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+Chiral life on a slab
+Sergey Alekseev1, Mykola Dedushenko2, and Mikhail Litvinov1
+1Department of Physics and Astronomy, Stony Brook University,
+Stony Brook, NY 11794-3800, USA
+2Simons Center for Geometry and Physics, Stony Brook
+University, Stony Brook, NY 11794-3636, USA
+January 3, 2023
+Abstract
+We study chiral algebra in the reduction of 3D N = 2 supersymmetric
+gauge theories on an interval with the N = (0, 2) Dirichlet boundary
+conditions on both ends. By invoking the 3D “twisted formalism” and
+the 2D βγ-description we explicitly ﬁnd the perturbative Q+ cohomology
+of the reduced theory. It is shown that the vertex algebras of boundary
+operators are enhanced by the line operators.
+A full non-perturbative
+result is found in the abelian case, where the chiral algebra is given by the
+rank two Narain lattice VOA, and two more equivalent descriptions are
+provided. Conjectures and speculations on the nonperturbative answer in
+the non-abelian case are also given.
+1
+arXiv:2301.00038v1  [hep-th]  30 Dec 2022
+
+Contents
+1
+Introduction
+2
+2
+Basics
+8
+2.1
+Basic Supersymmetry
+. . . . . . . . . . . . . . . . . . . . . . . .
+8
+2.2
+Holomorphic-topological twist . . . . . . . . . . . . . . . . . . . .
+9
+2.3
+βγ System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+10
+2.4
+(0,2) Cohomology And βγ System
+. . . . . . . . . . . . . . . . .
+11
+3
+3D Perspective
+13
+3.1
+Vector Multiplet
+. . . . . . . . . . . . . . . . . . . . . . . . . . .
+14
+3.2
+The non-perturbative corrections . . . . . . . . . . . . . . . . . .
+18
+3.3
+U(1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+19
+4
+2D Perspective or βγ System
+22
+4.1
+U(1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+23
+4.2
+SU(2)
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+28
+4.3
+Open Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+33
+5
+Conclusion
+34
+A De Rham cohomology
+35
+1
+Introduction
+The role of vertex operator algebras (VOA) in theoretical physics and math-
+ematics has vastly expanded over the recent decades, way beyond their origi-
+nal scope [BPZ84; Bor86; FLM88]. This includes, among others, applications
+of VOAs in diﬀerential topology of four- and three-manifolds [DGP17; FG20;
+Che+22], and in higher-dimensional QFT [Bee+15; BRR15; CG19; CCG19].
+While some of these topics are more developed [Bee+15], the existing literature
+only scratches the surface of the topological applications and some other top-
+ics, such as boundary algebras in [CDG20]. Our main interest in the current
+paper is an interplay between the older role of VOAs as chiral algebras in 2D
+N = (0, 2) theories [Wit94; SW94], and their modern appearance as boundary
+algebras supported by the (0, 2) boundary conditions in 3D N = 2 theories
+[CDG20]. As will be explained later, this is also motivated by applications to
+the four-manifolds, following [DGP17; FG20], see also earlier works [GGP16;
+ASW16].
+The main subject of this work is a 3D N = 2 theory placed on an interval
+with N = (0, 2) boundary conditions on both ends. This basic setup is also
+explored in a companion paper [DL22] from the more physical perspective, where
+we compute the eﬀective 2D N = (0, 2) action in the infrared (IR) limit of the
+interval-reduced 3D model. It is known that 2D N = (0, 2) theories contain
+VOAs as their chiral algebras in the Q+-cohomology of local observables [Wit94].
+Being invariant under the renormalization group (RG) ﬂow [Ded15], such a
+chiral algebra must admit a UV realization in a 3D theory on the interval. It
+consists of a pair of (possibly diﬀerent) VOAs, associated to the boundaries,
+2
+
+extended by a category of their bi-modules associated to the appropriate line
+operators stretched between the boundaries (see an illustration on Fig. 1).
+Bℓ
+Br
+L
+Figure 1: Two boundaries with a line operator connecting them.
+Generally speaking, one expects various types of line operators to appear,
+including the descendant lines compatible with our supercharge [CDG20]. In
+the simplest case of pure N = 2 super Yang-Mills (SYM) with Chern-Simons
+(CS) level, which will be our main focus in the bulk of this paper, we will fully
+describe the spectrum of relevant lines. In particular, Wilson lines (which are
+descendants of the ghost ﬁeld) play an important part in this story.
+This connects us to another topic that has seen a lot of interest recently,
+– line defects and their role in related or similar constructions. For example,
+holomorphic boundary conditions in 3D topological QFT (TQFT) may sup-
+port non-trivial (relative) rational CFT (RCFT), and such TQFT/RCFT cor-
+respondence [Wit89; FK89] has been studied in great detail [Fel+00; FRS02a;
+FRS02b; FRS04a; FRS04b; Frö+04]. Bulk topological lines that are parallel
+to the boundary lead in this case to the boundary topological lines, while the
+bulk topological lines piercing the boundary give modules of the boundary chiral
+algebra. Fusion of such lines corresponds to fusion of the VOA modules. Note
+that in this story one also naturally encounters interval reductions: A TQFT on
+an interval (with the appropriate boundary conditions) leads to the full (non-
+chiral) RCFT, and segments of line operators stretched between the boundaries
+generate its primary ﬁelds. A more sophisticated class of examples comes from
+the topologically twisted 3D N = 4 theories, whose categories of line defects
+were recently studied in [Dim+20], see also [Cre+21; Gar22a; Gar22b]. Such
+theories may also possess holomorphic boundary conditions [CG19] supporting
+certain boundary VOAs, and the bulk lines piercing boundaries generate their
+modules as well. This leads to interesting questions of determining categories
+of VOA modules corresponding to the bulk lines [BN22], and it has also been
+argued that moduli spaces of vacua of the underlying physical theory can be re-
+covered from the knowledge of boundary VOAs and their categories of modules
+[CCG19].
+We are interested in similar constructions applied to three-dimensional the-
+ories with N = 2 supersymmetry (including 3D N = 4 viewed as N = 2).
+The N = (0, 2) boundary conditions [GGP14; OY13; YS20] in such theories
+support non-trivial boundary VOAs in the Q+-cohomology, which is a direct
+3
+
+analog of the 2D chiral algebra construction. The N = (0, 2) boundary condi-
+tions are compatible with the holomorphic-topological (HT) twist in 3D N = 2
+[Aga+17; CDG20].
