Carrollian case, we have two choices: (2.24) and
+Ja
+n = ja
+n − ja
+−n,
+Ka
+n = ϵ(ja
+n + ja
+−n).
+in which case (2.26) remains same but we have
+F abc = 1
+2(fabc − ¯fabc),
+Gabc = 1
+2ϵ(fabc + ¯fabc)
+(2.31)
+– 9 –
+
+instead of (2.25). Since we will be considering the special case when fabc = ¯fabc, we will
+stick to the first choice in this paper. For the intrinsic analysis, which only uses the algebra
+(2.21), of course these choices do not matter. But when we derive results from the limit,
+in Sec. 7, it is important to state everything works for the other contraction as well. E.g.
+Sugawara construction through contraction (Sec 7.3) can be carried out for the second
+choice if we specialize to the case ¯fabc = −fabc in case of Galilean contraction.
+3
+An intrinsic Carrollian derivation
+Having derived the non-Lorentzian current algebra through a contraction, we now go on to
+present an intrinsically Carrollian derivation of the same, where we would not be alluding
+to a limiting procedure at all. This section heavily borrows from the machinery detailed in
+[43], some of the important features of which are described in Appendix A. Although we
+will try and be self consistent so that the section (with the help of the related appendix A)
+stands on its own, for any details that we may have inadvertently skipped in what follows,
+the reader is referred back to [43].
+3.1
+An Infinity of Conserved Quantities
+A 2D Carrollian CFT on the flat Carrollian background (t, x) is invariant under an infinite
+number of 2D Carrollian conformal (CC) transformations whose infinitesimal versions are
+given as:
+x′ = x + ϵxf(x) ,
+t′ = t + ϵxtf′(x) + ϵtg(x)
+(3.1)
+As a consequence, the EM tensor components classically satisfy the following conditions
+[43]:
+∂µT µ
+ν = 0 ;
+T x
+t = 0 ;
+T µ
+µ = 0
+=⇒ ∂tT t
+t = 0 ;
+∂tT t
+x = ∂xT t
+t.
+(3.2)
+This allows for an infinite number of Noether currents. The two conserved currents corre-
+sponding to the symmetry transformation (3.1) are noted below:
+jµ
+t =
+�
+jt
+t , jx
+t
+�
+=
+�
+g(x)T t
+t , 0
+�
+,
+jµ
+x =
+�
+jt
+x , jx
+x
+�
+=
+�
+f(x)T t
+x + tf′(x)T t
+t , −f(x)T t
+t
+�
+.
+(3.3)
+Now, we suppose that there are some other pairs of fields {J a
+x , J a
+t } in the theory that obey
+the following conditions analogous to (3.2):
+∂tJ a
+t (t, x) = 0 ;
+∂tJ a
+x (t, x) = ∂xJ a
+t (t, x)
+(3.4)
+where a is to be thought of as a ‘flavor’ index (but t and x in subscript are not tensor
+indices). Using these fields, we can construct an infinite number of conserved quantities:
+kaµ =
+�
+kat , kax�
+= (g(x)J a
+t , 0) ,
+(3.5a)
+jaµ =
+�
+jat , jax�
+=
+�
+f(x)J a
+x + tf′(x)J a
+t , −f(x)J a
+t
+�
+.
+(3.5b)
+We shall regard these conserved quantities as the Noether currents associated to some
+‘internal’ symmetries of the action.
+– 10 –
+
+3.2
+Current Ward Identities
+We now consider an infinitesimal internal transformation of a (possibly multi-component)
+field Φ(t, x):
+Φ(t, x) → ˜Φ(t, x) = Φ(t, x) + ϵa (Fa · Φ) (t, x)
+(3.6)
+where Fa · Φ denotes the functional changes of the (multi-component) field Φ under in-
+finitesimal transformations labeled by ϵa. So, the generator GaΦ of this transformation is
+given by:
+−iϵaGaΦ(t, x) := ˜Φ(t, x) − Φ(t, x) = ϵa (Fa · Φ) (t, x)
+(3.7)
+We shall now find the Ward identity corresponding to this internal transformation which
+is assumed to be a symmetry of the 2D Carrollian CFT. For this purpose, we analytically
+continue the real space variable x ∈ R∪{∞} to the complex plane; thus, the Ward identity
+reads [43]:
+∂µ⟨jµ
+a(t, x)X⟩ ∼ −i
+n
+�
+i=1
+δ(t − ti)
+�
+�−(Fa)i · ⟨X⟩
+x − xi
++
+�
+k≥2
+⟨Y (k)
+a
+⟩i(x1, ...xn)
+(x − xi)k
+�
+�
+(3.8)
+where the yet unknown correlators ⟨Y (k)
+a
+⟩i depend on the transformation properties of the
+fields in the string-of-fields X and the transformation itself and ∼ denotes ‘modulo terms
+holomorphic in x inside [. . .]’. We also use the shorthand x1 = (t1, x1). All the correlators
+are time-ordered.
+Let us now assume that the symmetry transformation (3.6) is associated to the following
+conserved current operator:
+kaµ = (J a
+t , 0) .
+(3.9)
+The corresponding Ward identity then is:
+∂t⟨J a
+t (t, x)X⟩ ∼ −i
+n
+�
+i=1
+δ(t − ti)
+�
+�−(Fa)i · ⟨X⟩
+x − xi
++
+�
+k≥2
+⟨Y (k)
+a
+⟩i(x1, ...xn)
+(x − xi)k
+�
+�
+⇒ ⟨J a
+t (t, x)X⟩ = −i
+n
+�
+i=1
+θ(t − ti)
+�
+�−(Fa)i · ⟨X⟩
+x − xi
++
+�
+k≥2
+⟨Y (k)
+a
+⟩i(x1, ...xn)
+(x − xi)k
+�
+�
+(3.10)
+where, following [43], the initial condition has been taken to be:
+lim
+t→−∞⟨J a
+t (t, x)X⟩ = 0
+(3.11)
+and as in 2D relativistic CFT [4], ⟨J a
+t (t, x)X⟩ is assumed to be finite whenever x ̸= {xi};
+this condition makes the holomorphic terms inside [. . .] vanish in this Ward identity.
+– 11 –
+
+Similarly, if the conserved current:
+jaµ = (J a
+x , −J a
+t )
+(3.12)
+is associated to another internal symmetry transformation:
+Φ(t, x) → ˜Φ(t, x) = Φ(t, x) + ϵa (Ga · Φ) (t, x)
+(3.13)
+the corresponding Ward identity is:
+⟨(∂tJ a
+x (t, x) − ∂xJ a
+t (t, x)) X⟩ ∼ −i
+n
+�
+i=1
+δ(t − ti)
+�
+�−(Ga)i · ⟨X⟩
+x − xi
++
+�
+k≥2
+⟨Y (k)
+a
+⟩i(x1, ...xn)
+(x − xi)k
+�
+�
+(3.14)
+Assuming the initial condition:
+lim
+t→−∞⟨J a
+x (t, x)X⟩ = 0
+(3.15)
+and the finite-ness property of ⟨J a
+x (t, x)X⟩ for x ̸= {xi}, this Ward identity together with
+(3.10) lead us to:
+⟨J a
+x (t, x)X⟩ = −i
+n
+�
+i=1
+θ(t − ti)
+�
+�−(Ga)i · ⟨X⟩
+x − xi
++
+�
+k≥2
+⟨Z(k)
+a ⟩i(x1, ...xn)
+(x − xi)k
+−(t − ti)
+�
+�−(Fa)i · ⟨X⟩
+(x − xi)2
++
+�
+k≥2
+k⟨Y (k)
+a
+⟩i(x1, ...xn)
+(x − xi)k+1
+�
+�
+�
+�
+(3.16)
+Again, the correlators ⟨Z(k)
+a ⟩i can not be determined without knowing the explicit internal
+transformation properties of the fields in X.
+For future references, we note the iϵ-form [43] of the Ward identities (3.10) and (3.16)
+below with ∆˜xp := x − xp − iϵ(t − tp) 1 :
+i⟨J a
+t (t, x)X⟩ = lim
+ϵ→0+
+n
+�
+i=1
+�
+�−(Fa)i · ⟨X⟩
+∆˜xi
++
+�
+k≥2
+⟨Y (k)
+a
+⟩i
+(∆˜xi)k
+�
+�
+(3.17)
+i⟨J a
+x (t, x)X⟩ = lim
+ϵ→0+
+n
+�
+i=1
+�
+�−(Ga)i · ⟨X⟩
+∆˜xi
++
+�
+k≥2
+⟨Z(k)
+a ⟩i
+(∆˜xi)k
+−(t − ti)
+�
+�−(Fa)i · ⟨X⟩
+(∆˜xi)2
++
+�
+k≥2
+k ⟨Y (k)
+a
+⟩i
+(∆˜xi)k+1
+�
+�
+�
+� (3.18)
+1We hope the reader does not confuse the delta appearing in ∆˜xp with the conformal weight ∆. The ∆
+appearing in the difference of coordinates would always appear with a coordinate.
+– 12 –
+
+Thus, a general (possibly multi-component) 2D Carrollian conformal field Φ(t, x) has the
+following OPEs (in the iϵ-form) with the current-vector components (∆˜x′ := x′−x−iϵ(t′−
+t)):
+iJ a
+t (t′, x′)Φ(t, x) ∼ lim
+ϵ→0+
+�
+. . . + −(Fa · Φ)(t, x)
+∆˜x′
+�
+(3.19)
+iJ a
+x (t′, x′)Φ(t, x) ∼ lim
+ϵ→0+
+�
+. . . + −(Ga · Φ)(t, x)
+∆˜x′
+− (t′ − t)
+�
+. . . + −(Fa · Φ)(t, x)
+(∆˜x′)2
+��
+(3.20)
+where . . . represents higher order poles at x′ = x and ∼ denotes ‘modulo terms holomorphic
+in x′ that have vanishing VEVs’.
+In this work, we shall only consider currents with scaling dimension ∆ = 1. Comparing
+the behavior of both sides of the OPEs (3.19) and (3.20) under dilatation, we then infer
+that the scaling dimensions of the local fields Fa · Φ and Ga · Φ must be same as that of Φ.
+Since this is true for any arbitrary local field Φ, we conclude that Fa · Φ and Ga · Φ must
+be linear combinations of the components of the multi-component field Φ that ‘internally’
+transforms under a bi-matrix representation of the global current symmetry algebra; all of
+these components must have an equal scaling dimension. Our goal is to find this global
+algebra and its infinite extension.
+We now express Fa · Φ and Ga · Φ as explicit linear combinations:
+(Fa · Φ)ii′ = (ta
+K)i
+j Φji′ ;
+(Ga · Φ)ii′ = Φij′ (ta
+J) i′
+j′
+(3.21)
+where ta
+J and ta
+K are just two matrices as of now.
+Later, we shall relate them to the
+generators of the internal symmetry algebra.
+3.3
+Current-primary fields
+In the operator formalism of a QFT, the conserved charge QA is the generator of an
+infinitesimal symmetry transformation on the space of the quantum fields:
+˜Φ(x) − Φ(x) = −iϵA[QA , Φ(x)]
+(3.22)
+In 2D Carroll CFT, the above generator equation for any conserved charge operator QA is
+related to the following contour integral prescription involving an OPE [43]2:
+QA =
+1
+2πi
+�
+Cu
+dx jt
+A(t, x)
+generates
+[QA , Φ(t, x)] =
+1
+2πi
+�
+x
+dx′ jt
+A(t+, x′)Φ(t, x)
+(3.23)
+where t+ > t and the counter-clockwise contour Cu encloses the upper half-plane along
+with the real line. The contour around x must not enclose any possible singularities of the
+vector field.
+2Section 5 of this reference contains a derivation.
+– 13 –
+
+The conserved quantum charges Qa
+t [g] and Qa
+x[f] of the respective currents (3.5a) and
+(3.5b) are thus given by:
+Qa
+t [g] =
+1
+2πi
+�
+Cu
+dx g(x)J a
+t (t, x)
+(3.24)
+Qa
+x[f] =
+1
+2πi
+�
+Cu
+dx
+�
+f(x)J a
+x (t, x) + tf′(x)J a
+t (t, x)
+�
+(3.25)
+The conserved charges of all flavors collectively induce the following infinitesimal changes
+to a generic quantum field, as deduced from the OPEs (3.19) and (3.20):
+−i
+�
+a
+ϵa [Qa
+t [ga] , Φ(x)] = − 1
+2π
+�
+a
+ϵa
+�
+x
+dx′ ga(x′)J a
+t (t+, x′)Φ(t, x)
+=
+�
+a
+ϵa [ga(x) (ta
+K · Φ) (t, x) + (h.d.t.)]
+(3.26)
+−i
+�
+a
+ϵa [Qa
+x[fa] , Φ(x)] = − 1
+2π
+�
+a
+ϵa
+�
+x
+dx′ �
+fa(x′)J a
+x (t+, x′) + t+fa′(x′)J a
+t (t+, x′)
+�
+Φ(t, x)
+=
+�
+a
+ϵa �
+fa(x) (Φ · ta
+J) (t, x) + tfa′(x) (ta
+K · Φ) (t, x) + (h.d.t.)
+�
+(3.27)
+where h.d.t. denotes terms necessarily containing derivatives (of order at least 1) of ga(x)
+and fa(x).
+For the currents (3.9) and (3.12), we simply have f(x) = 1 = g(x). For any arbitrary field
+Φ(t, x) this immediately leads to:
+− i
+�
+a
+ϵa [Qa
+t [1] , Φ(x)] =
+�
+a
+ϵa (ta
+K · Φ) (t, x)
+(3.28a)
+− i
+�
+a
+ϵa [Qa
+x[1] , Φ(x)] =
+�
+a
+ϵa (Φ · ta
+J) (t, x)
+(3.28b)
+Thus, the finite internal transformation that is generated by the charges {Qa
+k[1]} is:
+Φ(t, x) → ˜Φ(t, x) = e−i �
+a ϵaQa
+t [1]Φ(t, x)ei �
+a ϵaQa
+t [1] = e
+�
+a ϵata
+K · Φ(t, x)
+(3.29)
+which is obtained by using the BCH lemma. Similarly, the charges {Qa
+x[1]} generate the
+following finite transformation:
+Φ(t, x) → ˜Φ(t, x) = Φ(t, x) · e
+�
+a ϵata
+J
+(3.30)
+On the other hand, as derived from (3.26), an arbitrary field finitely transforms under the
+action of the charges {Qa
+t [ga]} as:
+Φ(t, x) → ˜Φ(t, x) = e
+�
+a ϵaga(x)ta
+K · Φ(t, x) + extra terms
+(3.31)
+– 14 –
+
+while (3.27) leads to the following finite action of the charges {Qa
+x[fa]}:
+Φ(t, x) → ˜Φ(t, x) = e
+�
+a ϵatfa′(x)ta
+K · Φ(t, x) · e
+�
+a ϵafa(x)ta
+J + extra terms
+(3.32)
+In view of the above discussion, we emphasize that while a generic field transforms co-
+variantly under the action of the charges associated to the conserved currents (3.9) and
+(3.12), that is not the case for the generic conserved currents (3.5a) and (3.5b). A field
+that transforms covariantly (i.e. for which the extra terms in (3.31) and (3.32) vanish)
+even under the action of the charges of any currents of the form (3.5a) and (3.5b) is called
+a current-primary field. Consequently, there is no h.d.t. in (3.26) and (3.27) appropriate
+for a current-primary field. This enables us to completely specify the pole structures of the
+current-primary OPEs for a primary field Φ(t, x):
+J a
+t (t′, x′)Φ(t, x) ∼ lim
+ϵ→0+ ita
+K · Φ(t, x)
+∆˜x′
+(3.33)
+J a
+x (t′, x′)Φ(t, x) ∼ lim
+ϵ→0+ i
+�Φ(t, x) · ta
+J
+∆˜x′
+− (t′ − t)ta
+K · Φ(t, x)
+(∆˜x′)2
+�
+(3.34)
+that immediately imply the following Ward identities for a string X of primary fields:
+⟨J a
+t (t, x)X⟩ = lim
+ϵ→0+ i
+n
+�
+i=1
+(ta
+K)i · ⟨X⟩
+∆˜xi
+(3.35)
+⟨J a
+x (t, x)X⟩ = lim
+ϵ→0+ i
+n
+�
+i=1
+�⟨X⟩ · (ta
+J)i
+∆˜xi
+− (t − ti)(ta
+K)i · ⟨X⟩
+(∆˜xi)2
+�
+(3.36)
+where (ta
+J)i and (ta
+K)i denotes transformation-matrices appropriate for the i-th primary
+field in X.
+3.4
+Current-Current OPEs
+To derive the current-current OPEs using the machinery just developed, we shall assume
+that no field in the theory has negative scaling dimension with the identity being the only
+field with ∆ = 0.