+Thus, one may focus entirely on such HT-twisted theo-
+ries, which often simpliﬁes the VOA-related questions. The HT twist is also
+compatible with complexiﬁed Wilson lines, vortex lines in abelian [DOP12] and
+non-abelian [HS21] cases and their generalizations. Unlike in the topologically
+twisted theories, these line operators are not fully topological and cannot have
+arbitrary shape. They are supported along a straight line in the topological
+direction and are point-like in the holomorphic plane.
+In general, one could choose diﬀerent left and right boundary conditions
+Bℓ and Br, leading to the left and right boundary chiral algebras Vℓ and Vr
+and relations between them. For example, often Bℓ = B admits a counterpart
+Br = B⊥, such that the interval theory is trivially gapped in the IR, with
+the trivial chiral algebra. This would imply an interesting “duality” between
+the boundary chiral algebras, perhaps of relevance to some of the questions
+studied in [Che+22]. The only pairs of N = (0, 2) boundary conditions on the
+interval that appear in the literature so far include: Neumann-Neumann for
+gauge theories in [SY20], and Dirichlet-Neumann in [DN21; BZ22].
+In this paper we will focus on the simplest nontrivial case of 3D N = 2
+pure SYM with group G and CS level k, which is placed on the interval with
+N = (0, 2) Dirichlet boundary conditions. One of the main goals of this paper
+is to elucidate both the structure of chiral algebra and the underlying physics
+in this setup. Perturbatively, each boundary supports the aﬃne VOA Vn(g),
+where g = Lie(G) and the level is n = −h∨ ± k for the left/right boundary,
+respectively, with h∨ being the dual Coxeter number.
+The full perturbative
+VOA (that one may compare to the IR one) is given by V−h∨−k(g)⊗V−h∨+k(g)
+extended by a series of bimodules, realized in this case via SUSY Wilson lines
+(in all representations) stretched between the boundaries (as in Fig. 1). Non-
+perturbatively, this answer is signiﬁcantly modiﬁed, and we have a complete
+understanding for the abelian G. Partial results, conjectures, and challenges of
+the non-abelian case will be discussed as well.
+The IR description of the interval-reduced theory is identiﬁed in the com-
+panion paper [DL22] as an N = (0, 2) non-linear sigma-model (NLSM) into the
+complexiﬁcation of the gauge group GC ≡ exp(g ⊗ C), with the Wess-Zumino
+(WZ) level k. This is essentially a non-compact N = (0, 2) version of the Wess-
+Zumino-Witten (WZW) model, which exists for arbitrary Lie group. 1 The HT
+twist in 3d reduces to the purely holomorphic twist in 2d N = (0, 2) theories,
+sometimes called the half-twist2. The perturbative chiral algebra in such models
+is related to the theory of chiral diﬀerential operators (CDO) [GMS99; GMS01]
+on the target or, synonymously, curved βγ systems [Nek05], as explored in de-
+tail in [Wit07]. The VOA extracted from the sheaf of CDO on GC is denoted
+Dk[GC] and is the perturbative chiral algebra in the IR sigma-model. Here k
+is the modulus of the CDO corresponding to the B-ﬁeld on GC (proportional
+to the connection on the canonical WZ gerbe on GC, with k appearing as the
+1Note that this diﬀers from the compact N = (0, 2) WZW models that were previously con-
+structed for even-dimensional target groups, such as U(1) × SU(2), which all possess complex
+structure [Spi+88a; Spi+88b; Sev+88; RSS91; Roc+91; Ang+18].
+2However, more often than not, the term refers to something else, usually in the context
+of N = (0, 2) deformations of the N = (2, 2) theories, see [KS06; Sha09; MS08; GS09; MM09;
+Mel09; Kre+11; MP11; MSS12; GJS15; GS17; GGS21]
+4
+
+proportionality factor). From the CDO perspective, i.e. in perturbation theory,
+k does not have to be quantized, and indeed it labels the complex B-ﬁeld ﬂux in
+H3(GC, C). In our interval model, however, the B-ﬂux is non-generic and related
+to the CS level in 3D. For convenience, in this paper the B-ﬂux k is parameter-
+ized in such a way that it is precisely equal to the CS level. It is a mathematical
+fact that V−h∨−k(g) ⊗ V−h∨+k(g) is a sub-VOA of Dk[GC] [GMS01], thus the
+latter is expected to be an extension of the former by bi-modules. For generic k
+(which here means k ∈ C\Q) such an extension is indeed known to hold [AG02;
+FS06; Zhu08; CG20; CKM22; Mor22]:
+Dk[GC] ∼=
+�
+λ∈P+
+Vλ,−h∨+k (g) ⊗ Vλ∗,−h∨−k (g) ,
+(1.1)
+where one sums over the dominant weights λ. Physically, the bi-modules in
+(1.1) come from the Wilson lines connecting the boundaries, as stated earlier.
+The decomposition (1.1) makes perfect sense in perturbation theory, where
+we are allowed to treat the CS level as a generic complex number. At physi-
+cal integer values of k, however, the equation (1.1) no longer holds, since the
+structure of Dk[GC] as a V−h∨−k(g)⊗V−h∨+k(g) bi-module becomes much more
+intricate [Zhu08] (for non-abelian G). Not only that, but also the nonperturba-
+tive corrections are expected to signiﬁcantly modify it. Indeed, Dk[GC] contains
+universal aﬃne sub-VOAs V−h∨−k(g) and V−h∨+k(g) [GMS01], not their simple
+quotients. Now, the algebra V−h∨−k(g) for k > 0, (i.e. at the below-critical
+level,) is already simple, see [KL93, Proposition 2.12] in the simply-laced case.
+However, for k > h∨, the aﬃne VOA V−h∨+k(g) living on the “positive” bound-
+ary is not simple: It is very well understood [KK79], has the proper maximal
+ideal, and the simple quotient denoted L−h∨+k(g). It was argued in [CDG20]
+that the nonperturbative boundary monopoles implement this simple quotient,
+at least on the “positive” boundary, turning V−h∨+k(g) into L−h∨+k(g).
+Thus we expect the exact chiral algebra (for k > h∨) to contain V−h∨−k(g)⊗
+L−h∨+k(g), not V−h∨−k(g) ⊗ V−h∨+k(g), which already diﬀers from Dk[GC].
+Furthermore, the 3D bulk is expected to admit only ﬁnitely many line operators
+in the IR. Indeed, it is gapped (at least if the interval is long enough) and given
+by the Gk−h∨ CS for levels3 k > h∨, which only admits ﬁnitely many Wilson
+lines labeled by the integrable representations of L−h∨+k(g) [Wit89; Kac95].