+Under these assumptions, the OPEs between the EM tensor components were derived using
+only symmetry arguments in [43]; the results are:
+T t
+t(t′, x′)T t
+t(t, x) ∼ 0
+T t
+x(t′, x′)T t
+t(t, x) ∼ lim
+ϵ→0+ −i
+� −i cM
+2
+(∆˜x′)4 + 2T t
+t(t, x)
+(∆˜x′)2
++ ∂xT t
+t(t, x)
+∆˜x′
+�
+T t
+t(t′, x′)T t
+x(t, x) ∼ lim
+ϵ��0+ −i
+� −i cM
+2
+(∆˜x′)4 + 2T t
+t(t, x)
+(∆˜x′)2
++ ∂tT t
+x(t, x)
+∆˜x′
+�
+(3.37)
+T t
+x(t′, x′)T t
+x(t, x) ∼ lim
+ϵ→0+ −i
+� −i cL
+2
+(∆˜x′)4 + 2T t
+x(t, x)
+(∆˜x′)2
++ ∂xT t
+x(t, x)
+∆˜x′
+−(t′ − t)
+�−2icM
+(∆˜x′)5 + 4T t
+t(t, x)
+(∆˜x′)3
++ ∂tT t
+x(t, x)
+(∆˜x′)2
+��
+.
+– 15 –
+
+The constants cL and cM are the central charges of the 2D Carrollian conformal QFT.
+We shall use the same technique to find the current-current OPEs below. Keeping in mind
+that here we are dealing with currents with scaling dimension ∆ = 1, below we note the
+most general allowed form of the Jt − Jt OPE compatible with the general OPE (3.19):
+J a
+t (t′, x′)J b
+t (t, x) ∼ lim
+ϵ→0+ i
+�
+Aab
+(∆˜x′)2 + (ta
+K · Jt)b (t, x)
+∆˜x′
+�
+(3.38)
+where Aab is a field proportional to identity so that it has vanishing scaling dimension.
+Clearly, (ta
+K · Jt)b (t, x) must have scaling dimension ∆ = 1. So, in a generic 2D CCFT,
+(ta
+K · Jt)b must be a linear combination of {J a
+x , J a
+t }.
+Correspondingly to the classical conservation equation ∂tJ a
+t (t, x) = 0 , in the QFT we
+should have:
+J a
+t (t′, x′)∂tJ b
+t (t, x) ∼ 0
+⇒
+∂t (ta
+K · Jt)b (t, x) = 0
+(3.39)
+which means that {J a
+x } can not contribute to the linear combination (ta
+K · Jt)b.
+Since the currents have scaling dimension ∆ = 1, they must satisfy the bosonic3 exchange
+property. For the J a
+t (t, x) field, it is:
+J a
+t (t′, x′)J b
+t (t, x) = J b
+t (t, x)J a
+t (t′, x′)
+(3.40)
+This condition implies the following restrictions:
+Aab = Aba
+and
+(ta
+K · Jt)b = −
+�
+tb
+K · Jt
+�a
+(3.41)
+Looking at (3.20), we write the allowed form of a Jx − Jt OPE:
+J a
+x (t′, x′)J b
+t (t, x) ∼ lim
+ϵ→0+ i
+�
+Bab
+(∆˜x′)2 + (Jt · ta
+J)b (t, x)
+∆˜x′
+− (t′ − t)
+�
+2Aab
+(∆˜x′)3 + (ta
+K · Jt)b (t, x)
+(∆˜x′)2
+��
+(3.42)
+where Bab are some constants. Now, we have the following restrictions:
+J a
+x (t′, x′)∂tJ b
+t (t, x) ∼ 0
+=⇒
+Aab = 0
+and
+(ta
+K · Jt)b = 0
+and
+∂t (Jt · ta
+J)b (t, x) = 0
+(3.43)
+Thus, again {J a
+x } do not contribute to the linear combination (Jt · ta
+J)b.
+On the other hand, from (3.19), we get the following Jt − Jx OPE:
+J b
+t (t′, x′)J a
+x (t, x) ∼ lim
+ϵ→0+ i
+�
+Cba
+(∆˜x′)2 +
+�
+tb
+K · Jx
+�a (t, x)
+∆˜x′
+�
+(3.44)
+3Since, as will be shown later, the currents are 2D CC primary fields with integer scaling dimension,
+this statement is justified [43].
+– 16 –
+
+with Cab being constants. Using the following bosonic exchange property:
+J b
+t (t′, x′)J a
+x (t, x) = J a
+x (t, x)J b
+t (t′, x′)
+to compare the OPE (3.42) with (3.44), we get the following conditions:
+Bab = Cba
+and
+(Jt · ta
+J)b = −
+�
+tb
+K · Jx
+�a
+(3.45)
+which implies that {J a
+x } do not appear also in the linear combination
+�
+tb
+K · Jx
+�a.
+Finally, we write the Jx − Jx OPE in accordance with the general form (3.20):
+J a
+x (t′, x′)J b
+x(t, x) ∼ lim
+ϵ→0+ i
+�
+Dab
+(∆˜x′)2 + (Jx · ta
+J)b (t, x)
+∆˜x′
+− (t′ − t)
+�
+2Cab
+(∆˜x′)3 + (ta
+K · Jx)b (t, x)
+(∆˜x′)2
+��
+(3.46)
+from which, the bosonic exchange property:
+J b
+x(t′, x′)J a
+x (t, x) = J a
+x (t, x)J b
+x(t′, x′)
+leads to the following conditions:
+Dab = Dba ;
+Cab = Cba
+(3.47)
+(Jx · ta
+J)b = −
+�
+Jx · tb
+J
+�a
+;
+(ta
+K · Jx)b = −
+�
+tb
+K · Jx
+�a
+;
+∂t (Jx · ta
+J)b = ∂x (ta
+K · Jx)b
+(3.48)
+It can be readily checked that these conditions are compatible with the quantum versions
+(in the OPE language) of the classical conservation laws (3.4).
+We now explicitly write the allowed forms of the linear combinations appearing in the
+above OPEs:
+(Jt · ta
+J)b = (ta
+K · Jx)b = F abcJ c
+t
+with
+F abc = −F bac,
+(Jx · ta
+J)b = F abcJ c
+x + GabcJ c
+t
+with
+Gabc = −Gbac.
+(3.49)
+Thus, the final forms of the current-current OPEs are:
+J a
+t (t′, x′)J b
+t (t, x) ∼ 0
+J a
+x (t′, x′)J b
+t (t, x) ∼ lim
+ϵ→0+ i
+� Cab
+(∆˜x′)2 + F abcJ c
+t (t, x)
+∆˜x′
+�
+J a
+t (t′, x′)J b
+x(t, x) ∼ lim
+ϵ→0+ i
+� Cab
+(∆˜x′)2 + F abcJ c
+t (t, x)
+∆˜x′
+�
+(3.50)
+J a
+x (t′, x′)J b
+x(t, x) ∼ lim
+ϵ→0+ i
+�
+Dab
+(∆˜x′)2 +
+�
+F abcJ c
+x + GabcJ c
+t
+�
+(t, x)
+∆˜x′
+− (t′ − t)
+� 2Cab
+(∆˜x′)3 + F abcJ c
+t (t, x)
+(∆˜x′)2
+��
+These OPEs imply that the currents themselves are not current-primary fields in general.
+– 17 –
+
+3.5
+Global internal symmetry
+All the correlation functions in the theory must be invariant under the global internal
+transformations associated to which are the conserved currents (3.9) and (3.12). This fact
+will put constraints on Cab and Dab, as we will now see.
+We begin by noting the following 2-point correlators between the currents, from the above
+OPEs:
+�
+J a
+t (t′, x′)J b
+t (t, x)
+�
+= 0
+�
+J a
+t (t′, x′)J b
+x(t, x)
+�
+=
+�
+J a
+x (t′, x′)J b
+t (t, x)
+�
+= lim
+ϵ→0+ i Cab
+(∆˜x′)2
+(3.51)
+�
+J a
+x (t′, x′)J b
+x(t, x)
+�
+= lim
+ϵ→0+ i
+� Dab
+(∆˜x′)2 − (t′ − t) 2Cab
+(∆˜x′)3
+�
+since the currents, with ∆ = 1, must have vanishing VEVs.
+Next, due to the global internal symmetry, an arbitrary n-point correlator in the theory
+must satisfy:
+n
+�
+i=1
+⟨Φ1(t1, x1) . . . (Φi · ta
+J) (ti, xi) . . . Φn(tn, xn)⟩ = 0
+n
+�
+i=1
+⟨Φ1(t1, x1) . . . (ta
+K · Φi) (ti, xi) . . . Φn(tn, xn)⟩ = 0
+Thus the 2-point current correlators explicitly satisfy:
+�
+(Jx · ta
+J)b (t1, x1)J c
+x(t2, x2)
+�
++
+�
+J b
+x(t1, x1) (Jx · ta
+J)c (t2, x2)
+�
+= 0
+=⇒ F abdDdc + F acdDdb + GabdCdc + GacdCdb = 0
+and
+F abdCdc + F acdCdb = 0
+(3.52)
+The analogues relations obtained from the invariance of the 3-point current correlators are:
+F abeF ecfCfd + F caeF ebfCfd + F adeF bcfCfe = 0
+F abeF ecfDfd + F caeF ebfDfd + F adeF bcfDfe +
+�
+F abeGecf + GabeF ecf�
+Cfd
++
+�
+F caeGebf + GcaeF ebf�
+Cfd +
+�
+F adeGbcf + GadeF bcf�
+Cfe = 0
+(3.53)
+In what follows, we shall see that {F abc} and {Gabc} must also obey the following constraints
+arising as the Jacobi identity of the infinite-dimensional Lie algebra of the current-modes,
+which we had previously described in our initial algebraic description from the contraction
+in (2.21) and also will derive independently later in this section (3.80):
+F abeF ecd + F caeF ebd + F bceF ead = 0
+– 18 –
+
+F abeGecd + GabeF ecd + F caeGebd + GcaeF ebd + F bceGead + GbceF ead = 0
+(3.54)
+No new constraint for {F abc} and {Gabc} arises from the global internal invariance of
+n-point current correlators for n ≥ 4 .
+Upon comparison, we notice that (3.53) reduces to (3.54) if we choose:
+Dab = −ik1δab
+and
+Cab = −ik2δab with k2 ̸= 0
+(3.55)
+In that case, F abc and Gabc are anti-symmetric in all indices, as seen from (3.52).
+3.6
+EM tensor-current OPEs
+We now show under the assumption that no field in the theory has negative scaling dimen-
+sion with the identity field being the only one with ∆ = 0 , that the currents J a
+x (t, x) and
+J a
+t (t, x) must transform as a rank-1
+2 primary multiplet under 2D CC transformations.
+From [43], we recall the OPEs of a general 2D CC (multi-component) field Φ(l)(t, x) having
+scaling dimension ∆, Carrollian boost-charge ξ and boost rank l with the EM tensor
+components:
+T t
+x(t′, x′)Φ(l)(t, x) ∼ lim
+ϵ→0+ −i
+�
+. . . + ∆Φ(l)(t, x)
+(∆˜x′)2
++ ∂xΦ(l)(t, x)
+∆˜x′
+−(t′ − t)
+�
+. . . + 2
+�
+ξ · Φ(l)
+�
+(t, x)
+(∆˜x′)3
++ ∂tΦ(l)(t, x)
+(∆˜x′)2
+��
+T t
+t(t′, x′)Φ(l)(t, x) ∼ lim
+ϵ→0+ −i
+�
+. . . +
+�
+ξ · Φ(l)
+�
+(t, x)
+(∆˜x′)2
++ ∂tΦ(l)(t, x)
+∆˜x′
+�
+(3.56)
+The defining feature of 2D CC primary fields is the vanishing of the higher order poles in
+the above OPEs that leads to:
+T t
+x(t′, x′)Φ(l)(t, x) ∼ lim
+ϵ→0+ −i
+�∆Φ(l)(t, x)
+(∆˜x′)2
++ ∂xΦ(l)(t, x)
+∆˜x′
+−(t′ − t)
+�
+2
+�
+ξ · Φ(l)
+�
+(t, x)
+(∆˜x′)3
++ ∂tΦ(l)(t, x)
+(∆˜x′)2
+��
+T t
+t(t′, x′)Φ(l)(t, x) ∼ lim
+ϵ→0+ −i
+��
+ξ · Φ(l)
+�
+(t, x)
+(∆˜x′)2
++ ∂tΦ(l)(t, x)
+∆˜x′
+�
+(3.57)
+Now, along with the above assumption, the classical time-independence of the field J a
+t
+restricts the general OPE (3.56) to the following form:
+T t
+t(t′, x′)J a
+t (t, x) ∼ lim
+ϵ→0+ −i
+�
+Aa
+1
+(∆˜x′)3 + (ξ · J a
+t ) (t, x)
+(∆˜x′)2
+�
+(3.58)
+where the field Aa
+1 is proportional to the identity field. Similarly, we write the most general
+allowed form of the following OPE from (3.56):
+T t
+x(t′, x′)J a
+t (t, x) ∼ lim
+ϵ→0+ −i
+�
+Aa
+2
+(∆˜x′)3 + J a
+t (t, x)
+(∆˜x′)2 + ∂xJ a
+t (t, x)
+∆˜x′
+− (t′ − t)
+� 3Aa
+1
+(∆˜x′)4 + 2 (ξ · J a
+t ) (t, x)
+(∆˜x′)3
+��
+(3.59)
+– 19 –
+
+with Aa
+2 being another constant. But, the quantum counterpart of the conservation law
+(3.4) leads to:
+T t
+t(t′, x′)∂tJ a
+t (t, x) ∼ 0 =⇒ ∂t (ξ · J a
+t ) (t, x) = 0,
+T t
+x(t′, x′)∂tJ a
+t (t, x) ∼ 0 =⇒ Aa
+1 = 0
+and
+ξ · J a
+t = 0.
+(3.60)
+The OPEs for J a
+x (t, x) may have the most general form given below:
+T t
+t(t′, x′)J a
+x (t, x) ∼ lim
+ϵ→0+ −i
+�
+Aa
+3
+(∆˜x′)3 + (ξ · J a
+x ) (t, x)
+(∆˜x′)2
++ ∂tJ a
+x (t, x)
+∆˜x′
+�
+(3.61)
+with the conservation law (3.4) forcing:
+∂t (ξ · J a
+x ) (t, x) = 0.
+(3.62)
+The remaining one OPE is given below:
+T t
+x(t′, x′)J a
+x (t, x) ∼ lim
+ϵ→0+ −i
+�
+Aa
+4
+(∆˜x′)3 + J a
+x (t, x)
+(∆˜x′)2 + ∂xJ a
+x (t, x)
+∆˜x′
+(3.63)
+−(t′ − t)
+� 3Aa
+3
+(∆˜x′)4 + 2 (ξ · J a
+x ) (t, x)
+(∆˜x′)3
++ ∂tJ a
+x (t, x)
+(∆˜x′)2
+��
+where Aa
+3 and Aa
+4 are constants. From the quantum version of (3.4), one then obtains:
+T t
+x(t′, x′) [∂tJ a
+x (t, x) − ∂xJ a
+t (t, x)] ∼ 0
+=⇒
+ξ · J a
+x = J a
+t
+and
+Aa
+2 = Aa
+3
+(3.64)
+i.e. the currents J a
+x (t, x) and J a
+t (t, x) transform under Carrollian boost as a rank-1
+2 mul-
+tiplet with boost charge ξ = 1.
+The OPEs for the currents then are:
+T t
+t(t′, x′)J a
+t (t, x) ∼ 0
+T t
+x(t′, x′)J a
+t (t, x) ∼ lim
+ϵ→0+ −i
+�
+Aa
+3
+(∆˜x′)3 + J a
+t (t, x)
+(∆˜x′)2 + ∂xJ a
+t (t, x)
+∆˜x′
+�
+T t
+t(t′, x′)J a
+x (t, x) ∼ lim
+ϵ→0+ −i
+�
+Aa
+3
+(∆˜x′)3 + J a
+t (t, x)
+(∆˜x′)2 + ∂tJ a
+x (t, x)
+∆˜x′
+�
+(3.65)
+T t
+x(t′, x′)J a
+x (t, x) ∼ lim
+ϵ→0+ −i
+�
+Aa
+4
+(∆˜x′)3 + J a
+x (t, x)
+(∆˜x′)2 + ∂xJ a
+x (t, x)
+∆˜x′
+−(t′ − t)
+� 3Aa
+3
+(∆˜x′)4 + 2J a
+t (t, x)
+(∆˜x′)3
++ ∂tJ a
+x (t, x)
+(∆˜x′)2
+��
+On the other hand, applying the bosonic exchange property between the currents and the
+EM tensor components, we obtain:
+J a
+t (t′, x′)T t
+t(t, x) ∼ 0
+– 20 –
+
+J a
+t (t′, x′)T t
+x(t, x) ∼ lim
+ϵ→0+ −i
+�
+−
+Aa
+3
+(∆˜x′)3 + J a
+t (t, x)
+(∆˜x′)2
+�
+J a
+x (t′, x′)T t
+t(t, x) ∼ lim
+ϵ→0+ −i
+�
+−
+Aa
+3
+(∆˜x′)3 + J a
+t (t, x)
+(∆˜x′)2
+�
+(3.66)
+J a
+x (t′, x′)T t
+x(t, x) ∼ lim
+ϵ→0+ −i
+�
+−
+Aa
+4
+(∆˜x′)3 + J a
+x (t, x)
+(∆˜x′)2 − (t′ − t)
+�
+− 3Aa
+3
+(∆˜x′)4 + 2J a
+t (t, x)
+(∆˜x′)3
+��
+Comparing these with (3.19) and (3.20), we immediately note the following:
+�
+ta
+K · T t
+t
+�
+(t, x) =
+�
+ta
+K · T t
+x
+�
+(t, x) =
+�
+T t
+t · ta
+J
+�
+(t, x) =
+�
+T t
+x · ta
+J
+�
+(t, x) = 0
+(3.67)
+which implies that the EM tensor components actually transform under the singlet repre-
+sentation of the global internal symmetry algebra.
+Next we look at the consequences of the global internal symmetry on the 2-point correlators
+between the currents and the EM tensor components to find possible constraints on Aa
+3
+and Aa
+4. It suffices to consider the following:
+�
+(Jx · ta
+J)b (t1, x1)T t
+x(t2, x2)
+�
++
+�
+J b
+x(t1, x1)
+�
+T t
+x · ta
+J
+�
+(t2, x2)
+�
+= 0
+=⇒
+Aa
+3 = Aa
+4 = 0
+(3.68)
+Causing the vanishing of the poles of appropriate orders in the OPEs (3.65), we finally
+conclude, comparing with (3.57), that:
+the currents tranform as a 2D CC primary multiplet of rank-1
+2 with ∆ = ξ = 1 .
+Thus, the currents have the following infinitesimal 2D CC transformation property under
+(3.1), as is obtained using the prescription (3.23) for the corresponding conserved currents
+(3.3):
+− iGxJ a
+t (t, x) = −[f(x)∂x + f′(x)]J a
+t (t, x)
+;
+−iGtJ a
+t (t, x) = 0
+− iGtJ a
+x (t, x) = −g(x)∂tJ a
+x (t, x) − g′(x)J a
+t (t, x)
+(3.69)
+− iGxJ a
+x (t, x) = −[f(x)∂x + tf′(x)∂t + f′(x)]J a
+x (t, x) − tf′′(x)J a
+t (t, x)
+As an aside, from (3.66) we note down below the infinitesimal internal transformation
+properties of the EM tensor components, generated by the conserved charges of the currents
+(3.5a) and (3.5b):
+− iGJT t
+t(t, x) = −fa′(x)J a
+t (t, x)
+;
+−iGKT t
+t(t, x) = 0
+(3.70)
+− iGKT t
+x(t, x) = −ga′(x)J a
+t (t, x)
+;
+−iGJT t
+x(t, x) = −fa′(x)J a
+x (t, x) − tfa′′(x)J a
+t (t, x)
+– 21 –
+
+3.7
+The Algebra of Modes
+The EM tensor components have the following mode-expansions:
+T t
+t(t, x) = −i
+�
+n∈Z
+x−n−2Mn
+;
+T t
+x(t, x) = −i
+�
+n∈Z
+x−n−2 [Ln − (n + 2) t
+xMn]
+(3.71)
+=⇒ Mn = i
+�
+0
+dx
+2πi xn+1 T t
+t(t, x)
+;
+Ln = i
+�
+0
+dx
+2πi
+�
+xn+1T t
+x(t, x) + (n + 1)xnt T t
+t(t, x)
+�
+(3.72)
+In [43], it was shown from the EM tensor OPEs (3.37) that the EM tensor modes indeed
+generate the centrally extended BMS3 algebra:
+[Mn , Mm] = 0
+[Ln , Mm] = (n − m)Mn+m + cM
+12 (n3 − n)δn+m,0
+(3.73)
+[Ln , Lm] = (n − m)Ln+m + cL
+12(n3 − n)δn+m,0
+The infinitesimal 2D CC transformation properties of the currents are expressed in the
+operator language [43] using the prescription (3.23) for the charges in (3.72):
+[Ln , J a
+x (t, x)] = [xn+1∂x + t(n + 1)xn∂t + (n + 1)xn]J a
+x (t, x) + t(n + 1)nxn−1J a
+t (t, x)
+[Mn , J a
+x (t, x)] = xn+1∂tJ a
+x (t, x) + (n + 1)xnJ a
+t (t, x)
+(3.74)
+[Ln , J a
+t (t, x)] = xn+1∂xJ a
+t (t, x) + (n + 1)xnJ a
+t (t, x)
+;
+[Mn , J a
+t (t, x)] = 0
+Next, the classical conservation laws (3.4) imply the following space-time dependence of
+the fields {J a
+t } and {J a
+x }:
+∂tJ a
+t (t, x) = 0
+=⇒
+J a
+t (t, x) = J a
+t (x)
+∂tJ a
+x (t, x) = ∂xJ a
+t (x)
+=⇒
+J a
+x (t, x) = t∂xJ a
+t (x) + Ra(x)
+with Ra(x) being arbitrary functions.
+Guided by the above functional dependence and using the fact that the pair of the fields
+J a
+x and J a
+t forms a 2D CC primary rank-1
+2 multiplet with scaling dimension ∆ = 1, we
+write down the mode-expansions following [43]:
+J a
+t (x) =
+�
+n∈Z
+x−n−1Ka
+n
+;
+J a
+x (t, x) =
+�
+n∈Z
+x−n−1
+�
+Ja
+n − (n + 1) t
+xKa
+n
+�
+(3.75)
+Ka
+n =
+�
+Cu
+dx
+2πi xnJ a
+t (x)
+;
+Ja
+n =
+�
+Cu
+dx
+2πi
+�
+xnJ a
+x (t, x) + nxn−1tJ a
+t (x)
+�
+(3.76)
+where the counter-clockwise contour Cu encloses the upper half-plane and the real line.
+– 22 –
+
+Comparing (3.76) with (3.24) and (3.25), we immediately see that:
+Ja
+n is the conserved charge of the current jaµ =
+�
+xnJ a
+x + tnxn−1J a
+t , −xnJ a
+t
+�
+Ka
+n is the conserved charge of the current kaµ = (xnJ a
+t , 0)
+Thus, using the prescription (3.23) on the OPEs (3.33) and (3.34) we reach the definition
+of a current primary field Φ(t, x) in the operator formalism (for any n ∈ Z):
+[Ka
+n , Φ(t, x)] = ixn (ta
+K · Φ) (t, x),
+(3.77a)
+[Ja
+n , Φ(t, x)] = i
+�
+xn (Φ · ta
+J) + tnxn−1 (ta
+K · Φ)
+�
+(t, x)
+(3.77b)
+For the EM tensor components, the analogues commutation relations are found directly
+from (3.70):
+[Ka
+n , T t
+t(t, x)] = 0, [Ja
+n , T t
+t(t, x)] = −inxn−1J a
+t (t, x), [Ka
+n , T t
+x(t, x)] = −inxn−1J a
+t (t, x),
+[Ja
+n , T t
+x(t, x)] = −i
+�
+nxn−1J a
+x + tn(n − 1)xn−2J a
+t
+�
+(t, x)
+(3.78)
+Substituting the current mode-expansion (3.75) and EM tensor mode expansion (3.71) in
+these operator relations, we obtain the cross-commutation relations between the modes of
+the EM tensor and the currents:
+[Ln , Ja
+m] = −mJa
+m+n
+;
+[Mn , Ja
+m] = −mKa
+m+n = [Ln , Ka
+m] ;
+[Mn , Ka
+m] = 0
+(3.79)
+Similarly, from the current-current OPEs (3.50), using the condition (3.55) and the current
+mode-expansion (3.75), we reach the Lie algebra of the current modes:
+[Ja
+n , Jb
+m] = iF abcJc
+n+m + iGabcKc
+n+m + nk1δabδn+m,0
+[Ja
+n , Kb
+m] = iF abcKc
+n+m + nk2δabδn+m,0
+;
+[Ka
+n , Kb
+m] = 0
+(3.80)
+The global internal symmetry is governed by the subalgebra of the zero-modes:
+[Ja
+0 , Jb
+0] = iF abcJc
+0 + iGabcKc
+0
+;
+[Ja
+0 , Kb
+0] = iF abcKc
+0
+;
+[Ka
+0 , Kb
+0] = 0
+(3.81)
+This is precisely the algebra we have obtained for the NL currents in (2.21).
+– 23 –
+
+4
+Sugawara Construction
+As is well known, in 2d CFT, the Sugawara construction is employed to construct (the
+modes of) the energy momentum tensor in terms of (the modes of) the currents. In this
+section, here we will attempt to construct a NL version of the Sugawara construction and
+express Lm and Mm in terms of Jm and Km.
+We shall see that the Lm’s and Mm’s
+constructed in such manner indeed satisfy the BMS algebra. Similar constructions have
+been studied earlier in [44, 45].
+4.1
+Intrinsic construction from algebra
+We begin here by assuming we have only the algebra of the NL currents, i.e.
+[Ja
+m, Jb
+n] = ifabcJc
+m+n + mk1δabδm+n,0,
+(4.1a)
+[Ja
+m, Kb
+n] = ifabcKc
+m+n + mk2δabδm+n,0,
+[Ka
+m, Kb
+n] = 0.
+(4.1b)
+In the above, the sum over the group indices e.g. fabcJc
+m+n = �dim(¯g)
+c=1
+fabcJc
+m+n is implied.
+Comparing with (2.21), as stressed before, we work with the case where F abc = fabc, Gabc =
+0. Let us first consider the zero modes of J and K. Putting n = m = 0 in (4.1), the algebra
+for the zero modes become
+[Ja
+0 , Jb
+0] = ifabcJc
+0 ;
+[Ja
+0 , Kb
+0] = ifabcKc
+0 ;
+[Ka
+0, Kb
+0] = 0
+(4.2)
+Now let us look for quadratic Casimir operators for the above algebra.
+The possible
+combinations are
+�
+a
+Ja
+0 Ja
+0 ,
+�
+a
+Ja
+0 Ka
+0 ,
+�
+a
+Ka
+0Ja
+0 ,
+�
+a
+Ka
+0Ka
+0.
+(4.3)
+Keeping in mind that Casimir operators must commute with all the generators, we can
+exclude �
+a Ja
+0 Ja
+0 since it does not commute with Ka
+0
+� �
+a
+Ja
+0 Ja
+0 , Kb
+0
+�
+= i
+�
+a,c
+fabc(JaKc + KcJa) ̸= 0
+(4.4)
+All other combinations in (4.3) commute with all Ja
+0 ’s and Ka
+0’s. Hence a generic Casimir
+operator constructed from the zero modes of J and K will be a linear combination of the
+above three combinations of Ja
+0 ’s and Ka
+0’s. We therefore want to construct the zero level
+generator L0, M0 from these combinations.
+Since L0 and M0 are expected to be quadratic in terms of Ja
+n’s and Ka
+n’s, we can write
+down the following generic expression for them (The term � JJ is excluded since we have
+already seen that the term �
+a Ja
+0 Ja
+0 does not contribute to the zero mode part of L0 and
+M0)
+L0 = α
+�
+a,l,n
+Ja
+l Ka
+n + β
+�
+a,l,n
+Ka
+l Ja
+n + ρ
+�
+a,l,n
+Ka
+l Ka
+n
+(4.5a)
+M0 = µ
+�
+a,l,n
+Ja
+l Ka
+n + ν
+�
+a,l,n
+Ka
+l Ja
+n + η
+�
+a,l,n
+Ka
+l Ka
+n
+(4.5b)
+– 24 –
+
+Now putting the conditions that
+[L0, Ja
+n] = −nJa
+n, [M0, Ja
+n] = −nKa
+n
+and looking at just the level of current generators on the RHS, we can see that only (l = −n)
+terms should contribute, so we obtain
+L0 = α
+�
+a,n
+Ja
+nKa
+−n + β
+�
+a,n
+Ka
+nJa
+−n + ρ
+�
+a,n
+Ka
+nKa
+−n = αX0 + βY0 + ρZ0
+(4.6a)
+M0 = µ
+�
+a,l
+Ja
+l Ka
+−l + ν
+�
+a,l
+Ka
+l Ja
+−l + η
+�
+a,l
+Ka
+l Ka
+−l = µX0 + νY0 + ηZ0
+(4.6b)
+Now take
+Y0 =
+�
+a,l
+Ka
+l Ja
+−l =
+�
+a,l
+Ja
+−lKa
+l + k2
+dimg
+2
+∞
+�
+l=−∞
+l = X0 + c,
+(4.7)
+where c is a (possibly infinite) constant. Since we always have the independence of redefin-
+ing BMS generators by constant shifts, we can define level 0 BMS generators as
+L0 ≡ L′
+0 = α + β
+2
+(X0 + Y0) + ρZ0,
+M0 ≡ M′
+0 = µ + ν
+2
+(X0 + Y0) + ηZ0.
+(4.8)
+The normal ordering of the operators L0, M0 is achieved by individual normal ordering
+of X0, Y0, Z0. We generalise the definition to the other BMS generators as (with normal
+ordering)
+Lm =
+dimg
+�
+a=1
+�
+�α + β
+2
+�
+�
+�
+�
+l≤−1
+(Ja
+l Ka
+m−l + Ka
+l Ja
+m−l) +
+�
+l>−1
+(Ja
+m−lKa
+l + Ka
+m−lJa
+l )
+�
+�
+� + ρ
+�
+l
+Ka
+l Ka
+m−l
+�
+�
+Mm =
+dimg
+�
+a=1
+�
+�µ + ν
+2
+�
+�
+�
+�
+l≤−1
+(Ja
+l Ka
+m−l + Ka
+l Ja
+m−l) +
+�
+l>−1
+(Ja
+m−lKa
+l + Ka
+m−lJa
+l )
+�
+�
+� + η
+�
+l
+Ka
+l Ka
+m−l
+�
+�
+(4.9)
+Using the form of the BMS generators in (4.9), and substituting in the BMS-current cross
+commutators, i.e.
+[Lm, Ja
+n] = −nJa
+m+n ; [Lm, Ka
+n] = −nKa
+m+n ; [Mm, Ja
+n] = −nKa
+m+n,
+(4.10)
+we obtain the following values for the coefficients
+α + β = 1
+k2
+; ρ = −k1 + 2Cg
+2k2
+2
+;
+µ + ν = 0 ; η =
+1
+2k2
+.
+(4.11)
+So we obtain the final form for our NL Sugawara construction
+Lm =
+1
+2k2
+dim(g)
+�
+a=1
+�
+�
+�
+�
+l≤−1
+(Ja
+l Ka
+m−l + Ka
+l Ja
+m−l) +
+�
+l>−1
+(Ja
+m−lKa
+l + Ka
+m−lJa
+l ) − (k1 + 2Cg)
+k2
+�
+l
+Ka
+l Ka
+m−l
+�
+�
+�
+– 25 –
+
+Mm =
+1
+2k2
+dim(g)
+�
+a=1
+�
+l
+Ka
+l Ka
+m−l
+(4.12)
+As a check of the validity of the analysis, we compute the algebra of the L and M generators.
+These satisfy the following algebra (see Appendix B for detailed calculations)
+[Lm, Ln] = (m − n)Lm+n + dim(g)
+6
+(m3 − m)δm+n,0 ; [Lm, Mn] = (m − n)Mm+n
+(4.13)
+Hence, we see that by doing the NL Sugawara construction of NL currents, we can find
+BMS algebra with
+cL = 2dim(g),
+cM = 0.
+(4.14)
+Non-zero cM:
+We can obtain a non-zero cM with a slight modification to the Sugawara
+construction presented above. In case of Virasoro algebra with additional symmetries, we
+can define new Virasoro generators [46] as
+˜Ln = LS
+n + inθaja
+n + 1
+2kθ2δn,0
+(4.15)
+where LS
+n is the Virasoro generators obtained from Sugawara construction and θ = θata is a
+vector belonging to the Lie algebra with generators ta. It can be showed that (see Appendix
+C) if Ln satisfy Virasoro algebra with central charge c then ˜Ln will satisfy Virasoro algebra
+with shifted central charge ˜c where
+˜c = c + 12kθ2.
+(4.16)
+In case of NL Kac-Moody algebra, inspired by this, we introduce the following redefinitions
+˜Ln = LS
+n + inθaJa
+n + 1
+2k1θ2δn,0
+˜
+Mn = MS
+n + inθaKa
+n + 1
+2k2θ2δn,0
+(4.17)
+where LS
+n and MS
+n are given by (4.9). This redefinition will give us the following algebra
+(see Appendix C)
+[˜Lm, ˜Ln] = (m − n)˜Lm+n + cL
+12(m3 − m)δn+m,0
+[˜Lm, ˜
+Mn] = (m − n) ˜
+Mm+n + cM
+12 (m3 − m)δn+m,0
+(4.18)
+where the central charges are given by
+cL = 2dim(g) + 12k1θ2
+cM = 12k2θ2
+(4.19)
+Hence we can obtain the fully centrally extended BMS algebra.
+– 26 –
+
+4.2
+Consistency with OPEs
+We will recast the NL Sugawara construction in terms of the NL EM tensor that we
+introduced in Sec. 3 instead of the generators of the BMS: {Ln, Mn}. {Ln, Mn} are of
+course modes of the NL EM tensor. Hence the calculations in this subsection provide a
+sanity check for our analysis making sure the various formulations are consistent with each
+other.
+The NL Sugawara construction gave us (4.12).