+We, therefore, expect the exact VOA to be, roughly, V−h∨−k(g) ⊗ L−h∨+k(g)
+extended by such a ﬁnite set of bimodules. This appears to be signiﬁcantly
+diﬀerent from Dt[GC] (e.g., the latter is an inﬁnite extension for generic t).
+In this paper, we fully address the abelian case, G = U(1), and give par-
+tial results on the nonabelian G, including the perturbative treatment outlined
+earlier. We also speculate on the kind of nonperturbative corrections in the non-
+abelian GC NLSM that are expected to modify Dk[GC]. We leave the detailed
+study of such nonperturbative eﬀects for future work.
+In the abelian case, GC = C∗, so the IR regime is described by the NLSM
+into C∗, which was analyzed previously in [DGP17]. This theory of course has a
+C∗-valued βγ system for its perturbative chiral algebra Dt[C∗]. We think of it ei-
+ther multiplicatively, i.e., γ is C∗-valued, or additively, i.e., �γ ∼ �γ +2πiR, where
+R is the radius of the compact boson (to be determined by the CS level). Since
+the theory is free, the distinction between perturbative and non-perturbative
+3The IR behavior at arbitrary k is described in [Bas+18], see also [AHW82; Oht99; GKS18].
+5
+
+is slightly formal. In the case of the C∗ model, the natural nonperturbative
+completion amounts to including twisted sectors into the theory, which corre-
+spond to windings around the nontrivial one-cycle in the target. The twist ﬁelds
+are vortex-like disorder operators, whose 3D origin is, expectedly, the boundary
+monopoles. The presence of such sectors extends the βγ system by the so-called
+spectrally ﬂowed modules [RW14; AW22]. We show that this results in the lat-
+tice VOA for the simplest Narain lattice [Nar86; NSW87], i.e., Z2 ⊂ R2 with the
+scalar product whose Gram matrix is (0 1
+1 0). In addition to showing this, we also
+provide the 3D perspective, where each boundary supports a 1D lattice VOA
+(abelian WZW [DGP18]), and Wilson lines extend them by the bi-modules.
+In total, we ﬁnd three presentation of the same VOA: (1) as a Narain lattice
+VOA; (2) as an extension of the βγ system; (3) as an extension of the abelian
+WZWk ⊗ WZW−k by its bimodules.
+The nonabelian GC NLSM is interacting, which makes the non-perturbative
+corrections to the perturbative answer Dk[GC] much more challenging to quan-
+tify. We are certain from the previous discussion, however, that such corrections
+must be present. The known instanton eﬀects in 2D N = (0, 2) theories are vor-
+tices that are best understood in the gauged linear sigma model case [SW95;
+BS03; BP15] (see also [LNS00] for the N = (2, 2) case). In the NLSMs they
+are captured by the worldsheet wrapping compact holomorphic curves in the
+target [Din+86; Din+87; BW06], which is notoriously hard to compute except
+for the simplest models [TY08]. In our case, however, GC does not support any
+compact holomorphic curves, which basically implies that there must be some
+novel nonperturbative corrections at play. They correspond to the boundary
+monopoles that exist in the parent 3D gauge theory, and can be described as new
+“noncompact” vortices in 2D. Indeed, monopoles are labeled by the cocharacters
+eibϕ : U(1) → G, up to conjugation, which are complexiﬁed to zb : C∗ → GC.
+This allows to deﬁne a natural vortex singularity for the NLSM ﬁeld φ(z, z):
+φ ∼ zb,
+as z → 0.
+(1.2)
+Since it is meromorphic, it will deﬁne a half-BPS defect.
+Dynamics in the
+presence of such defects is expected to modify the chiral algebra Dt[GC] ap-
+propriately. We will explore this elsewhere, while here we only focus on the
+perturbative aspects when G is non-abelian.
+Another subtle point is that the presence of such disorder operators, and
+hence the non-perturbative corrections, in principle, depends on the UV com-
+pletion.
+We will assume that the 3D gauge theory with compact G admits
+monopoles.4 Without them, the Dirichlet boundary on the “positive” end would
+appear in tension with unitarity [DGP18; CDG20] (the same issue is not present
+on the “negative” boundary since it supports a non-compact CFT). Thus the
+NLSM originating from the gauge theory on the interval must admit the vortex-
+type disorder operators and the corresponding non-perturbative phenomena. On
+the other hand, if one completes the GC NLSM in the UV into the LG model as
+in [DL22], there is no room for any vortex defects. In this case the Dt[GC] is con-
+jectured to give the exact chiral algebra. Purely from the viewpoint of NLSM,
+it is conceivable that the knowledge of metric on the target (which is usually
+ignored in the BPS calculations, but which was computed in [DL22],) allows one
+4Which has far-reaching consequences for its dynamics, as, e.g., in Polyakov’s argument
+for conﬁnement in 3D [Pol77].
+6
+
+to tell apart the cases with and without the nonperturbative corrections. This
+question is likewise left for the future.
+Motivation via VOA[M4].
+One of the motivations behind this work is to de-
+velop a toolkit for computing VOA[M4] [DGP17].5 Recall that the VOA[M4], or
+more precisely VOA[M4, g], is deﬁned as the chiral algebra in the Q+-cohomology
+of the 2D N = (0, 2) theory T[M4, g], which is obtained by the twisted compact-
+iﬁcation of the 6d (2, 0) SCFT (of type g) on the four-manifold M4 [GGP16].
+Let us consider a class of four-manifolds M4 admitting a metric with S1
+isometry, such that the S1 action is not free. Note that when the S1 action
+is free, the four-manifold is an S1 bundle over some smooth three-manifold
+M3 (which is a relatively tame class of four-manifolds), and it is conceptually
+clear how the dimensional reduction simpliﬁes (ﬁrst reduce on S1 and then
+on M3). Of course this is still an interesting and nontrivial problem, but our
+motivation comes from the opposite case, when the S1 action is not free. Some
+examples of such four-manifolds include but are not limited to: (1) Σg × S2,
+where Σg is an arbitrary genus-g surface, and the S1 action rotates S2, which
+is equipped with an S1-invariant “sausage” metric; (2) the unique nontrivial
+sphere bundle over the Riamann surface of genus g, denoted as Σg �×S2; (3)
+CP2 with the standard Fubini-Study metric; (4) Hirzebruch surface, given by
+the connected sum CP2#CP
+2, which is actually isomorphic to the nontrivial
+sphere bundle over a sphere, S2 �×S2; (5) four-sphere S4. In fact, the general
+class of such four-manifolds is quite well understood, at least in the simply
+connected compact case. It was shown by Fintushel [Fin77; Fin78] that if a
+simply connected M4 admits an S1 action (not necessarily an isometry), it must
+be a connected sum of some number of S2 × S2, S4, CP2 and CP
+2. If S1 is an
+isometry and the corresponding simply-connected four-manifold is, in addition,
+non-negatively curved, [SY94] proved (see also [HK89]) that it belongs to the
+list {S4, CP2, S2 × S2, CP2#CP
+2, CP2#CP2}. If we only allow codimension-2
+ﬁxed loci, then the list is even shorter: {S2 × S2, CP2#CP
+2}. Such lists may
+appear utterly specialized, however, these manifolds present certain interest to
+us in view of conjectures in [FG20], which we aim to check in the future work.
+We may consider the twisted compactiﬁcation on a manifold with S1 isom-
+etry in two steps: (1) ﬁrst reduce the 6D theory along the S1 orbits; (2) then
+perform reduction along the remaining quotient space M4/U(1). The advantage
+of this procedure is that the ﬁrst step yields a relatively simple and concrete re-
+sult – the maximal 5D SYM (MSYM) with gauge group G (the simply-connected
+Lie group whose algebra is g), albeit placed on some curved space with bound-
+aries and, possibly, defects. For example, for M4 = Σg × S2, reducing along
+the parallels of S2 gives the Σg × I geometry, where I is an interval with the
+principal Nahm pole boundary conditions at both ends [CDT13]. Further re-
+duction of the 5D MSYM along Σg with the topological twist simply gives a 3D
+N = 4 SYM with g adjoint hypermultiplets. Thus we end up with the former
+3D theory on the interval, with the (0, 4) Nahm pole boundary conditions at
+both ends. In this case, the resulting eﬀective 2D theory in the IR has (0, 4)
+SUSY [PSY16]. Other examples would lead to the interval reductions of vari-
+5A few recent appearances of the interval compactiﬁcations in related contexts include
+[GR19; PR18; DP19].
+7
+
+ous 3D N = 2 gauge theories with matter and CS levels, which would ﬂow to
+N = (0, 2) theories in the 2D limit.
+We will explore various such examples in the future work [DL23]. However,
+it is natural to start with the most basic 3D N = 2 gauge theory, that is the pure
+SYM, and study the interval VOA in this case. This is one of the underlying
+motivations for the current paper.
+The rest of this paper is structured as follows.
+In Section 2 we review
+the necessary background material. Then we move on to computing the interval
+compactiﬁcation chiral algebra in the 3D N = 2 SYM theory with N = (0, 2)
+Dirichlet boundaries in Section 3. In Section 4 we compute the same chiral
+algebra from the 2D perspective and discuss some issues. In the abelian case, we
+end up with three diﬀerent presentations of the same VOA. In the nonabelian
+case, we make general statements when possible, but mostly work with the
+G = SU(2) example. Then we ﬁnish with some open questions and speculations,
+and conclude in Section 5.
+2
+Basics
+In this section, we set up the conventions and brieﬂy review the background
+material, including the holomorphic-topological (HT) twist. In particular, we
+discuss the 3D N = 2 supersymmetric theory and its twisted content.
+We
+will describe protected sectors, namely, the cohomology of a Q+ supercharge in
+3D and 2D theories. A βγ system will be brieﬂy discussed as well. Also the
+connection between the cohomology of N = (0, 2) theories and the Čech coho-
+mology of the βγ system is expounded upon, as it will be one of the important
+computational tools later.
+Conventions:
+We consider R2 × I with Euclidean signature and with coor-
+dinates xµ on R2 and t ∈ [0, L] on I. We choose γi
+αβ matrices to be the Pauli
+matrices σi, and the antisymmetric symbol ϵ12 = ϵ21 = 1 to lower and raise
+indices [IS13].
+2.1
+Basic Supersymmetry
+In Euclidean 3D space spinors lie in a 2-dimensional complex representation of
+SU (2) and the N = 2 supersymmetry algebra takes the following form:
+�
+QI, QJ�
+= δIJγµPµ.
+(2.1)
+By deﬁning a new combination of supercharges Q = Q1+iQ2 and Q = Q1−iQ2,
+one can obtain the following conventional form of the superalgebra:
+�
+Q, Q
+�
+= 2γµPµ.
+(2.2)
+Note that the supercharges are not conjugate to each other in Euclidean sig-
+nature contrary to Minkowski space, where minimal representations are real
+(Majorana). This algebra admits a U (1)R-charge, which is an automorphism
+of this algebra and acts by rotating Q-charges. Operators also have a charge J0
+8
+
+with respect to Spin(2)E rotation parallel to boundaries. Let us also deﬁne the
+combination J := R
+2 − J0, then all charges can be summarized by the following
+table:
+Q+
+Q+
+Q−
+Q−
+dz
+dz
+U(1)R
+1
+−1
+1
+−1
+0
+0
+Spin(2)E
+1
+2
+1
+2
+− 1
+2
+− 1
+2
+1
+−1
+U(1)J
+0
+−1
+1
+0
+−1
+1
+In what follows, we will be considering the cohomology of Q =
+def Q+. The Pz
+is the only Pµ which is not Q-exact. It makes our algebra into an algebra with
+only holomorphic dependence on the coordinates. We would consider boundary
+conditions that preserve a (0, 2)-part of the supersymmetry algebra generated
+by Q+ and Q+. We also want to leave U (1)R unbroken in the bulk and on the
+boundary.
+The only relevant 3D N = 2 multiplet for this paper is a vector multiplet
+for some gauge group G:
+V3D = θσmθAm + iθθσ − iθ2θλ − iθ
+2θλ + 1
+2θ2θ
+2D3d
+(WZ gauge) ,
+where all the ﬁelds lie in the Lie algebra g = Lie(G). It consists of a connection
+Am, a real scalar σ, a complex fermion λα, and a real auxiliary ﬁeld D3d. We
+can also deﬁne a covariant superﬁeld
+Σ3D = − i
+2ϵαβDαDβV3d = σ − θλ + θλ + θγµθϵµνρF νρ + iθθD + . . .
+that satisﬁes DαDαΣ3D = D
+αDαΣ3D = 0.
+2.2
+Holomorphic-topological twist
+In this section, we review some formulas of the HT-twisted formalism [Aga+17;
+CDG20]. The Q-cohomology of operators of the twisted theory and physical
+theory are the same. The convenience of this formalism is that some calculations
+have only a ﬁnite number of Feynman diagrams.