+We now normal order the products to
+rewrite this as
+Ln =
+1
+2k2
+�
+l,a
+�
+: Ja
+l Ka
+n−l : + : Ka
+l Ja
+n−l : −k1 + 2Cg
+k2
+: Ka
+l Ka
+n−l :
+�
+Mn =
+1
+2k2
+�
+l,a
+: Ka
+l Ka
+n−l :
+(4.20)
+Here, : Al : is a shorthand for normal ordered products. Now substituting (4.20) in (3.71)
+and rearranging the terms, we get uv to xt
+Tu(u, v) =
+1
+2k2
+�
+a
+��
+(J a
+u J a
+v )(u, v) + (J a
+v J a
+u )(u, v)
+�
+− k1 + 2Cg
+k2
+(J a
+v J a
+v )(v)
+�
+Tv(v) =
+1
+2k2
+�
+a
+(J a
+v J a
+v )(v)
+(4.21)
+Here (. . . ) means normal ordered products of fields, which can be expressed in terms of
+contour integrals.
+Here we are using u, v coordinates instead of x, t of Sec. 3 because
+this construction applies to both Galilean and Carrollian CFTs, because of the similarity
+between the 2 theories mentioned in the introduction. So by substituting u → x, v → t or
+u → t, v → x, we can obtain the Galilean or Carrollian theory expressions, respectively.
+So (4.21) gives the field expression for the NL Sugawara construction. For the modified
+Sugawara construction as defined in (4.17), the EM tensor fields turn out to be (by doing
+the mode expansion using the new generators)
+˜Tu(u, v) = Tu(u, v) − iθa∂vJ a
+u (u, v) − iθa J a
+u (u, v)
+v
+− iθa uJ a
+v (v)
+v2
++ 1
+2
+k1θ2
+v2
++ uk2θ2
+v3
+˜Tv(v) = Tv(v) − iθa∂vJ a
+v (v) − iθa J a
+v (v)
+v
++ 1
+2
+k2θ2
+v2
+(4.22)
+where, Tu(u, v), Tv(u, v) refers to the quantities in (4.21), i.e. the normal NL Sugawara
+construction expressions.
+T − J OPE:
+Now, for a consistency check of our analysis so far, we will rederive the
+OPEs from the above definitions of the NL Sugawara construction. We begin with the T-J
+OPEs. Taking the definitions as in (4.21) and contracting with the currents, we get the
+– 27 –
+
+following expressions (See Appendix D for detailed calculation):
+Tv(u1, v1)J b
+v (u2, v2) ∼ regular
+Tu(u1, v1)J b
+v (u2, v2) ∼ J b
+v (u2, v2)
+v2
+12
++ ∂vJ b
+v (u2, v2)
+v12
++ . . .
+Tv(u1, v1)J a
+u (u2, v2) ∼ J a
+v (u2, v2)
+v2
+12
++ ∂vJ a
+v (u2, v2)
+v12
++ . . .
+Tu(u1, v1)J a
+u (u2, v2) ∼ J a
+u (u2, v2)
+v2
+12
++ ∂vJ a
+u (u2, v2)
+v12
++ u12
+v2
+12
+∂uJ a
+u (u2, v2)
++ 2u12
+v12
+�J a
+v (u2, v2)
+v2
+12
++ ∂vJ a
+v (u2, v2)
+v12
+�
++ . . .
+(4.23)
+Which are equivalent to the [Lm, Ja
+n] type commutation relations given in (2.21). Clearly
+these relation satisfy the OPE relations (3.66) that the currents and the EM tensors of a
+theory are supposed to satisfy.
+T − T OPE:
+Next we use the definition of the Sugawara construction to compute the
+T-T type OPEs. Doing the calculations (see Appendix D for detailed Calculations), we
+get,
+Tu(u1, v1)Tu(u2, v2) ∼2Tu(u2, v2)
+v2
+12
++ ∂vTu(u2, v2)
+v12
++ u12
+v2
+12
+∂uTu(u2, v2)
++ 2u12
+v12
+�2Tv(u2, v2)
+v2
+12
++ ∂vTv(u2, v2)
+v12
+�
++ dim(g)
+v4
+12
++ . . .
+Tu(u1, v1)Tv(u2, v2) ∼2Tv(u2, v2)
+v2
+12
++ ∂vTv(u2, v2)
+v12
++ . . .
+(4.24)
+Tv(u1, v1)Tv(u2, v2) ∼ regular
+These results (4.24) match with the OPEs of the energy momentum tensors of a 2d Carrol-
+lian or Galilean CFT, as given in (3.37). So, the NL Sugawara construction really gives the
+EM tensors of a 2d NL CFT (proved both at algebra level and now at field level). Also from
+the OPE expressions, we can verify the earlier results of cL = 2dim(g), cM = 0 obtained ear-
+lier. Again if we start with the modified Sugawara construction (4.22), we can obtain same
+OPE relations as above, but with modified central charges cL = 2dim(g) + 12k1θ2, cM =
+12k2θ2. To see this, let’s look at, for example, the ˜Tu(u1, v1) ˜Tv(v2) OPE and focus our
+attention to a specific term
+˜Tu(u1, v1) ˜Tv(v2) ∼ −θaθb∂vJ a
+u (u1, v1)∂vJ a
+v (v2) + · · · ∼ −θ2∂v1∂v2
+k2
+v2
+12
++ · · · ∼ 6k2θ2
+v4
+12
++ . . .
+Matching the above expression with (3.37), we can see cM = 12k2θ2. Similarly we can
+obtain the other shifted central charge.
+– 28 –
+
+5
+Tensionless String as NLKM
+The tensionless limit of string theory is the limit that is diametrically opposite to the
+usual point particle limit where known supergravity appears from strings. This limit that
+explores the very strong gravity regime and, when quantized, the very highly quantum and
+highly stringy regime of string theory. In this section, we explore how this is connected
+to the NLKM structures we have discussed in this paper. We will see that the currents
+satisfying the U(1) NLKM algebra come up intrinsically when we are looking at tensionless
+strings propagating in flat spacetimes.
+As is very well known, the action for a tensile relativistic string propagating in a flat
+d-dimensional spacetime is given by the Polyakov action:
+SP = T
+�
+d2ξ √γγαβ∂αXµ∂βXνηµν.
+(5.1)
+In order to take the tensionless limit, it is helpful to work with the phase space action of
+string. One can then systematically take the limit [] and the resulting action can be cast
+in a Polyakov like form, which we will call the ILST action after the authors:
+SILST =
+�
+d2ξ V αV β∂αXµ∂βXνηµν.
+(5.2)
+Here Xµ are the coordinates in the background flat space ηµν which are scalar fields on
+the worldsheet parametrized by ξa = σ, τ. The worldsheet metric γαβ degenerates in the
+limit and the following replacement is made:
+T√γγαβ → V αV β,
+(5.3)
+where V α is a vector density. The action is invariant under worldsheet diffeomorphisms
+and hence one needs to fix a gauge. It is helpful to go to the equivalent of the conformal
+gauge
+V α = (v, 0).
+(5.4)
+There is some symmetry left over even after this gauge fixing. In the usual tensile string,
+the residual symmetry gives two copies of the Virasoro algebra and the appearance of
+a 2d CFT on the worldsheet is the central reason why we understand string theory as
+well as we do. Now in the tensionless case, the residual gauge symmetry that appears on
+the worldsheet is the BMS3 algebra. This now dictates the theory of tensionless strings.
+The reason behind the appearance of the BMS algebra is that the tensionless string in
+flat spacetimes is actually a null string. This is the string equivalent of a massless point
+particle which is constrained to travel on a null geodesic of the ambient spacetime. The
+string sweeps out a worldsheet which is null and since there are no mass terms, the resulting
+action has to have conformal Carrollian symmetry in d = 2 or equivalently BMS3.
+Now, let us connect to our discussion in this paper. In the favourable conformal gauge
+(5.4), the equations of motion and constraints arising from (5.2) take the form
+EOM:
+¨Xµ = 0 ;
+Constraints:
+˙X2 = 0 ,
+˙X.X′ = 0.
+(5.5)
+– 29 –
+
+A convenient form of the solution is given by
+Xµ(σ, τ) = xµ +
+√
+2c′�
+Jµ
+0 σ + Kµ
+0 τ + i
+�
+n̸=0
+1
+n(Jµ
+n − inτKµ
+n)e−inσ�
+(5.6)
+Here σ, τ are the coordinates on the cylinder, which are related to the planar Non-Lorentzian
+coordinates as
+u = e−iσ ; v = −τe−iσ
+(5.7)
+The periodic condition Xµ(σ + 2π, τ) = Xµ(σ, τ) forces us to have Jµ
+0 = 0. From (5.6), we
+obtain
+Jµ
+τ = ∂σXµ =
+√
+2c′ �
+n
+(Jµ
+n − inτKµ
+n)e−inσ,
+Jµ
+σ = −∂τXµ = −
+√
+2c′ �
+n
+Kµ
+ne−inσ
+(5.8)
+Clearly the currents in (5.8) satisfy similar relations as (3.4), i.e.
+∂τJµ
+σ = 0 ; ∂τJµ
+τ + ∂σJµ
+σ = 0.
+(5.9)
+If we canonically quantise the system by demanding [Xµ(τ, σ), Πν(τ, σ′)] = iδ(σ − σ′)ηµν,
+we obtain the following relations for the current modes
+[Jµ
+m, Jν
+n] = [Kµ
+m, Kν
+n] = 0 ; [Jµ
+m, Kν
+n] = 2mδm+n,0ηµν
+(5.10)
+Clearly, the current modes for the same spacetime index µ follow a special case of U(1) NL
+Kac-Moody algebra (look at (2.21) for comparison), with k(µν)
+1
+= 0, k(µν)
+2
+= 2ηµν.
+Next if we look into the (classical) energy momentum tensor of this theory, their mode
+expansion coefficients are given by
+Ln = 1
+2
+�
+m
+Jµ
+mKν
+n−mηµν ; Mn = 1
+4
+�
+m
+Kµ
+mKν
+n−mηµν.
+(5.11)
+This matches classically with the relation (4.12) for the NL Sugawara construction with
+k1 = 0,
+k2 = 2.
+(5.12)
+Note that Cg = 0 for U(1) algebra. The generators in (5.11) satisfy the (centreless) BMS3
+algebra, which is expected as the action (5.2) has BMS symmetry as a residual gauge
+symmetry.
+Thus we can see that the Non-Lorentzian U(1) Kac-Moody algebra and the associated NL
+Sugawara construction come up in the theory of tensionless or null strings propagating on
+a flat background geometry. An outstanding question is what happens when we look at the
+propagation of null strings on arbitrary curved manifolds. These can be viewed as group
+manifolds and there will be the NL equivalent of a Wess-Zumino-Witten model appearing
+for tensionless strings in this context. We expect that the non-abelian NLKM algebras
+explored in this work would naturally appear on these NL WZW models. This is work in
+progress and we hope to report on this in the near future.
+– 30 –
+
+6
+Non-Lorentzian Knizhnik Zamolodchikov Equations
+In usual Lorentzian Kac-Moody algebras, the Knzhnik-Zamolodchikov equations are the
+linear differential equations that are satisfied by the correlation functions of these CFTs
+endowed with additional affine Lie symmetry. In this section, we write down the non-
+Lorentzian analogues of these KZ equations based on our construction of the NLKM algebra
+and its representations in this paper.
+In what follows, we will outline the steps to get to the NL Knizhnik Zamolodchikov equa-
+tions. The details of the calculation are presented in a separate appendix E.
+We begin with the OPE definition of BMS primary field (3.57) and obtain
+∂uΦ(u′, v′) = −
+�
+v′
+dv
+2πi
+�
+u′
+du
+2πi(u − u′)−1Tv(u, v)Φ(u′, v′) = −
+�
+v′
+dv
+2πiTv(v)Φ(u′, v′)
+∂vΦ(u′, v′) =
+�
+v′
+dv
+2πi
+�
+u′
+du
+2πi(u − u′)−1Tu(u, v)Φ(u′, v′) =
+�
+v′
+dv
+2πiTu(u′, v)Φ(u′, v′)
+(6.1)
+Next, we take the correlation function of a string of primary fields as shown below
+⟨∂uΦ(u, v)Φ1(u1, v1) . . . Φn(un, vn)⟩ = ⟨∂uΦ(u, v)X({ui, vi})⟩ =
+�
+v
+dv′
+2πi⟨Tv(u′, v′)Φ(u, v)X({ui, vi})⟩
+=
+�
+v
+dv′
+2πi⟨ 1
+2k2
+�
+(J a
+v J a
+v )(u′, v′)
+�
+Φ(u, v)X({ui, vi})⟩
+and use relations (3.57, 6.1) to finally get the following expression (details in Ap. E)
+⟨∂uΦ(u, v)X({ui, vi})⟩ = − 1
+k2
+��
+j
+ta
+R,K ⊗ ta
+Rj,K
+(v − vj)
+�
+⟨Φ(u, v)X({ui, vi})⟩
+(6.2)
+which can be written as
+�
+∂ui + 1
+k2
+�
+j̸=i
+ta
+Ri,K ⊗ ta
+Rj,K
+(vi − vj)
+�
+⟨Φ1(u1, v1) . . . Φn(un, vn)⟩ = 0.
+(6.3)
+This is one of the Non-Lorentzian Knizhnik Zamolodchikov equations.
+Similarly we can start with
+⟨∂vΦ(u, v)Φ1(u1, v1) . . . Φn(un, vn)⟩ = ⟨∂vΦ(u, v)X({ui, vi})⟩ =
+�
+v
+dv′
+2πi⟨Tu(u′, v′)Φ(u, v)X({ui, vi})⟩
+=
+�
+v
+dv′
+2πi⟨ 1
+2k2
+�
+(J a
+v J a
+u )(u′, v′) + (J a
+u J a
+v )(u′, v′) − k1 + 2Cg
+k2
+(J a
+v J a
+v )(u′, v′)
+�
+Φ(u, v)X({ui, vi})⟩
+(6.4)
+From here, we can proceed in the same way as Appendix E to obtain the other equation
+�
+∂vi − 1
+k2
+�
+j̸=i
+1
+(vi − vj)
+dimg
+�
+a=1
+��
+ta
+Ri,J ⊗ ta
+Rj,K + ta
+Ri,K ⊗ ta
+Rj,J
+�
++
+�ui − uj
+vi − vj
+− k1 + 2Cg
+k2
+�
+ta
+Ri,K ⊗ ta
+Rj,K
+��
+⟨ΦR1(u1, v1)ΦR2(u2, v2)...ΦRn(un, vn)⟩ = 0
+(6.5)
+– 31 –
+
+This above equation (6.5) is the second of the Non-Lorentzian Knizhnik Zamolodchikov
+equations. The solutions to these two equations (6.3), (6.5) would give the correlation
+functions of the underlying NLKM theory. For usual relativistic theories, the KZ equations
+are difficult to solve in general, but one can do it for the four-point functions yielding a
+closed solution in terms of hypergeometric functions.
+It would be instructive to check
+whether something similar happens here. If one can find the solutions to the NLKM four
+point functions, it would also be a nice exercise to check whether these can be arrived
+at as a limit of the relativistic answers. This process of attempting to generate the non-
+Lorentzian answers from the relativistic ones is something we outline in detail in the section
+that follows.
+7
+NLKM from Contractions
+In this section, we rederive various results we have obtained earlier in the paper through
+a systematic limit on the algebraic structures obtained in the relativistic set-up. Before
+moving to recovering the answers, we spend a bit of time understanding which limit is
+appropriate for our purposes.
+7.1
+A brief detour to representations of BMS
+We said in the introduction that there were two distinct contractions that land us up on
+the BMS3 algebra starting from two copies of the Virasoro algebra.
+One of them was
+a Carrollian or ultra-relativistic limit (2.17), where there was a mixing of positive and
+negative Virasoro modes creating the BMS generators, while the other was a Galilean or
+non-relativistic (2.13), where no mixing took place. This mixing of modes is critical for the
+understanding of the representations in the limit.
+We begin with the Galilean contraction.
+In the parent relativistic CFT, the theory is
+best described in terms of the highest weight representation. The states of the theory are
+labelled by the zero modes:
+L0|h, ¯h⟩ = h|h, ¯h⟩,
+¯L0|h, ¯h⟩ = ¯h|h, ¯h⟩
+(7.1)
+There is a class of states called primary states which are annihilated by all positive modes:
+Ln|h, ¯h⟩p = 0,
+¯Ln|h, ¯h⟩p = 0,
+∀n > 0.
+(7.2)
+The Virasoro modules are built on these primary states by acting with raising operators
+L−n. Now, looking back at the Galilean contraction (2.13), we see that
+Ln = 1
+2
+�
+Ln + 1
+ϵ Mn
+�
+,
+¯Ln = 1
+2
+�
+Ln − 1
+ϵ Mn
+�
+(7.3)
+The 2d CFT primary conditions then boil down to:
+L0|∆, ξ⟩ = ∆|∆, ξ⟩, M0|∆, ξ⟩ = ξ|∆, ξ⟩;
+Ln|∆, ξ⟩p = 0, Mn|∆, ξ⟩p = 0, ∀n > 0.
+(7.4)
+– 32 –
+
+In the above, the assumption is that the state |h, ¯h⟩ goes to the state |∆, ξ⟩ in the limit. So
+we see that highest weight states map to highest weight states in the Galilean contraction.
+This analysis holds in a similar way when we consider the full NLKM algebra. We will be
+using this for the analysis in the rest of the section.
+But before we get there, let us point out why we are not using the Carroll limit (2.17) for
+this purpose. In the Carroll contraction, we can read off
+Ln = 1
+2
+�
+Ln + 1
+ϵ Mn
+�
+,
+¯Ln = 1
+2
+�
+−L−n + 1
+ϵ M−n
+�
+(7.5)
+The 2d CFT primary conditions in the Carroll limit become:
+M0|M, s⟩ = M|M, s⟩,
+L0|M, s⟩ = s|M, s⟩,
+Mn|M, s⟩ = 0,
+∀n ̸= 0.
+(7.6)
+In the above, the state |h, ¯h⟩ in the Carroll limit becomes the state |M, s⟩. This is clearly
+not a highest weight state. The set of these states form what is called the induced repre-
+sentation. The story is again similar for the full NLKM. In the analysis that follows, we
+will not focus on the induced representations. It would be of interest to consider them in
+future work.
+We should stress however, that the answers we have obtained in the intrinsic Carrollian
+form earlier are for highest weight representations. The Galilean limit would be an effective
+way of reproducing these answers with a flip of space and time directions at the end of the
+analysis.
+7.2
+Contraction of the affine parameters
+In this section we will establish the mode expansion of the NLKM currents (3.76) from
+another approach, by doing a contraction of the original loop extended algebra. Starting
+from the finite Lie algebra
+[ja, jb] = ifabcjc ; [¯ja, ¯jb] = ifabc¯jc
+(7.7)
+we can define loop extended generators
+ja
+n = ja ⊗ zn ; ¯ja
+n = ¯ja ⊗ ¯zn,
+(7.8)
+which satisfy
+[ja
+n, jb
+m] = ifabcjc ⊗ zn+m = ifabcjc
+n+m.
+(7.9)
+Loop extended algebra obtained above admits a central extension to give us the current
+algebra in (2.12). Now, we can do a contraction of the affine parameter as z = t + ϵx and
+¯z = t − ϵx in the Galilean limit. We then obtain (keeping upto first order in ϵ),
+ja
+n = ja ⊗ (tn + nϵxtn−1 + ...) ; ¯ja
+n = ¯ja ⊗ (tn − nϵxtn−1 + ...)
+(7.10)
+Now we can define new contracted generators of the finite algebra as follows:
+Ja = ja + ¯ja ; Ka = ϵ(¯ja − ja)
+(7.11)
+– 33 –
+
+Above definition can be extended to general modes of J and K using (7.10),
+Ja
+n = lim
+ϵ→0(ja
+n + ¯ja
+n) = Ja ⊗ tn − nKa ⊗ xtn−1
+(7.12a)
+Ka
+n = lim
+ϵ→0 ϵ(¯ja
+n − ja
+n) = Ka ⊗ tn
+(7.12b)
+In this way, we get the structure of the power series expansion of J and K in terms of x
+and t, which is the Galilean analog of (3.76), or identical upto the flip of temporal and
+spatial directions.
+If we define primary field φ(x, t) as a representation of the finite algebra, i.e. in terms of
+the action of zero modes of the generators J and K,
+[Ja
+0 , φ(x, t)] ≡ [Ja, φ(x, t)] = ta
+Jφ(x, t)
+[Ka
+0, φ(x, t)] ≡ [Ka, φ(x, t)] = ta
+Kφ(x, t)
+(7.13)
+We can get the action of general modes using (7.12),
+[Ja
+n, φ(x, t)] = ta
+Jφ(x, t) ⊗ tn − nta
+Kφ(x, t) ⊗ xtn−1 ≡ tnta
+Jφ(x, t) − nxtn−1ta
+Kφ(x, t)
+[Ka
+n, φ(x, t)] = ta
+Kφ(x, t) ⊗ tn ≡ tnta
+Kφ(x, t)
+(7.14)
+This definition of a primary field agrees with our earlier result (3.77).
+This action of Ja
+n and Ka
+n on primary fields also appears when we draw motivation from
+[41]. We can define the primary field as,
+[Ja
+0 , φ(0, 0)] = ta
+Jφ(0, 0) ; [Ka
+0, φ(0, 0)] = ta
+kφ(0, 0),
+[Ja
+n, φ(0, 0)] = 0 ; [Ka
+n, φ(0, 0)] = 0
+∀ n > 0.
+(7.15)
+For n ≥ 0 and U = etL−1−xM−1,
+[Ja
+n, φ(x, t)] = [Ja
+n, Uφ(0)U −1] = U[U −1Ja
+nU, φ(0)]U −1
+(7.16)
+Consider,
+U −1Ja
+nU = e−tL−1+xM−1Ja
+netL−1−xM−1
+=
+n
+�
+k=0
+tk
+k!
+n!
+(n − k)!Ja
+n−k − nx
+n−1
+�
+k=0
+tk
+k!
+(n − 1)!
+(n − k − 1)!Ka
+n−k−1
+(7.17)
+where we have used the Baker-Campbell-Hausdorff(BCH) formula twice. Putting this back
+in (7.16), we finally get,
+[Ja
+n, φ(x, t)] = U
+� n
+�
+k=0
+tk
+k!
+n!
+(n − k)![Ja
+n−k, φ(0)] − nx
+n−1
+�
+k=0
+tk
+k!
+(n − 1)!
+(n − k − 1)![Ka
+n−k−1, φ(0)]
+�
+U −1
+= (tnta
+J − nxtn−1ta
+K)φ(x, t)
+(7.18)
+where only the terms corresponding to k = n and k = n − 1 survived respectively in the
+first and the second sum because of (7.15). Similarly for Ka
+n, we get,
+[Ka
+n, φ(x, t)] = U
+� n
+�
+k=0
+tk
+k!
+n!
+(n − k)![Ka
+n−k, φ(0)]
+�
+U −1 = tnta
+Kφ(x, t)
+(7.19)
+These results verify (7.14) again from a different perspective.
+– 34 –
+
+7.3
+Contracting the Sugawara construction
+In section 4.1, we constructed the BMS generators by taking quadratic products of the
+NL currents and ended up with (4.12). In this subsection, we shall reproduce the same
+from the Galilean limit of Sugawara construction for CFT. We start from the following
+expression for virasoro modes Lm and ¯Lm in relativistic sugawara construction.
+Lm = γ
+dim(g)
+�
+a=1
+(
+�
+l≤−1
+ja
+l ja
+m−l +
+�
+l>−1
+ja
+m−lja
+l )
+(7.20a)
+¯Lm = ¯γ
+dim(g)
+�
+a=1
+(
+�
+l≤−1
+¯ja
+l ¯ja
+m−l +
+�
+l>−1
+¯ja
+m−l¯ja
+l )
+(7.20b)
+where γ =
+1
+2(k + Cg),¯γ =
+1
+2(¯k + Cg) and Cg = −
+1
+2dim(g)
+�
+b,c
+fbacfbcd
+(7.20c)
+We can now take the Galilean limit by using the inverted version of relations (2.20) and
+get the following form of Virasoro Generators by collecting the terms with same order in ϵ,
+Lm = γ
+4(Am − Bm
+ϵ
++ Cm
+ϵ2 ) ; ¯Lm = ¯γ
+4(Am + Bm
+ϵ
++ Cm
+ϵ2 )
+(7.21)
+where,
+Am =
+dim(g)
+�
+a=1
+�
+l≤−1
+Ja
+l Ja
+m−l +
+�
+l>−1
+Ja
+m−lJa
+l
+(7.22a)
+Bm =
+dim(g)
+�
+a=1
+{
+�
+l≤−1
+(Ja
+l Ka
+m−l + Ka
+l Ja
+m−l) +
+�
+l>−1
+(Ja
+m−lKa
+l + Ka
+m−lJa
+l )}
+(7.22b)
+Cm =
+dim(g)
+�
+a=1
+�
+l
+Ka
+l Ka
+m−l
+(7.22c)
+where we have employed the commutativity of K’s in order to write 1
+ϵ2 term(Cm) as a sum
+from l = −∞ to ∞. We can use (2.13) and the above relations to obtain:
+Lm = 1
+4
+�
+(γ + ¯γ)Am − (γ − ¯γ)
+ϵ
+Bm + (γ + ¯γ)
+ϵ2
+Cm
+�
+(7.23a)
+Mm = −1
+4
+�
+ϵ(γ − ¯γ)Am − (γ + ¯γ)Bm + (γ − ¯γ)
+ϵ
+Cm
+�
+(7.23b)
+Inverting the relations (2.20), we can write k and ¯k in terms of k1 and k2,
+k = 1
+2(k1 − k2
+ϵ ) ; ¯k = 1
+2(k1 + k2
+ϵ )
+(7.24)
+Using the definitions of γ and ¯γ in (7.20) and using (7.24), we can write,
+lim
+ϵ→0
+γ + ¯γ
+ϵ2
+= −2(k1 + 2Cg)
+k2
+2
+and lim
+ϵ→0
+γ − ¯γ
+ϵ
+= − 2
+k2
+(7.25)
+– 35 –
+
+Hence, in limit ϵ → 0 (7.23) can be written as,
+Lm =
+1
+2k2
+�
+Bm − (k1 + 2Cg)
+k2
+Cm
+�
+and Mm =
+1
+2k2
+Cm
+(7.26)
+which agrees with (4.12).
+As we have already seen in the previous section that (4.12) satisfies BMS algebra with
+central charges cL = 2dim(g) and cM = 0. This fact can also be verified using the following
+definitions of central charges in the relativistic Sugawara construction,
+c = kdim(g)
+k + Cg
+; ¯c =
+¯kdim(g)
+¯k + Cg
+(7.27)
+Using (7.27) and following similar steps as before using the (7.24), we can get the following,
+cL = lim
+ϵ→0(c + ¯c) = lim
+ϵ→0
+�
+dim(g)
+�
+1
+2(k1 − k2
+ϵ )
+1
+2(k1 − k2
+ϵ ) + Cg
++
+1
+2(k1 + k2
+ϵ )
+1
+2(k1 + k2
+ϵ ) + Cg
+��
+⇒ cL = 2dim(g)
+(7.28)
+Similarly, we can get,
+cM = lim
+ϵ→0 ϵ(¯c − c) = 0
+(7.29)
+These are the same values of central charges appeared in the L, M commutation relations
+as we got in (4.13).
+The modification we have done in order to get non-zero cM too can be retrieved from
+contraction. For this, we can start from (4.15) and its conjugate. Now if we take the limit
+in (2.20), we shall again retrieve the redefined Lns and Mns as we have seen in (4.17), and
+the commutators will be same as (4.18) with central charges (4.19).
+7.4
+NLKZ equations from Contraction
+Finally, we show how to obtain the non-Lorentzian Knizhnik Zamolodchikov equations
+from a limit of the ones for a 2d CFT with additional symmetry. Some of the details of the
+analysis are contained in Appendix F. We start with the original Knizhnik Zamolodchikov
+equations:
+�
+�∂wi − 2γ
+�
+j̸=i
+�
+a(ta
+Ri)ri
+si(ta
+Rj)rj
+sj
+wi − wj
+�
+� ⟨...φsi
+Ri(wj)...φsj
+Rj(wj)...⟩ = 0
+(7.30a)
+�
+�∂ ¯wi − 2¯γ
+�
+j̸=i
+�
+a(¯ta¯Ri)¯ri¯si(¯ta¯Rj)¯rj
+¯sj
+¯wi − ¯wj
+�
+� ⟨...¯φ¯si¯Ri( ¯wj)...¯φ¯sj
+¯Rj( ¯wj)...⟩ = 0
+(7.30b)
+where
+(ta
+Ri)ri
+si(ta
+Rj)rj
+sj⟨...φsi
+Ri(wj)...φsj
+Rj(wj)...⟩ = ((ta
+Ri ⊗ ta
+Rj)⟨...φRi(wj)...φRj(wj)...⟩)ri,rj
+(7.31)
+– 36 –
+
+Using the separability of the primary fields, Φr,¯r
+R, ¯R(w, ¯w) = φr
+R(w) ⊗′ ¯φ¯r¯R( ¯w), we can write,
+⟨Φr1,¯r1
+R1, ¯R1(w1, ¯w1)...ΦrN,¯rN
+RN, ¯RN (wN, ¯wN)⟩ = ⟨φr1
+R1(w1)...φrN
+RN (wN)⟩⟨¯φ¯r1¯R1( ¯w1)...¯φ¯rN
+¯RN ( ¯wN)⟩ (7.32)
+where the primed tensor product(⊗′) ensures independent action of operators on chiral and
+anti-chiral sectors whereas the unprimed tensor product(⊗) ensures independent action on
+ith and jth insertion in the n-point function.
+Using the following linear combinations in limit ϵ → 0,
+(7.30a) × ⟨¯φ¯r1¯R1( ¯w1)...¯φ¯rN
+¯RN ( ¯wN)⟩ + (7.30b) × ⟨φr1
+R1(w1)...φrN
+RN (wN)⟩ = 0
+ϵ{(7.30a) × ⟨¯φ¯r1¯R1( ¯w1)...¯φ¯rN
+¯RN ( ¯wN)⟩ − (7.30b) × ⟨φr1
+R1(w1)...φrN
+RN (wN)⟩} = 0
+(7.33)
+We get (shown in details in Appendix F),
+�
+∂ti − 1
+k2
+�
+j̸=i
+��
+a(ta
+Ri,J ⊗ ta
+Rj,K + ta
+Ri,K ⊗ ta
+Rj,J)
+tij
++
+�xij
+t2
+ij
+− (k1 + 2Cg)
+k2tij
+� �
+a
+(ta
+Ri,K ⊗ ta
+Rj,K)
+��
+⟨...ΦRi, ¯Ri(xi, ti)...ΦRj, ¯
+Rj(xj, tj)...⟩ = 0
+�
+∂xi + 1
+k2
+�
+j̸=i
+�
+a(ta
+Ri,K ⊗ ta
+Rj,K)
+tij
+�
+⟨...ΦRi, ¯Ri(xi, ti)...ΦRj, ¯Rj(xj, tj)...⟩ = 0
+(7.34)
+where, ta
+Ri,J = ta
+Ri ⊗′ ¯I +I ⊗′ ¯ta¯Ri and ta
+Ri,J = ϵ(I ⊗′ ¯ta¯Ri −ta
+Ri ⊗′ ¯I) in limit ϵ → 0. The above
+equations are same as what we got form intrinsic analysis i.e. (6.3) and (6.5), with t → v
+and x → u as its supposed to for a Galilean contracted result.
+– 37 –
+
+8
+Conclusions
+8.1
+Summary
+In this paper we have explored aspects of Non-Lorentzian CFTs with additional Lie Al-
+gebraic symmetries. First we have reproduced the Non-Lorentzian Kaˇc-Moody algebra by
+taking singular limit from the Virasoro Kaˇc-moody algebra.
+After this we attempt to construct the same from intrinsic viewpoint without any knowledge
+of the parent algebra. We see that 2d Carrollian Conformal symmetry allows an infinite
+number of Noether currents. We then take a few pairs of conserved currents (introducing
+flavour indices to distinguish them) satisfying the conditions for EM tensor components in
+2d Carrollian Conformal symmetry. After this we derive the Ward identities associated with
+those conserved currents. We also derive OPEs of a general 2d Carrollian Conformal field
+with the current. After this we find the conserved charge operators associated with these
+currents. The transformation generated by these charges on a generic field is also derived.
+Here we first encounter the transformation matrices which we encounter in relativistic
+CFT with Kac-Moody algebra. We introduce current primary fields which turn out to
+be analogous to the Virasoro Kaˇc-Moody primary fields. After this the current current
+OPEs are derived. While doing so, the structure constants emerge from the action of the
+transformation matrices on the currents. After this, global internal symmetry is applied on
+two point and three point correlation and it turns out that the structure constants we have
+encountered while calculating the current current OPEs satisfy Jacobi identity. After this
+we derive the OPE between the Energy Momentum Tensor components and the current
+components. Using all these OPEs we derive the algebra of the current modes and the
+Energy Momentum tensor modes. The algebra transpire to be identical to the algebra
+obtained from limits of Virasoro Kaˇc-Moody algebra.
+Later in the paper, we attempt to construct the EM tensor modes (which forms the BMS
+algebra) from the current modes through Sugawara construction. Here we see that the
+Sugawara construction only takes us to the BMS algebra with one of the central charges
+to be zero. We needed another modification to the Sugawara construction in order to get
+a fully centrally extended BMS algebra. Using the expression of the EM tensor modes in
+terms of the current modes, we calculate the OPEs of EM tensor fields with themselves and
+with currents and see that the OPEs thus derived matches with the OPEs in the earlier
+section.
+After this we have a brief look at the tensionless string.
+When we look at the mode
+expansion of the coordinates (treating them as scalar fields), we see that the modes satisfy
+a special case of Non Lorentzian U(1) Kac-Moody algebra. We also look at the expression
+of modes of the classical energy momentum tensor in terms of these U(1) modes and see
+that classically this matches with the expression we have earlier derived for Non-Lorentzian
+Sugawara construction. Hence we see that U(1) NLKM currents are intrinsically present
+in the tensionless string.