+The twisted formalism is reviewed nicely in [CDG20, Section 3.2]. Let us
+consider a dg-algebra:
+Ω• = C∞ �
+R3�
+[dt, dz] ,
+(2.3)
+where the multiplication is the multiplication of diﬀerential forms. One also
+needs to consider forms with values in the k-th power of the canonical line
+bundle in the z-direction:
+Ω•,k = Ω• ⊗ Kk = C∞ �
+R3�
+[dt, dz] dzk.
+(2.4)
+There is cohomological charge R, which is related to the original R-charge
+in the physical theory by adding a ghost charge to it, and a twisted spin charge
+J. In the holomorphic-topological twisted 3D N = 2 theory the ﬁelds can be
+organized into the following BV superﬁelds:
+A = c + (At dt + Az dz) + B∗
+zt dz dt ∈ Ω• ⊗ g [1] ,
+B =
+�
+B + A∗
+µ dxµ + c∗ dz dt
+�
+dz ∈ Ω•,1 ⊗ g∗,
+(2.5)
+9
+
+where the superﬁelds A and B are obtained from the vector multiplet, and we
+also introduced ghosts. For example, the ﬁeld
+A := Az dz + At dt
+is just a connection with complexiﬁed At = At − iσ and ordinary Az. The
+ﬁeld B ≡ Bz is identiﬁed with
+1
+g2 Fzt =
+1
+g2 Fzt + . . . on shell.
+The bracket
+[1] indicates a shift of cohomological degree by one. The forms dt and dz are
+treated as Grassmann variables, so they anticommute with fermionic ﬁelds, and
+superﬁelds can be regarded as either bosonic or fermionic.
+The action of the Q-charge in the twisted formalism can be written as follows:
+QA = F(A),
+QB = dAB −
+k
+2π∂A.
+(2.6)
+Here the diﬀerentials are deﬁned as follows:
+dA = d′ − iA,
+d′ = dt ∂t + dz ∂z,
+∂ = dz ∂z
+(2.7)
+and the curvature is
+F(A) = id2
+A = d′A − iA2.
+(2.8)
+It will be also useful to keep in mind the following tables of the R and J charges
+of the operators:
+c
+At
+Az
+R
+1
+0
+0
+J
+0
+0
+-1
+B
+A⋆
+t
+A⋆
+z
+c⋆
+R
+0
+-1
+-1
+-2
+J
+1
+1
+0
+0
+Table 1:
+The charges of the ﬁelds.
+The following charges are assigned to the diﬀerential forms:
+dt
+dz
+dz
+R
+1
+1
+0
+J
+0
+1
+-1
+Table 2:
+The charges of the diﬀerential forms.
+2.3
+βγ System
+In this section we review the βγ system, or as it is usually called in mathematical
+literature, a sheaf of chiral diﬀerential operators. All formulas can be found in
+[GMS99; GMS01; Nek05; Wit07]
+Classically, consider a complex manifold M, a map γ : Σ → M, and a (1, 0)-
+form β on Σ with values in the pullback γ∗(T ∗M), governed by the following
+action:
+�
+Σ
+βi∂γi,
+(2.9)
+where γi and βi are the holomorphic components of γ and β, respectively.
+10
+
+Quantum mechanically, the situation is more interesting as we want to pre-
+serve the OPE’s locally. On each patch we have the usual βγ system with the
+OPE:
+γi (z) βj (w) ∼ δi
+j
+dw
+z − w,
+(2.10)
+which in the physics notation yields:
+�
+γi
+n, βj k
+�
+= δi
+jδn+k,0.
+(2.11)
+The normal ordering prescription for polynomials is deﬁned by the point-splitting
+procedure and depends on a chosen complex structure.
+To get a global theory, we need to learn how to glue ﬁelds on diﬀerent patches
+together. First, let us choose two sets of local coordinates γi and �γb on some
+open set. The gluing is done by a local automorphisms and γ is transformed as
+in the classical theory. As we mentioned before, β is transformed classically as
+β �→ �β = f ∗β, where f is a local holomorphic diﬀeomorphism. The quantum
+version of this transformation law is given by the following general formula
+[Nek05]:
+�βa = βi
+∂γi
+∂�γa − 1
+2
+�
+∂jgi
+a∂igj
+b
+� ∂�γb
+∂γk ∂γk
+�
+��
+�
+quantum part
++
+1
+2µab∂�γb
+�
+��
+�
+moduli parameter
+,
+(2.12)
+where the Jacobian of the transformation is gi
+a = ∂γi
+∂�γa . The “quantum part”
+appears because we want to keep the right OPE on both patches after gluing.
+There is an intrinsic ambiguity associated to solving for the OPE equations.
+Moreover, the moduli space of the βγ system is parametrized by µ or, stating
+it simply, diﬀerent ways of gluing our system globally are in one to one corre-
+spondence with the possible choices of µ. The parameter µ takes values in the
+ﬁrst Čech cohomology group with coeﬃcients in the sheaf of closed holomorphic
+two-forms, i.e. H1 �
+Ω2,cl, M
+�
+.
+This algebra becomes VOA if c1(M) = 0. The global stress energy tensor is
+T = −βi∂γi − 1
+2(log w)′′,
+(2.13)
+where w is the coeﬃcient of the holomorphic top form ω = wdγ1 ∧ . . . ∧ dγn.
+2.4
+(0,2) Cohomology And βγ System
+One of the physics applications of the curved βγ system is in the realm of (0, 2)
+theories. As discussed in [Wit07] and will be reviewed shortly, the βγ system
+describes the perturbative cohomology of half-twisted (0, 2) theories.
+Let us ﬁrst discuss a general (0, 2) sigma model. The Lagrangian is con-
+structed locally by introducing a (1, 0)-form K = Ki dφi, with complex conju-
+gate K = Ki dφ
+i, and setting
+I =
+� ��d2z
+�� dθ
++ dθ+
+�
+− i
+2Ki(Φ, Φ)∂zΦi + i
+2Ki(Φ, Φ)∂zΦ
+i�
+,
+11
+
+where Φi is a chiral superﬁeld whose bottom component φi deﬁnes a map from
+a Riemann surface Σ to a target complex manifold X.
+The cohomology of
+the supercharge Q+ can be deformed by H = 2i∂ω ∈ H1 �
+M, Ω2,cl�
+, where
+ω = i
+2
+�
+∂K − ∂K
+�
+. Not only that but H must be of type (2, 1) to preserve (0, 2)
+supersymmetry. Note that it is the same class that parametrizes the βγ system
+moduli.