+In section 6, we take a correlation function of a string of BMS current primary fields. Using
+the OPE definition of the BMS primary field, we arrive at the Non-Lorentzian version of the
+– 38 –
+
+Knizhnik Zamolodchikov equations. Finally, in section 7, we derive all the earlier results
+by taking limits from the parent algebra.
+8.2
+Discussions and future directions
+We mentioned in the introduction that our results in this paper lay the groundwork for a
+large number of applications, most importantly to the construction of a holographic dictio-
+nary for asymptotic flat spacetimes and also the understanding of tensionless strings. We
+have built the underlying algebraic structures in this paper which would be of importance
+to the quantum field theories that are at the heart of these problems.
+One of the most important and immediate next steps is to construct a Non-Lorentzian
+Wess-Zumino-Witten model that realises these symmetries. This would be central to the
+understanding of tensionless null strings moving on arbitrarily curved manifolds. The work
+on this is currently underway.
+The tensionless limit of string theory on arbitrary backgrounds is intimately related to this
+and as mentioned in the introduction, we wish to revisit the construction of [36] in the
+light of our findings in this paper. We believe that this limit would lead to null tensionless
+strings in AdS. This is to be contrasted with tensionless strings in AdS considered in e.g.
+[47] and subsequent work in this direction, which are tensionless but not null. A better
+understanding of the differences and perhaps similarities between the two approaches would
+be important.
+A generalisation of the methods outlined in this paper would be carried out for 3d Carrollian
+and 3d Galilean theories. The structures are expected to remain similar for the 3d Galilean
+theories, as the infinite dimensional structure of the algebra without the extra currents
+remains intact when generalised to higher dimensions. But for Carrollian theories, owing
+to the fundamental difference between BMS3 and BMS4, where one copy of the Virasoro
+in the 3d case gets enhanced to two Virasoros in the 4d case and supertranslations develop
+two legs instead of one, the construction of the quantum field theories with additional
+non-abelian currents would be more involved.
+Acknowledgements
+We thank Aritra Banerjee, Rudranil Basu and Niels Obers for interesting discussions and
+comments on an initial version of the manuscript.
+The work of AB is partially supported by a Swarnajayanti fellowship (SB/SJF/2019-
+20/08) from the Science and Engineering Research Board (SERB) India, the SERB grant
+(CRG/2020/002035), and a visiting professorship at ´Ecole Polytechnique Paris. AB also
+acknowledges the warm hospitality of the Niels Bohr Institute, Copenhagen during later
+stages of this work. RC is supported by the CSIR grant File No: 09/092(0991)/2018-EMR-
+I. AS is financially supported by a PMRF fellowship, MHRD, India. RK acknowledges the
+support of the Department of Atomic Energy, Government of India, under project number
+RTI4001. DS would like to thank ICTS, Bengaluru for hospitality during the course of this
+project.
+– 39 –
+
+APPENDICES
+A
+Carroll Multiplets
+In this appendix, we review the construction of Carrollian boost multiplets in two dimen-
+sions.
+In two space-time dimensions, the Carrollian boost (CB) transformation is defined as:
+x → x′ = x , t → t′ = t + vx ; or equivalently, as:
+�
+x
+t
+�
+−→
+�
+x′
+t′
+�
+=
+�
+exp
+��
+0 0
+v 0
+��� �
+x
+t
+�
+⇐⇒ xµ → x′µ =
+�
+evB(2)�µ
+ν xν
+(A.1)
+with
+B(2) :=
+�
+0 0
+1 0
+�
+(A.2)
+being the 2D representation of the CB generator B that is clearly not diagonalizable.
+Taking a cue from the Lorentz covariance of Lorentz tensors, it was postulated in [] that
+a rank-n Carrollian Cartesian tensor field Φ with ‘boost-charge’ ξ transforms under the
+Carrollian boost as:
+Φµ1...µn(t, x) −→ ˜Φµ1...µn(t′, x′) =
+�
+e−ξvB(2)
+�µ1
+ν1...
+�
+e−ξvB(2)
+�µn
+νnΦν1...νn(t, x)
+⇐⇒
+Φ(t, x) −→ ˜Φ(t′, x′) =
+� n
+�
+i=1
+e−ξvB(2)
+�
+Φ(t, x) = e
+−ξv
+n
+�
+i=1
+B(2)Φ(t, x)
+(A.3)
+where µi, νi are Carrollian space-time indices and for matrices, the left index denotes row
+while the right one denotes column; repeated indices are summed over and, in (A.3), indices
+are suppressed. It is to be noted that the up/down appearance of a tensor-index does not
+matter; only the left/right ordering is important.
+Clearly, the Carrollian Cartesian tensors defined above are decomposible.
+So, we now
+construct indecomposible Carrollian multiplets from these tensors. We begin by recognizing
+that:
+B(2) = J−
+(l= 1
+2 )
+(A.4)
+which is the lowering ladder operator in the SU(2) spin-1
+2 representation. Thus,
+n�
+i=1
+B(2) in
+(A.3) can be decomposed into indecomposable representations of J− using the technique
+of ‘addition of n spin-1
+2 angular momenta’ in quantum mechanics, such that:
+B(d) ≡ J−
+(l= d−1
+2 )
+(A.5)
+It is evident that the representations B(d) of the classical CB generator are indecomposable
+since their only generalized eigenvalue is 0 and it has geometric multiplicity 1. But, these
+representations are reducible for d ≥ 2.
+– 40 –
+
+A multi-component field transforming under the d-dimensional representation of CB, B(d),
+will be called a Carrollian multiplet of rank d−1
+2
+with d components, denoted by
+Φm
+(l= d−1
+2 )
+with
+m = 1 − d
+2
+, 3 − d
+2
+, ..., d − 1
+2
+By treating the µ = t index as spin-1
+2 up-state and the µ = x index as spin-1
+2 down-state,
+components Φm
+(l) of a Carrollian multiplet arise precisely as such linear combinations (with
+proper Clebsch-Gordon coefficients) of the components of a Cartesian tensor of an allowed
+rank n that would appear while expanding the |l, m⟩ states in an allowed |s1, s2, ..., sn⟩
+basis (where |si⟩ are Jz
+( 1
+2 ) eigenstates). So, as a linear combination of the components of
+a rank-n Cartesian tensor, one can obtain multipltes of ranks: 0, 1, 2, ..., n
+2 for even n and
+1
+2, 3
+2, 5
+2, ..., n
+2 for odd n. As an example, we see how Carrollian multiplets of ranks 1
+2 and 3
+2
+are constructed from a rank-3 Cartesian tensor:
+Φ
+3
+2
+( 3
+2 )(t, x) := Φttt(t, x)
+Φ
+1
+2
+( 3
+2 )(t, x) :=
+1
+√
+3
+�
+Φttx + Φtxt + Φxtt�
+(t, x)
+Φ
+− 1
+2
+( 3
+2 )(t, x) :=
+1
+√
+3
+�
+Φtxx + Φxtx + Φxxt�
+(t, x)
+Φ
+− 3
+2
+( 3
+2 )(t, x) := Φxxx(t, x)
+Φ
+1
+2
+( 1
+2 )(t, x) :=
+1
+√
+a2 + b2 + c2
+�
+aΦttx + bΦtxt + cΦxtt�
+(t, x)
+with a + b + c = 0
+Φ
+− 1
+2
+( 1
+2 )(t, x) :=
+1
+√
+a2 + b2 + c2
+�
+aΦxxt + bΦxtx + cΦtxx�
+(t, x)
+(As the tuple (a, b, c) in R3 lies on the plane a + b + c = 0 which is spanned by two basis
+vectors, two linearly independent rank-1
+2 multiplets arise.)
+A rank-l Carrollian multiplet with boost-charge ξ thus transforms under the 2l + 1 dimen-
+sional representation of the CB as:
+Φm
+(l)(t, x) −→ ˜Φm
+(l)(t′, x′) =
+�
+e−ξvJ−
+(l)
+�m
+m′Φm′
+(l)(t, x)
+(A.6)
+After constructing the Carrollian multiplets from the Cartesian tensors as demonstrated
+above, the components of the multiplets can always be redefined such that:
+in (A.6), J−
+(l) is replaced by M(l) := sub-diag (1, 1, ..., 1)2l+1 .
+Hence, instead of the actual J−
+(l) matrix, only the indecomposable Jordan-block structure
+is important for defining the CB transformation property of the Carrollian multiplets.
+We conclude this appendix with the following observation. Since the finite dimensional
+indecomposable representations of B are not symmetric (or Hermitian), one can start
+with:
+�
+t
+x
+�
+−→
+�
+t′
+x′
+�
+=
+�
+exp
+��
+0 v
+0 0
+��� �
+t
+x
+�
+⇐⇒ xµ → x′µ =
+�
+evB′
+(2)
+�µ
+ν xν
+– 41 –
+
+where
+B′
+(2) :=
+�
+0 1
+0 0
+�
+and follow the preceding argument to construct
+B′
+(d) ≡ J+
+(l= d−1
+2 )
+which is the SU(2) raising ladder operator. But, as J+
+(l) = (J−
+(l))T, the raising and lowering
+operators’ representation matrices are related to each other by the similarity transforma-
+tion:
+S = anti-diag (1, 1, ..., 1)2l+1
+and consequently, B and B′ furnish two equivalent representations of the CB generator.
+B
+Calculation of Sugawara Construction Commutators
+In this appendix, we provide the details of the calculation of commutators of the NLKM
+algebra from the Sugawara construction.
+Calculating [Mm, Jb
+n]
+[Mm, Jb
+n] =
+1
+2k2
+�
+a
+�
+l
+[Ka
+l Ka
+m−l, Jb
+n]
+(Using (4.12))
+=
+1
+2k2
+�
+a
+�
+l
+�
+[Ka
+l , Jb
+n]Ka
+m−l + Ka
+l [Ka
+m−l, Jb
+n]
+�
+=
+1
+2k2
+�
+a
+�
+l
+�
+(−i
+�
+c
+fbacKc
+n+l − nk2δn+l,0δab)Ka
+m−l
++ Ka
+l (−i
+�
+c
+fbacKc
+m+n−l − nk2δm+n−l,0δab)
+�
+(Using (2.21))
+=
+i
+2k2
+�
+a,c
+�
+l
+fabc�
+Kc
+n+lKa
+m−l + Ka
+l Kc
+m+n−l
+�
+− nKb
+m+n
+(∵ −fbac = fabc)
+where we have omitted the limits in summation over a and c and it is understood that
+it runs over 1 to dim(g). Now, �
+l Kc
+n+lKa
+m−l = �
+l Kc
+m+n−lKa
+l simply by translating l.
+Hence,
+[Mm, Jb
+n] = i
+k2
+�
+a,c
+�
+l
+fabcKa
+l Kc
+m+n−l − nKb
+m+n
+(B.1)
+Now, again using antisymmetry property of fabc
+�
+a,c
+�
+l
+fabcKa
+l Kc
+m+n−l =
+�
+a,c
+�
+l
+fabcKa
+m+n−lKc
+l
+– 42 –
+
+=
+�
+a,c
+�
+l
+fcbaKc
+m+n−lKa
+l
+= −
+�
+a,c
+�
+l
+fabcKa
+l Kc
+m+n−l
+⇒
+�
+a,c
+�
+l
+fabcKa
+l Kc
+m+n−l = 0
+(B.2)
+Hence, (B.1) can be written as,
+[Mm, Jb
+n] = −nKb
+m+n
+(B.3)
+Calculating [Lm, Kb
+n]
+Again, using (4.12) and (2.21), we can get the following,
+[Lm, Kb
+n] =
+1
+2k2
+�
+a
+� �
+l≤−1
+([Ja
+l , Kb
+n]Ka
+m−l + Ka
+l [Ja
+m−l, Kb
+n])
++
+�
+l>−1
+([Ja
+m−l, Kb
+n]Ka
+l + Ka
+m−l[Ja
+l , Kb
+n])
+�
+=
+1
+2k2
+�
+a
+� �
+l≤−1
+�
+i
+�
+c
+fabc(Kc
+l+nKa
+m−l + Ka
+l Kc
+m+n−l) + k2lKa
+m−lδl+n,0δab
++ k2(m − l)Ka
+l δm+n−l,0δab
+�
++
+�
+l>−1
+�
+i
+�
+c
+fabc(Kc
+m+n−lKa
+l + Ka
+m−lKc
+l+n)
++ k2(m − l)Ka
+l δm+n−l,0δab + k2lKa
+m−lδl+n,0δab
+��
+= i
+k2
+�
+a,c
+�
+l
+fabc(Kc
+m+n−lKa
+l + Ka
+m−lKc
+l+n)
++
+1
+2k2
+�
+a
+�
+l
+�
+k2(m − l)Ka
+l δm+n−l,0δab + k2lKa
+m−lδl+n,0δab
+�
+(B.4)
+First term vanishes again because of the antisymmetry of f and commutativity of K’s.
+Therefore,
+[Lm, Kb
+n] = −nKb
+m+n
+(B.5)
+Calculating [Lm, Jb
+n]
+Following similar steps as before, we get,
+[Lm, Jb
+n] = 1
+2k2
+�
+a
+� �
+l≤−1
+([Ja
+l Ka
+m−l, Jb
+n] + [Ka
+l Ja
+m−l, Jb
+n]) +
+�
+l>−1
+([Ja
+m−lKa
+l , Jb
+n]
++ [Ka
+m−lJa
+l , Jb
+n])
+�
+− (k1 + 2Cg)
+k2
+[Mm, Jb
+n]
+(B.6)
+– 43 –
+
+First term can be simplified as,
+�
+a
+�
+l≤−1
+([Ja
+l Ka
+m−l, Jb
+n] + [Ka
+l Ja
+m−l, Jb
+n])
+=
+�
+a
+�
+l≤−1
+�
+Ja
+l [Ka
+m−l, Jb
+n] + [Ja
+l , Jb
+n]Ka
+m−l + Ka
+l [Ja
+m−l, Jb
+n] + [Ka
+l , Jb
+n]Ja
+m−l
+�
+=
+�
+a
+�
+l≤−1
+�
+i
+�
+c
+fabcJa
+l Kc
+m+n−l + i
+�
+c
+fabcJc
+l+nKa
+m−l + i
+�
+c
+fabcKa
+l Jc
+m+n−l
++ i
+�
+c
+fabcKc
+l+nJa
+m−l − k2nJa
+l δm+n−l,0δab + k1lKa
+m−lδl+n,0δab
++ k1(m − l)Ka
+l δm+n−l,0δab − k2nJa
+m−lδl+n,0δab
+�
+=
+�
+l≤−1
+�
+i
+�
+a,c
+fabcJa
+l Kc
+m+n−l + i
+�
+a,c
+fabcJc
+l+nKa
+m−l + i
+�
+a,c
+fabcKa
+l Jc
+m+n−l
++ i
+�
+a,c
+fabcKc
+l+nJa
+m−l
+�
+− n
+�
+l≤−1
+(k2Jb
+m+n + k1Kb
+m+n)(δm+n−l,0 + δl+n,0) (B.7)
+Similarly, the second term looks like,
+�
+a
+�
+l>−1
+([Ka
+m−lJa
+l , Jb
+n] + [Ja
+m−lKa
+l , Jb
+n])
+=
+�
+l>−1
+�
+i
+�
+a,c
+fabcKa
+m−lJc
+l+n + i
+�
+a,c
+fabcKc
+m+n−lJa
+l + i
+�
+a,c
+fabcJa
+m−lKc
+l+n
++ i
+�
+a,c
+fabcJc
+m+n−lKa
+l
+�
+− n
+�
+l>−1
+(k2Jb
+m+n + k1Kb
+m+n)(δm+n−l,0 + δl+n,0) (B.8)
+Hence, we have,
+[Lm, Jb
+n] =
+1
+2k2
+�
+i
+�
+a,c
+fabc
+�
+�
+0≤l≤n−1
+Jc
+l Ka
+m+n−l −
+�
+0≤l≤n−1
+Ka
+l Jc
+m+n−l −
+�
+0≤l≤n−1
+Ka
+m+n−lJc
+l
++
+�
+0≤l≤n−1
+Jc
+m+n−lKa
+l
+��
+− nJb
+m+n + 2Cg
+k2
+nKb
+m+n
+=
+1
+2k2
+�
+i
+�
+a,c
+fabc
+�
+�
+0≤l≤n−1
+[Jc
+l , Ka
+m+n−l] +
+�
+0≤l≤n−1
+[Jc
+m+n−l, Ka
+l ]
+��
+− nJb
+m+n + 2Cg
+k2
+nKb
+m+n
+=
+1
+2k2
+�
+− 2n
+�
+a,c,d
+fabcfcadKd
+m+n + 2ik2
+�
+a,c
+�
+0≤l≤n−1
+fabcδm+n,0δab
+�
+− nJb
+m+n + 2Cg
+k2
+nKb
+m+n
+= −nJb
+m+n + n
+k2
+(2CgKb
+m+n −
+�
+a,c,d
+fabcfcadKd
+m+n)
+– 44 –
+
+= −nJb
+m+n
+Hence, we have,
+[Lm, Jb
+n] = −nJb
+m+n
+(B.9)
+Calculating [Lm, Mn]
+[Lm, Mn] =
+1
+2k2
+�
+a
+�
+l
+�
+[Lm, Ka
+l ]Ka
+n−l + Ka
+l [Lm, Ka
+n−l]
+�
+=
+1
+2k2
+�
+a
+�
+l
+�
+− lKa
+m+lKa
+n−l − (n − l)Ka
+l Ka
+m+n−l
+�
+(Using (B.5))
+=
+1
+2k2
+�
+a
+�
+l
+{−(l − m)Ka
+l Ka
+n+m−l − (n − l)Ka
+l Ka
+m+n−l}
+= (m − n) 1
+2k2
+�
+a
+�
+l
+Ka
+l Ka
+n+m−l
+where , we have done the re-labelling l → l − m in the second step. Hence, we get,
+[Lm, Mn] = (m − n)Mm+n
+(B.10)
+Calculating [Lm, Ln]
+It can be carried out in similar fashion by writing one of the L’s in terms of J’s and K’s
+using (4.12) and then using (B.5), (B.9) and (B.10), we can do the following,
+[Lm, Ln] = 1
+2k2
+�
+a
+� �
+l≤−1
+([Lm, Ja
+l ]Ka
+n−l + Ja
+l [Lm, Ka
+n−l] + [Lm, Ka
+l ]Ja
+n−l + Ka
+l [Lm, Ja
+n−l])
++
+�
+l>−1
+([Lm, Ja
+n−l]Ka
+l + Ja
+n−l[Lm, Ka
+l ] + [Lm, Ka
+n−l]Ja
+l + Ka
+n−l[Lm, Ja
+l ])
+�
+− (k1 + 2Cg)
+k2
+[Lm, 1
+2k2
+�
+l
+Ka
+l Ka
+n−l]
+=
+1
+2k2
+�
+a
+� �
+l≤−1
+(−lJa
+m+lKa
+n−l + lJa
+l Ka
+m+n−l − lKa
+m+lJa
+n−l + lKa
+l Ja
+m+n−l)
++
+�
+l>−1
+(lJa
+m+n−lKa
+l − lJa
+n−lKa
+m+l + lKa
+m+n−lJa
+l − lKa
+n−lJa
+m+l)
+�
+− m(k1 + 2Cg)
+k2
+1
+2k2
+�
+a
+�
+l
+Ka
+l Ka
+m+n−l − nLm+n
+(B.11)
+Changing the index l → (l − m) in the negative terms in the curly brackets and then
+simplifying, we can get,
+[Lm, Ln] =
+1
+2k2
+�
+a
+� �
+l≤−1
+m(Ja
+l Ka
+n+m−l + Ka
+l Ja
+m+n−l) +
+�
+l>−1
+m(Ja
+n+m−lKa
+l + Ka
+n+m−lJa
+l )
+– 45 –
+
++
+m−1
+�
+l=0
+(m − l)([Ja
+l , Ka
+n+m−l] − [Ja
+n+m−l, Ka
+l ])
+�
+− m(k1 + 2Cg)
+k2
+1
+2k2
+�
+a
+�
+l
+Ka
+l Ka
+m+n−l − nLm+n
+=
+1
+2k2
+�
+a
+� m−1
+�
+l=0
+(m − l)(k2lδm+n,0 − k2(n + m − l)δm+n,0)
+�
++ (m − n)Lm+n
+(B.12)
+which upon further simplification gives,
+[Lm, Ln] = (m − n)Lm+n + dim(g)
+6
+m(m2 − 1)δm+n,0
+(B.13)
+Other commutation relations of type [Mm, Kb
+n] and [Mm, Mn] vanish trivially because of
+the vanishing [Ka
+m, Kb
+n] commutator.