+If we set αi = −
+√
+2ψ
+i
++, ρi = −iψi
++/
+√
+2 and twist the theory then ρ is an
+element of Ω0,1 (Σ) ⊗ φ∗ (TX) and α is from φ∗ �
+TX
+�
+. Both α and ρ are Grass-
+mann variables. After the twisting, Q+ becomes a worldsheet scalar with the
+following action on the ﬁelds:
+Qφi = 0,
+Qφ
+i = αi,
+Qρi
+z = −∂zφi,
+Qαi = 0,
+(2.14)
+and the action is given by:
+I =
+�
+d2z
+�
+gij∂zφi∂zφ
+j + gijρi
+z∂zαj − gij,kαkρi
+z∂zφ
+j�
++ ST ,
+(2.15)
+where ST = −
+�
+d2z
+�
+Tij∂zφi∂zφj − Tij,kαkρi∂zφj�
+and H = dT should be of
+the type described above. We also note that T is not a 2-form but a 2-gauge
+ﬁeld.
+Locally, the structure of the Q-cohomology can be understood easily with
+the help of the βγ system. Consider an open ball Uα:
+I = 1
+2π
+�
+Uα
+��d2z
+�� �
+i,j
+δi,j
+�
+∂zφi∂zφ
+j + ρi∂zαj
+�
+.
+(2.16)
+All the sections in the cohomology can be written as (for details refer to [Wit07]):
+F
+�
+φ, ∂zφ, . . . ; ∂zφ, ∂2
+zφ, . . .
+�
+∈ H0 �
+Ops2d, Q
+2d
++
+�
+.
+(2.17)
+If we set βi = δij∂zφ
+j, which is an operator of dimension (1, 0), and γi = φi of
+dimension (0, 0), the bosonic part of the action can be rewritten as:
+Iβγ
+Uα = 1
+2π
+� ��d2z
+�� �
+i
+βi∂zγi
+(2.18)
+and the space of all sections of this theory is
+F
+�
+γ, ∂zγ, ∂2
+zγ, . . . ; β, ∂zβ, ∂2
+zβ . . .
+�
+.
+(2.19)
+So, locally, the space of sections of the βγ system and the Q-cohomology of
+the (0, 2) sigma model coincide. Globally, things are a little more complicated
+and we are required to consider Čech cohomology to ﬁnd the operators with all
+possible R-charges. The R-charge in the sigma model description is matched
+with the cohomological degree:
+H• �
+Ops2d, Q
+2d
++
+�
+∼= H•
+ˇCech(X, �A),
+(2.20)
+where �A is a sheaf of free βγ systems.
+12
+
+3
+3D Perspective
+In this section, we discuss the Q-cohomology from the 3D N = 2 point of view.
+There are a few constructions one could consider. Firstly, the Q-cohomology of
+local operators in the bulk is a commutative vertex algebra (VA) V intrinsic to
+the theory [CDG20; OY20]. Secondly, the Q-cohomology of local operators at
+the boundary preserving (0, 2) SUSY (explored in the same reference) is, gener-
+ally speaking, a noncommutative VA. Thirdly, — and this is the new structure
+that we study here, — one can deﬁne the Q-cohomology on the interval, or
+the chiral algebra of the interval compactiﬁcation. If both the 3D theory and
+its (0, 2) boundary conditions preserve the R-symmetry, this is a vertex op-
+erator algebra (VOA), as opposed to just VA, i.e., it necessarily contains the
+stress energy tensor. This is obvious since in the IR limit, the theory becomes
+eﬀectively two-dimensional [DL22], and the chiral algebra of an R-symmetric
+2D N = (0, 2) theory always has the Virasoro element, as can be seen from
+the general R-multiplet structure [Ded15]. In fact, one can also prove this by
+constructing the (0, 2) R-multiplet from the integrated currents directly in 3D
+[BST19].
+Intuitively, the interval VOA contains all 3D observables that look like local
+operators in the 2D limit. These includes 3D local operators and lines, thus
+eﬀectively enhancing the Q-cohomology of local operators by the line operators
+stretched between the boundaries. The line operators can additionally be dec-
+orated by local operators in the Q-cohomology. We are allowed to move them
+to the boundaries, as follows from the properties of Q [CDG20]. Additionally,
+the two ends of the line operator can support some other boundary operators
+that are stuck there and cannot be shifted into the bulk. Thus the most gen-
+eral conﬁguration in the Q-cohomology consists of a line stretched between the
+boundaries with some local operators sitting at its two endpoints.
+This in-
+cludes the possibility of colliding a boundary operator from the boundary VA
+mentioned earlier with the endpoint of a line.
+On the half-space, the latter
+implies that lines ending at the boundary engineer modules for the boundary
+VA [CCG19]. In our case, i.e. on the interval, this similarly means that the line
+operators give bi-modules of the pair of boundary VAs supported at the two
+ends of the interval.
+Examples of line operators that appear here include descendants of the Q-
+closed local operators integrated over the interval [CDG20]. Things like Wilson
+and vortex lines or their generalizations [Dim+20] may appear as well (the
+Wilson line can be also viewed as a descendent of the ghost ﬁeld). We are striving
+to compute the OPE involving such operators. In fact, we will compute the exact
+chiral algebra in the abelian case and the perturbative one in the nonabelian
+case, that is the OPE of both local and line operators, for gauge theories with
+the Dirichlet boundary conditions preserving (0, 2) supersymetry. We will ﬁnd
+that the order line operators, i.e. Wilson lines, create representations for the
+boundary operator algebras. They are naturally included into the perturbative
+interval VOA. The disorder or vortex lines (when allowed), on the other hand,
+together with the boundary monopoles should be viewed as manifestation of the
+non-perturbative phenomena. We claim to fully understand them in the abelian
+case but only brieﬂy discuss in the nonabelian setting.
+Generally speaking, we have chiral algebras on the left and right boundaries
+denoted by Vℓ and Vr, respectively. There is also the bulk algebra (commutative
+13
+
+VA) V, which includes only local operators. Moreover, V maps naturally into
+the left and right algebras via the bulk-boundary maps, allowing to deﬁne their
+tensor product over V. There are two maps, which are deﬁned by pushing the
+local operators from V to the two boundaries:
+ρℓ,r : V → Vℓ,r.
+(3.1)
+Let us ﬁrst deﬁne the algebra that only includes the local operators in 3D:
+Vℓ ⊗V Vr.