+C
+Modified Sugawara Construction
+In this appendix, we give details of the modified Sugawara construction. First we start
+with the modified construction in the relativistic case and then explain the construction in
+the non-Lorentzian case.
+We begin by calculating [ ˜Lm, ˜Ln] with ˜Lm given in (4.15)
+[ ˜Lm, ˜Ln] = [LS
+m + imθaja
+m + 1
+2kθ2δm,0, LS
+n + inθbjb
+n + 1
+2kθ2δn,0]
+= [LS
+m, LS
+n] + inθb[LS
+m, jb
+n] + imθa[ja
+m, LS
+n] − mnθaθb[ja
+m, jb
+n]
+= (m − n)LS
+m+n − in2θbjb
+m+n + im2θaja
+m+n − mnθaθb�
+ifabcjc
+m+n + mkδm+nδab
+�
+= (m − n) ˜Lm+n − i(m2 − n2)θaja
+m+n − 1
+2kθ2(m − n)δm+n,0 + i(m2 − n2)θaja
+m+n
+− imnθaθbfabcjc
+m+n + m3kθ2δm+n,0 + c
+12(m3 − m)δm+n,0
+= (m − n) ˜Lm+n +
+� c
+12 + kθ2�
+(m3 − m)δm+n,0
+= (m − n) ˜Lm+n + ˜c
+12(m3 − m)δm+n,0
+(C.1)
+Here ˜c = c + 12kθ2. In the fifth line we have used the fact that θaθbfabc vanishes due to
+antisymmetry of fabc over indices a and b, also the fact that nδm+n,0 = −mδm+n,0.
+Now defining ˜L and ˜
+M as in (4.17) we would like to calculate [˜Lm, ˜Ln] and [˜Lm, ˜
+Mn]. The
+calculation of [˜Lm, ˜Ln] will be exactly similar to that of [ ˜Lm, ˜Ln], while that of [˜Lm, ˜
+Mn]
+will be as following
+[˜Lm, ˜
+Mn] = [LS
+m + imθaJa
+m + 1
+2k2θ2δn,0, MS
+n + inθbKb
+n + 1
+2k2θ2δn,0]
+= [LS
+m, MS
+n ] + imθa[LS
+m, Ka
+n] + inθb[Ja
+m, MS
+n ] − mnθaθb[Jb
+m, Ka
+n]
+– 46 –
+
+= (m − n)MS
+m+n − in2θbKb
+m+n + im2θaKa
+m+n − mnθaθb�
+ifabcKc
+m+n + mk2δm+nδab
+�
+= (m − n)MS
+m+n − i(m2 − n2)θaKa
+m+n − 1
+2k2θ2(m − n)δm+n,0 + i(m2 − n2)θaKa
+m+n
+− imnθaθbfabcKc
+m+n + m3k2θ2δm+n,0
+= (m − n)Mm+n + k2θ2(m3 − m)δm+n,0
+(C.2)
+Hence, just by doing a slight modification to the Sugawara construction we end up with
+fully centrally extended BMS algebra with central charges given in (4.19).
+D
+Details of OPE calculations
+Calculation of T-J OPE
+First consider
+J a
+v (u1, v1)
+�
+b
+�
+(J b
+uJ b
+v )(u2, v2)
+�
+=
+�
+v2
+dv′
+v′ − v2
+�
+u2
+du′
+u′ − u2
+�
+b
+�
+J a
+v (u1, v1)J b
+u(u′, v′)J b
+v (u2, v2) + J b
+u(u′, v′)J a
+v (u1, v1)J b
+v (u2, v2)
+�
+=
+�
+v2
+dv′
+v′ − v2
+�
+u2
+du′
+u′ − u2
+�
+b
+�
+ifabc J c
+v (u′, v′)J b
+v (u2, v2)
+(v1 − v′)
++ k2δab J b
+v (u2, v2)
+(v1 − v′)2 + J b
+u(u′, v′) × 0
+�
+=
+�
+v2
+dv′
+v′ − v2
+�
+u2
+du′
+u′ − u2
+��
+b
+ifabc (J c
+v J b
+v )(u2, v2)
+(v1 − v′)
++ k2
+J a
+v (u2, v2)
+(v1 − v′)2
+�
+=
+�
+b
+ifabc (J c
+v J b
+v )(u2, v2)
+v12
++ k2
+J a
+v (u2, v2)
+v2
+12
+(D.1)
+Similarly we get
+J a
+v (u1, v1)
+�
+b
+�
+(J b
+v J b
+u)(u2, v2)
+�
+=
+�
+v2
+dv′
+v′ − v2
+�
+u2
+du′
+u′ − u2
+��
+b
+ifabc (J b
+v J c
+v )(u2, v2)
+v12
++ k2
+J a
+v (u′, v′)
+v2
+12
+�
+=
+�
+b
+iF abc (J b
+v J c
+v )(u2, v2)
+v12
++ k2
+J a
+v (u2, v2)
+v2
+12
+(D.2)
+Summing up (3.66) and (D.2), we see that the first terms in the expressions cancel each
+other due to the antisymmetry of the structure constant, so we get
+J a
+v (u1, v1)
+�
+b
+�
+(J b
+uJ b
+v )(u2, v2) + (J b
+v J b
+u)(u2, v2)
+�
+= 2k2
+J a
+v (u2, v2)
+v2
+12
+= 2k2
+�J a
+v (u1, v1)
+v2
+21
++ ∂vJ a
+v (u1, v1)
+v21
+�
+(D.3)
+– 47 –
+
+Also it can be trivially shown that
+J a
+v (u1, v1)
+�
+b
+�
+(J a
+v J a
+v )(u2, v2)
+�
+= 0
+(D.4)
+From (D.3) and (D.4), we can determine
+Tv(u1, v1)J b
+v (u2, v2) ∼ regular
+Tu(u1, v1)J b
+v (u2, v2) ∼ J b
+v (u2, v2)
+v2
+12
++ ∂vJ b
+v (u2, v2)
+v12
++ . . .
+(D.5)
+Similarly we get
+Tv(u1, v1)J a
+u (u2, v2) ∼ J a
+v (u2, v2)
+v2
+12
++ ∂vJ a
+v (u2, v2)
+v12
++ . . .
+Tu(u1, v1)J a
+u (u2, v2) ∼ J a
+u (u2, v2)
+v2
+12
++ ∂vJ a
+u (u2, v2)
+v12
++ u12
+v2
+12
+∂uJ a
+u (u2, v2)
++ 2u12
+v12
+�J a
+v (u1, v1)
+v2
+12
++ ∂vJ a
+v (u2, v2)
+v2
+12
+�
++ . . .
+(D.6)
+Calculation of T-T OPE
+First consider
+Tu(u1, v1)(J a
+v J a
+v )(u2, v2)
+=
+�
+v2
+dv′
+v′ − v2
+�
+u2
+du′
+u′ − u2
+�
+Tu(u1, v1)J a
+v (u, v)J a
+v (u2, v2) + J a
+v (u, v)Tu(u1, v1)v2, u2
+�
+=
+�
+v2
+dv′
+v′ − v2
+�
+u2
+du′
+u′ − u2
+�J a
+v (u, v)J a
+v (u2, v2)
+(v1 − v′)2
++ ∂v′J a
+v (u, v)J a
+v (u2, v2)
+(v1 − v′)
++ J a
+v (u, v)J a
+v (u2, v2)
+v2
+12
++ J a
+v (u, v)∂v2J a
+v (u2, v2)
+v12
+�
+= 2(J a
+v J a
+v )(u2, v2)
+v2
+12
++
+�
+v2
+dv′
+v′ − v2
+�
+u2
+du′
+u′ − u2
+�∂v′[(J a
+v J a
+v )(u2, v2) + (∂vJ a
+v J a
+v )(u2, v2) + . . . ]
+(v1 − v′)
++ ∂v2[(J a
+v J a
+v )(u2, v2) + (∂vJ a
+v J a
+v )(u2, v2) + . . . ]
+v12
+�
+= 2(J a
+v J a
+v )(u2, v2)
+v2
+12
++ ∂v(J a
+v J a
+v )(u2, v2)
+v12
+(D.7)
+So we obtain
+Tu(u1, v1)Tv(u2, v2) ∼ 2(J a
+v J a
+v )(u2, v2)
+v2
+12
++ ∂v(J a
+v J a
+v )(u2, v2)
+v12
++ . . .
+(D.8)
+Similarly, we can check
+Tv(u1, v1)Tv(u2, v2) ∼ regular,
+(D.9)
+– 48 –
+
+and
+Tu(u1, v1)Tu(u2, v2) ∼ 2Tu(u2, v2)
+v2
+12
++∂vTu(u2, v2)
+v12
++ u12
+v2
+12
+∂uTu(u2, v2)
++ 2u12
+v12
+�
+2Tv(u2, v2)
+v2
+12
++ ∂vTv(u2, v2)
+v12
+�
++ dim(g)
+v4
+12
++ . . .
+(D.10)
+E
+K-Z equation in field theory approach
+We start with
+⟨∂uΦ(u, v)Φ1(u1, v1) . . . Φn(un, vn)⟩ = ⟨∂uΦ(u, v)X⟩
+= −
+�
+v
+dv′
+2πi⟨Tv(u′, v′)Φ(u, v)X⟩
+= −
+�
+v
+dv′
+2πi
+1
+2k2
+⟨(J a
+v J a
+v )(u′, v′)Φ(u, v)X⟩
+= −
+�
+v
+dv′
+2πi
+1
+2k2
+�
+v′
+dv′′
+2πi
+1
+(v′′ − v′)⟨
+�
+J a
+v (u′′, v′′)Φ(u, v)J a
+v (u′, v′) + J a
+v (u′′, v′′)J a
+v (u′, v′)Φ(u, v)
+�
+X⟩
+−
+�
+v
+dv′
+2πi
+1
+2k2
+⟨(J a
+v J a
+v Φ)(u, v)X⟩ + . . .
+= −
+�
+v
+dv′
+2πi
+1
+2k2
+�
+v′
+dv′′
+2πi
+1
+(v′′ − v′)⟨
+�
+ta
+k
+(v′′ − v)Φ(u, v)J a
+v (u′, v′) +
+ta
+K
+(v′ − v)J a
+v (u′′, v′′)Φ(u, v)
+�
+X⟩ + 0
+= I1 + I2
+(E.1)
+(Where, second line is (6.1), third line is the Non-Lorentzian Sugawara construction, fourth
+line uses definition of normal ordering and the fact that OPE = contractions + normal
+ordered product + . . . )
+Now,
+I1 = − ta
+k
+2k2
+�
+v
+dv′
+2πi
+�
+v′
+dv′′
+2πi
+1
+(v′′ − v′)(v′′ − v)⟨Φ(u, v)J a
+v (u′, v′)X⟩
+= − ta
+k
+2k2
+�
+v
+dv′
+2πi
+1
+(v′ − v)⟨Φ(u, v)J a
+v (u′, v′)X⟩
+= ta
+k
+2k2
+�
+j
+�
+vj
+dv′
+2πi
+1
+(v′ − v)⟨Φ(u, v)Φ1(u1, v1)...
+�
+J a
+v (u′, v′)Φj(uj, vj)
+�
+. . . Φn(un, vn)⟩
+=
+1
+2k2
+�
+j
+�
+vj
+dv′
+2πi
+ta
+K ⊗ ta
+K,j
+(v′ − v)(v′ − vj)⟨Φ(u, v)Φ1(u1, v1)...Φj(uj, vj) . . . Φn(un, vn)⟩
+=
+1
+2k2
+�
+j
+ta
+K ⊗ ta
+K,j
+(vj − v) ⟨Φ(u, v)Φ1(u1, v1)...Φj(uj, vj) . . . Φn(un, vn)⟩
+(E.2)
+– 49 –
+
+(third line involves a change in contour)
+and similarly,
+I2 = − ta
+k
+2k2
+�
+v
+dv′
+2πi
+�
+v′
+dv′′
+2πi
+1
+(v′′ − v′)(v′ − v)⟨J a
+v (v′′, x′′)Φ(u, v)X⟩
+=
+1
+2k2
+�
+j
+ta
+K ⊗ ta
+K,j
+(vj − v) ⟨Φ(u, v)Φ1(u1, v1)...Φj(uj, vj) . . . Φn(un, vn)⟩
+= I1
+(E.3)
+Using the above two results,we get one of the Carrollian K-Z equations
+�
+∂ui + 1
+k2
+�
+j̸=i
+ta
+Ri,K ⊗ ta
+Rj,K
+vij
+�
+⟨Φ1(u1, v1) . . . Φn(un, vn)⟩ = 0
+(E.4)
+The other equation can be obtained similarly.