+(3.2)
+The tensor product over V involves the identiﬁcation of operators that can be ob-
+tained from the same operator in the bulk. The next step is to extend this alge-
+bra by modules that correspond to the Q-closed line operators stretched between
+the boundaries. We will denote the resulting 3D cohomology as H•(Ops3d, Q).
+Last but not least, let us note explicitly that in the non-abelian case, we
+will be mostly discussing the CS level k > h∨. The IR physics of a 3D N = 2
+YM-CS is known for all values of k [Bas+18], and for 0 < |k| < h∨ it exhibits
+spontaneous SUSY breaking [Ber+99; Oht99], and runaway for k = 0 [AHW82].
+What happens on the interval in the range 0 ≤ |k| < h∨ will be addressed
+elsewhere, while the k ≥ h∨ case is more straightforward. Yet, it is interesting
+enough, as we see in this work.
+3.1
+Vector Multiplet
+Consider a vector multiplet sandwiched between the Dirichlet boundary condi-
+tions. In the twisted formalism this amounts to choosing A
+�� = 0[CDG20] at
+both ends. This, in turn, is equivalent to setting c| = 0 and Az| = 0
+The transformation rules for c, A, B, and A∗ in the bulk follow from (2.6):
+Q (c) = −ic2,
+Q (A) = dAc,
+QB = −i[c, B] −
+k
+2π∂zc,
+QA∗ = d′B − i [A, B] −
+k
+2π∂zA.
+(3.3)
+where dA = d′ − iA.
+Before diving deeper, we review what is known about the perturbative alge-
+bra on the boundary. Recall that the action in the HT twist takes the following
+form:
+�
+BF (A) + k
+4π
+�
+A∂A.
+(3.4)
+In the twisted formalism, the propagator connects A with B, as follows from
+the kinetic energy B d′A. The rest of terms, including the Chern-Simons, are
+treated as interactions, which induces the bivalent and the trivalent vertices:
+1. the vertex connecting two A,
+2. the vertex connecting two A and one B.
+This form of Feynmann rules is very restrictive, and there can only be a ﬁnite
+number of diagrams for a given number of external legs [GW19; CDG20].
+For the group G the ﬁeld B lies in g∗. The gauge group is broken on the
+boundary and becomes a global symmetry there. There is also a non-trivial
+14
+
+boundary anomaly due to the bulk Chern-Simons term and the fermions in
+the gauge multiplet. The former contributes ±k to the anomaly and the latter
+contributes −h∨.
+Thus, we expect to get two boundary aﬃne algebras, one for each bound-
+ary global symmetry, with levels dictated by the anomaly. From (3.3), on the
+boundary we have QB = 0. So, B is in the cohomology and its OPE with itself
+was obtained in [CDG20, section 7.1]:
+Ba (z) Bb (w) ∼ (−h∨ ± k) κab
+(z − w)2
++
+ifabc
+(z − w) Bc (w) ,
+(3.5)
+where Ba are the components of B in some basis, κab is the standard bilinear
+form equal to
+1
+2h∨ times the Killing form in that basis, and h∨ is the dual
+Coxeter number. This expression can be obtained from the charge conservation
+and anomalies alone. The J charge of B is 1 (see Table 1). Thus, only z up to
+the second power can contribute. The ﬁrst term is the anomaly term explained
+above.
+All half-BPS line operators hitting the boundaries are expected to create
+modules for the boundary algebras, and we will show that it is true perturba-
+tively for the Wilson line momentarily. A Wilson line can be written as Pe
+�
+t A.
+Observe that it is indeed Q-invariant in the usual formalism, or in the twisted
+formalism by invoking Q (A) = dAc and c| = 0. The kinetic term for B and A
+can be written as:
+Tr B (∂zAt − ∂tAz) .
+(3.6)
+There is a gauge symmetry associated to this term:
+At → At + ∂tη,
+Az → Az + ∂zη.
+(3.7)
+The propagator for B and At with the appropriate gauge ﬁxing [CDG20] is just
+G(z, z; t) ∝
+z
+|x2|
+3
+2 ,
+where
+x2 = zz + t2,
+(3.8)
+where we do not keep track of a proportionality constant.
+Next, we can calculate the OPE of the boundary operator B(0) with the
+Wilson line segment W (λ)(z) in the irreducible representation of g labeled by
+the dominant weight λ. Expanding the Wilson line in a Taylor series, one ﬁnds
+that there is only one diagram that can possibly contribute, where a single A
+from the Wilson line is directly connected to the operator B at the boundary,
+see Fig. 2. This is proven simply by looking at the two vertices mentioned
+earlier and realizing that they cannot contribute. The diagram evaluates to
+�
+Aa
+t (t, z, z)Bb (0) dt ∝ δa
+b
+� L
+0
+z dt
+(t2 + zz)
+3
+2 = δa
+b
+L
+z
+√
+L2 + zz
+= δa
+b
+1
+z + . . . ,
+(3.9)
+where we keep only the singular term in the z → 0 expansion on the right. This
+computation already includes corrections to the propagator in the presence of
+15
+
+W
+B
+Figure 2: The diagram connecting single A in the Wilson line to the operator
+B at the boundary.
+boundaries. Indeed, such corrections can be accounted for using the method of
+images. Since we are interested in the singular term in the OPE, it is enough
+to only include the ﬁrst image At(−t, z, z) = At(t, z, z), as the other ones never
+get close to B(0). This simply doubles the contribution of the original insertion
+At(t, z, z). Combining everything together, this computation shows that the
+OPE with the Wilson line is
+Ba (z) W (λ) (w) ∼ TaW (λ) (w)
+z − w
+,
+(3.10)
+where Ta denotes the Lie algebra generator in the same representation λ as the
+Wilson line, and TaW (λ)(w) means the matrix product. Looking at the Table 1,
+we immediately see that this is the only possible OPE as W (λ) is not charged.
+More precisely, there are two copies of the aﬃne generators on the interval,
+denoted Bℓ
+a and Br
+a for the left and right boundaries, respectively. Assuming
+that the Wilson line performs parallel transport from the right to the left, we
+ﬁnd the following OPE’s on the interval:
+Bℓ
+a (z) W (λ) (w) ∼ TaW (λ) (w)
+z − w
+,
+Br
+a (z) W (λ) (w) ∼ W (λ) (w) Ta
+z − w
+.
+(3.11)
+For completeness, note that the OPE of Wilson line’s matrix elements is regular:
+W (λ)
+ij (z)W (µ)
+kl (w) ∼ 0,
+(3.12)
+simply because no Feynmann diagram can connect two At’s.