+F
+NL KZ equation as a limit
+Taking the following linear combination of the equations (7.30a) and (7.30b),
+(7.30a) × ⟨¯φ¯r1¯R1( ¯w1)...¯φ¯rN
+¯RN ( ¯wN)⟩ + ((7.30b)) × ⟨φr1
+R1(w1)...φrN
+RN (wN)⟩ = 0
+(F.1)
+gives us,
+�
+�∂wi + ∂ ¯wi − 2γ
+�
+j̸=i
+�
+a(ta
+Ri ⊗′ ¯I)ri,¯ri
+si,¯si(ta
+Rj ⊗′ ¯I)rj,¯rj
+sj,¯sj
+wi − wj
+− 2¯γ
+�
+j̸=i
+�
+a(I ⊗′ ¯ta¯Ri)ri,¯ri
+si,¯si(I ⊗′ ¯ta¯Rj)rj,¯rj
+sj,¯sj
+¯wi − ¯wj
+�
+�
+⟨...Φsi,¯si
+Ri, ¯Ri(wi, ¯wi)...Φsj,¯sj
+Rj, ¯Rj(wj, ¯wj)...⟩ = 0
+(F.2)
+We have (wi, ¯wi) = t ± ϵx ⇒ ∂wi = 1
+2(∂ti + ∂xi
+ϵ ); ∂ ¯wi = 1
+2(∂ti − ∂xi
+ϵ ) such that the Carrollian
+limit is achieved by taking ϵ → 0. Using these we get,
+⇒
+�
+�∂ti − 2
+�
+j̸=i
+X
+�
+� ⟨...Φsi,¯si
+Ri, ¯Ri(xi, ti)...Φsj,¯sj
+Rj, ¯Rj(xj, tj)...⟩ = 0
+(F.3)
+where(with tij = ti − tj and xij = xi − xj), we have,
+X =
+�
+γ
+tij + ϵxij
+�
+a
+(ta
+Ri ⊗′ ¯I)ri,¯ri
+si,¯si(ta
+Rj ⊗′ ¯I)rj,¯rj
+sj,¯sj +
+¯γ
+tij − ϵxij
+�
+a
+(I ⊗′ ¯ta¯Ri)ri,¯ri
+si,¯si(I ⊗′ ¯ta¯Rj)rj,¯rj
+sj,¯sj
+�
+(F.4)
+Inverting relations (1.11), we have,
+ta
+Ri ⊗′ ¯I = 1
+2(ta
+Ri,J −
+ta
+Ri,K
+ϵ
+) ; I ⊗′ ¯ta¯Ri = 1
+2(ta
+Ri,J +
+ta
+Ri,K
+ϵ
+)
+(F.5)
+– 50 –
+
+Hence,
+X = 1
+4
+�
+a
+�
+γ
+tij + ϵxij
+�
+ta
+Ri,J −
+ta
+Ri,K
+ϵ
+�ri,¯ri
+si,¯si
+�
+ta
+Rj,J −
+ta
+Rj,K
+ϵ
+�rj,¯rj
+sj,¯sj
++
+¯γ
+tij − ϵxij
+�
+ta
+Ri,J +
+ta
+Ri,K
+ϵ
+�ri,¯ri
+si,¯si
+�
+ta
+Rj,J +
+ta
+Rj,K
+ϵ
+�rj,¯rj
+sj,¯sj
+�
+= 1
+4
+�
+a
+�
+γ
+tij + ϵxij
+�
+ta
+Ri,J ⊗ ta
+Rj,J − 1
+ϵ (ta
+Ri,J ⊗ ta
+Rj,K + ta
+Ri,K ⊗ ta
+Rj,J)
++ 1
+ϵ2 ta
+Ri,K ⊗ ta
+Rj,K
+�ri,¯ri;rj,¯rj
+si,¯si;sj,¯sj
++
+¯γ
+tij − ϵxij
+�
+ta
+Ri,J ⊗ ta
+Rj,J + 1
+ϵ (ta
+Ri,J ⊗ ta
+Rj,K + ta
+Ri,K ⊗ ta
+Rj,J)
++ 1
+ϵ2 ta
+Ri,K ⊗ ta
+Rj,K
+�ri,¯ri;rj,¯rj
+si,¯si;sj,¯sj
+�
+(F.6)
+We introduce notation for convenience,
+�
+a
+(ta
+Ri,A ⊗ ta
+Rj,B)ri,¯ri;rj,¯rj
+si,¯si;sj,¯sj = tij
+AB (where A, B = J, K)
+(F.7)
+Therefore,
+X =1
+4
+� γ
+tij
+(1 − ϵxij
+tij
++ ϵ2 x2
+ij
+t2
+ij
++ ...)(tij
+JJ − 1
+ϵ (tij
+JK + tij
+KJ) + 1
+ϵ2 tij
+KK)
++ ¯γ
+tij
+(1 + ϵxij
+tij
++ ϵ2 x2
+ij
+t2
+ij
++ ...)(tij
+JJ + 1
+ϵ (tij
+JK + tij
+KJ) + 1
+ϵ2 tij
+KK)
+�
+⇒ X =
+1
+4tij
+�
+− (γ − ¯γ)
+ϵ
+(tij
+JK + tij
+KJ) + γ + ¯γ
+ϵ2
+tij
+KK) − xij
+4t2
+ij
+(γ − ¯γ)
+ϵ
+tij
+KK
+�
+(F.8)
+In limit ϵ → 0 and using (7.25), we get,
+⇒ X =
+1
+4tij
+� 2
+k2
+(tij
+JK + tij
+KJ) − 2(k1 + 2Cg)
+k2
+2
+tij
+KK) + xij
+4t2
+ij
+2
+k2
+tij
+KK
+�
+⇒ X =
+1
+2k2
+�(tij
+JK + tij
+KJ)
+tij
++ (xij
+t2
+ij
+− (k1 + 2Cg)
+k2tij
+)tij
+KK
+�
+(F.9)
+Therefore, (F.1) can be finally written as( in limit ϵ → 0),
+⇒
+�
+∂ti −
+�
+j̸=i
+1
+k2
+�(tij
+JK + tij
+KJ)
+tij
++
+�xij
+t2
+ij
+− (k1 + 2Cg)
+k2tij
+�
+tij
+KK
+��
+⟨...Φsi,¯si
+xi,ti(xi, ti)...Φsj,¯sj
+Rj, ¯Rj(xj, tj)...⟩ = 0
+⇒
+�
+∂ti − 1
+k2
+�
+j̸=i
+��
+a(ta
+Ri,J ⊗ ta
+Rj,K + ta
+Ri,K ⊗ ta
+Rj,J)ri,¯ri;rj,¯rj
+si,¯si;sj,¯sj
+tij
+– 51 –
+
++
+�xij
+t2
+ij
+− (k1 + 2Cg)
+k2tij
+� �
+a
+(ta
+Ri,K ⊗ ta
+Rj,K)ri,¯ri;rj,¯rj
+si,¯si;sj,¯sj
+��
+⟨...Φsi,¯si
+Ri, ¯Ri(xi, ti)...Φsj,¯sj
+Rj, ¯Rj(xj, tj)...⟩ = 0
+⇒
+�
+∂ti − 1
+k2
+�
+j̸=i
+��
+a(ta
+Ri,J ⊗ ta
+Rj,K + ta
+Ri,K ⊗ ta
+Rj,J)
+tij
++
+�xij
+t2
+ij
+− (k1 + 2Cg)
+k2tij
+� �
+a
+(ta
+Ri,K ⊗ ta
+Rj,K)
+��
+⟨...ΦRi, ¯Ri(xi, ti)...ΦRj, ¯
+Rj(xj, tj)...⟩ = 0
+(F.10)
+which is in agreement with (6.3).
+Now,we consider another linear combination,
+ϵ{(7.30a) × ⟨¯φ¯r1¯R1( ¯w1)...¯φ¯rN
+¯RN ( ¯wN)⟩ − (7.30b) × ⟨φr1
+R1(w1)...φrN
+RN (wN)⟩} = 0
+(F.11)
+Again following similar steps as before,
+⇒
+�
+∂xi − 2
+�
+j̸=i
+Y
+�
+⟨...Φsi,¯si
+Ri, ¯Ri(xi, ti)...Φsj,¯sj
+Rj, ¯Rj(xj, tj)...⟩ = 0
+(F.12)
+where,
+Y = ϵ
+�
+γ
+tij + ϵxij
+�
+a
+(ta
+Ri ⊗′ ¯I)ri,¯ri
+si,¯si(ta
+Rj ⊗′ ¯I)rj,¯rj
+sj,¯sj −
+¯γ
+tij − ϵxij
+�
+a
+(I ⊗′ ¯ta¯Ri)ri,¯ri
+si,¯si(I ⊗′ ¯ta¯Rj)rj,¯rj
+sj,¯sj
+�
+(F.13)
+Similar to what we did for X, we get an equation analogous to (F.8) using (F.5) and (F.7),
+Y = ϵ
+4{ γ
+tij
+(1 − ϵxij
+tij
++ ϵ2 x2
+ij
+t2
+ij
++ ...)(tij
+JJ − 1
+ϵ (tij
+JK + tij
+KJ) + 1
+ϵ2 tij
+KK)
+− ¯γ
+tij
+(1 + ϵxij
+tij
++ ϵ2 x2
+ij
+t2
+ij
++ ...)(tij
+JJ + 1
+ϵ (tij
+JK + tij
+KJ) + 1
+ϵ2 tij
+KK)}
+(F.14)
+Again using (7.25) and collecting the finite terms in the limit ϵ → 0, we get,
+Y = − 1
+2k2
+tij
+KK
+tij
+(F.15)
+Putting back in (F.12),
+�
+�∂xi + 1
+k2
+�
+j̸=i
+�
+a(ta
+Ri,K ⊗ ta
+Rj,K)
+tij
+�
+� ⟨...ΦRi, ¯Ri(xi, ti)...ΦRj, ¯Rj(xj, tj)...⟩ = 0
+(F.16)
+which is in agreement with (6.5).
+– 52 –
+
+References
+[1] A. A. Belavin, A. M. Polyakov and A. B. Zamolodchikov, Infinite Conformal Symmetry in
+Two-Dimensional Quantum Field Theory, Nucl. Phys. B241 (1984) 333.
+[2] D. Friedan, E. J. Martinec and S. H. Shenker, Conformal Invariance, Supersymmetry and
+String Theory, Nucl. Phys. B 271 (1986) 93.
+[3] D. Gepner and E. Witten, String Theory on Group Manifolds, Nucl. Phys. B 278 (1986) 493.
+[4] A. B. Zamolodchikov, Infinite Additional Symmetries in Two-Dimensional Conformal
+Quantum Field Theory, Theor. Math. Phys. 65 (1985) 1205.
+[5] D. Das, S. Datta and S. Pal, Charged structure constants from modularity, JHEP 11 (2017)
+183 [1706.04612].
+[6] A. Bagchi, Correspondence between Asymptotically Flat Spacetimes and Nonrelativistic
+Conformal Field Theories, Phys. Rev. Lett. 105 (2010) 171601 [1006.3354].
+[7] C. Duval, G. W. Gibbons and P. A. Horvathy, Conformal Carroll groups and BMS
+symmetry, Class. Quant. Grav. 31 (2014) 092001 [1402.5894].
+[8] M. Bondi, H. van der Burg and A. Metzner, Gravitational waves in general relativity. 7.
+Waves from axisymmetric isolated systems, Proc. Roy. Soc. Lond. 21 (1962) .
+[9] R. Sachs, Asymptotic symmetries in gravitational theory, Phys. Rev. 128 (1962) 2851.
+[10] A. Bagchi and R. Fareghbal, BMS/GCA Redux: Towards Flatspace Holography from
+Non-Relativistic Symmetries, JHEP 10 (2012) 092 [1203.5795].
+[11] A. Bagchi, R. Basu, A. Kakkar and A. Mehra, Flat Holography: Aspects of the dual field
+theory, JHEP 12 (2016) 147 [1609.06203].
+[12] A. Bagchi, S. Detournay, R. Fareghbal and J. Sim´on, Holography of 3D Flat Cosmological
+Horizons, Phys. Rev. Lett. 110 (2013) 141302 [1208.4372].
+[13] G. Barnich, Entropy of three-dimensional asymptotically flat cosmological solutions, JHEP
+10 (2012) 095 [1208.4371].
+[14] G. Barnich, A. Gomberoff and H. A. Gonzalez, The Flat limit of three dimensional
+asymptotically anti-de Sitter spacetimes, Phys. Rev. D86 (2012) 024020 [1204.3288].
+[15] A. Bagchi, R. Basu, D. Grumiller and M. Riegler, Entanglement entropy in Galilean
+conformal field theories and flat holography, Phys. Rev. Lett. 114 (2015) 111602 [1410.4089].
+[16] A. Bagchi, D. Grumiller and W. Merbis, Stress tensor correlators in three-dimensional
+gravity, Phys. Rev. D 93 (2016) 061502 [1507.05620].
+[17] J. Hartong, Holographic Reconstruction of 3D Flat Space-Time, JHEP 10 (2016) 104
+[1511.01387].
+[18] H. Jiang, W. Song and Q. Wen, Entanglement Entropy in Flat Holography, JHEP 07 (2017)
+142 [1706.07552].
+[19] A. Bagchi, S. Banerjee, R. Basu and S. Dutta, Scattering Amplitudes: Celestial and
+Carrollian, Phys. Rev. Lett. 128 (2022) 241601 [2202.08438].
+[20] L. Cornalba and M. S. Costa, A New cosmological scenario in string theory, Phys. Rev. D 66
+(2002) 066001 [hep-th/0203031].
+– 53 –
+
+[21] R. Basu, S. Detournay and M. Riegler, Spectral Flow in 3D Flat Spacetimes, JHEP 12
+(2017) 134 [1706.07438].
+[22] A. Bagchi, R. Chatterjee, R. Kaushik, S. Pal, M. Riegler and D. Sarkar, BMS Field Theories
+with u(1) Symmetry, 2209.06832.
+[23] A. Strominger, Lectures on the Infrared Structure of Gravity and Gauge Theory, 1703.05448.
+[24] S. Pasterski, Lectures on celestial amplitudes, Eur. Phys. J. C 81 (2021) 1062 [2108.04801].
+[25] A.-M. Raclariu, Lectures on Celestial Holography, 2107.02075.
+[26] L. Donnay, A. Fiorucci, Y. Herfray and R. Ruzziconi, Carrollian Perspective on Celestial
+Holography, Phys. Rev. Lett. 129 (2022) 071602 [2202.04702].
+[27] L. Donnay, A. Fiorucci, Y. Herfray and R. Ruzziconi, Bridging Carrollian and Celestial
+Holography, 2212.12553.
+[28] S. Banerjee, S. Ghosh and P. Paul, MHV graviton scattering amplitudes and current algebra
+on the celestial sphere, JHEP 02 (2021) 176 [2008.04330].
+[29] S. Banerjee and S. Ghosh, MHV gluon scattering amplitudes from celestial current algebras,
+JHEP 10 (2021) 111 [2011.00017].
+[30] A. Strominger, w1+∞ Algebra and the Celestial Sphere: Infinite Towers of Soft Graviton,
+Photon, and Gluon Symmetries, Phys. Rev. Lett. 127 (2021) 221601.
+[31] A. Schild, Classical Null Strings, Phys. Rev. D16 (1977) 1722.
+[32] J. Isberg, U. Lindstrom, B. Sundborg and G. Theodoridis, Classical and quantized
+tensionless strings, Nucl. Phys. B411 (1994) 122 [hep-th/9307108].
+[33] A. Bagchi, Tensionless Strings and Galilean Conformal Algebra, JHEP 05 (2013) 141
+[1303.0291].
+[34] A. Bagchi, S. Chakrabortty and P. Parekh, Tensionless Strings from Worldsheet Symmetries,
+JHEP 01 (2016) 158 [1507.04361].
+[35] A. Bagchi, A. Banerjee, S. Chakrabortty, S. Dutta and P. Parekh, A tale of three —
+tensionless strings and vacuum structure, JHEP 04 (2020) 061 [2001.00354].
+[36] U. Lindstrom and M. Zabzine, Tensionless strings, WZW models at critical level and
+massless higher spin fields, Phys. Lett. B584 (2004) 178 [hep-th/0305098].
+[37] T. Harmark, J. Hartong, L. Menculini, N. A. Obers and Z. Yan, Strings with
+Non-Relativistic Conformal Symmetry and Limits of the AdS/CFT Correspondence, JHEP
+11 (2018) 190 [1810.05560].
+[38] A. Bagchi, A. Banerjee, R. Basu, M. Islam and S. Mondal, Magic Fermions: Carroll and
+Flat Bands, 2211.11640.
+[39] P.-x. Hao, W. Song, X. Xie and Y. Zhong, A BMS-invariant free scalar model, 2111.04701.
+[40] A. Bagchi and R. Gopakumar, Galilean Conformal Algebras and AdS/CFT, JHEP 07 (2009)
+037 [0902.1385].
+[41] A. Bagchi and I. Mandal, On Representations and Correlation Functions of Galilean
+Conformal Algebras, Phys. Lett. B675 (2009) 393 [0903.4524].
+[42] A. Bagchi, R. Gopakumar, I. Mandal and A. Miwa, GCA in 2d, JHEP 08 (2010) 004
+[0912.1090].
+– 54 –
+
+[43] A. Saha, Intrinsic Approach to 1 + 1D Carrollian Conformal Field Theory, 2207.11684.
+[44] J. Rasmussen and C. Raymond, Higher-order Galilean contractions, Nucl. Phys. B 945
+(2019) 114680 [1901.06069].
+[45] E. Ragoucy, J. Rasmussen and C. Raymond, Multi-graded Galilean conformal algebras, Nucl.
+Phys. B 957 (2020) 115092 [2002.08637].
+[46] R. Caroca, P. Concha, E. Rodr´ıguez and P. Salgado-Rebolledo, Generalizing the bms3 and
+2D-conformal algebras by expanding the Virasoro algebra, Eur. Phys. J. C 78 (2018) 262
+[1707.07209].
+[47] M. R. Gaberdiel and R. Gopakumar, Tensionless string spectra on AdS3, JHEP 05 (2018)
+085 [1803.04423].
+– 55 –
+