+Let us pause and contemplate on what we have obtained so far. We found
+that Bℓ,r and W (λ) are elements of the extended cohomology, and we claim
+that they generate the perturbative chiral algebra. The boundary B’s satisfy
+the OPE relations of the aﬃne Kac-Moody vertex algebras V−h∨±k(g). They
+also act on W (λ) as on a primary ﬁeld of the highest weight representation
+of the aﬃne algebra (i.e., a Weyl module Vλ,−h∨±k = Ind�g
+�bVλ, where Vλ is a
+ﬁnite-dimensional module for the underlying Lie algebra g). Thus, we obtain
+the following result for the perturbative chiral algebra:
+Ck[GC] :=
+�
+λ∈P+
+Vλ,−h∨+k ⊗ Vλ∗,−h∨−k,
+(3.13)
+16
+
+where λ runs over the set P+ of dominant weights, and λ∗ = −w(λ), where
+w is the longest element of the Weyl group of G. It is not a coincidence that
+Ck[GC] looks like Dk[GC] from the equation (1.1) in the Introduction.
+This
+object is well known in the mathematical literature [AG02; FS06; Zhu08; CG20;
+CKM22; Mor22], and away from the rational values of k, Ck[GC] is a simple VOA
+isomorphic to the VOA Dk[GC] of chiral diﬀerential operators on GC, with the
+deformation parameter (perturbative B-ﬁeld ﬂux) k ∈ C = H3(G, C). At the
+rational points k ∈ Q, we can encounter singular vectors, and life is getting
+much more interesting, e.g., Ck[GC] and Dk[GC] are no longer the same [Zhu08].
+We can also ask what happens to the stress-energy tensor in our setup. We
+know that it does not exist as a local operator in the bulk chiral algebra [CDG20,
+section 2.2], and generally, the boundary VA does not have to possess a stress-
+energy tensor as well. At the same time, we have the current that generates
+holomorphic translations. It acts on the boundary operators as:
+∂wO (w) =
+�
+HS2 ∗(Tzµ dxµ)O(w).
+(3.14)
+There is no boundary part in this expression as we do not introduce any non-
+trivial degrees of freedom at the boundary. We can create a line stress-energy
+operator by stretching the integration surface HS2 to a cylinder in a way that
+is shown in Fig. 3. This allows to act with the holomorphic translations also on
+=
+O1(x)
+O2(x)
+O2(x)
+O1(x)
+Figure 3: Possible codimension one surfaces over which Tzν is integrated to
+generate holomorphic translations along the boundary.
+line operators by enclosing them with such a cylinder. The integration over this
+tube can be separated into two parts. The integral over dt deﬁnes the integrated
+stress-energy tensor,
+T int
+zz =
+� L
+0
+dt Tzz,
+T int
+zz =
+� L
+0
+dt Tzz,
+(3.15)
+which behaves as a 2D stress tensor generating the holomorphic translations.
+The remaining integration over the contour in the boundary plane is reminiscent
+of the 2D CFT setup. In fact, it was shown in [BST19] that such integrated 3D
+currents (the stress tensor, the R-current and the supercurrents) ﬁt precisely in
+the 2D N = (0, 2) R-multiplet. The presence of this multiplet automatically
+17
+
+implies existence of the stress-energy tensor in the cohomology [Ded15]. In the
+IR regime, as t collapses, T int of course becomes the 2D stress tensor, and the
+2D N = (0, 2) arguments are applicable directly. In either case, we see again
+that the interval chiral algebra has the stress-energy tensor that follows from
+the integrated currents.
+Outside of critical levels, the boundary algebras are VOAs and have well-
+deﬁned Sugawara stress tensors. Physically, we expect that the interval stress-
+energy tensor becomes the sum of T sug
+ℓ
+and T sug
+r
+as an element in the 2D chiral
+algebra. It is clear that they act in the same way on the boundary operators.
+It indeed turns out to be true as we will argue in the next section using the 2D
+perspective.
+3.2
+The non-perturbative corrections
+What are the possible non-perturbative corrections to the above? One comes
+from the boundary monopole operators discussed in [Bul+16; DGP18; CDG20].
+Another possibility is the vortex line connecting the two boundaries. General
+gauge vortices discussed in [KWY13; DOP14; HS21] are characterized by a
+singular gauge background A = b dϕ close to the vortex locus, where ϕ is an
+angular coordinate in the plane orthogonal to the line, and b is from the Cartan
+subalgebra. This deﬁnes a line defect that is local in the plane orthogonal to
+the vortex, at least away from the boundary, because gauge invariant objects
+do not feel the gauge holonomy. This still holds near the Neumann boundary,
+where the gauge symmetry is unbroken. However, if such a vortex ends at the
+Dirichlet boundary, it creates a non-trivial monodromy e2πib for the boundary
+global symmetry. Hence for generic b, it is not a local operator from the 2D
+boundary point of view as it can be detected far away from the insertion point.
+On the interval with the Dirichlet boundaries, such vortices do not lead to local
+operators in the IR, they become the twist ﬁelds that are not included in the
+VOA.
+A less general possibility is a vortex characterised by a nontrivial background
+A = b dϕ, yet its monodromy is trivial, e2πib = 1. The latter means that b is
+a co-weight (in fact, a co-character, because the gauge symmetry forces us to
+consider the Weyl group orbit of b). Thus such a vortex has its magnetic charge
+labeled by a cocharacter of G, or a subgroup U(1) �→ G taken up to conjugation.
+This is the same as magnetic charges of the monopoles, and we can think of the
+vortex as an inﬁnitesimal tube of magnetic ﬂux, with the same amount of ﬂux as
+created by the charge-b magnetic monopole. This also suggests that monopoles
+can be located at the endpoints or at the junctions of vortices.
+Such vortex lines, however, are not expected to be independent line operators
+in the IR, at least for a non-zero CS level there. When k > h∨, our 3D theory
+becomes the level k − h∨ CS theory at large distances. It has been argued in
+[MS89] that such vortex lines are equivalent to Wilson lines in a CS theory
+(for the abelian case, see the argument in [KWY13].) Thinking of the CS level
+as a pairing K : Γ × Γ → Z, where Γ ⊂ t is the co-weight lattice of G, the
+representation of the Wilson line is determined precisely by the weight K(·, b).
+This is also consistent with the well-known fact (at least in the abelian case
+[KS11]) that in the presence of CS level, monopoles develop electric charges,
+and so Wilson lines can end on them. In particular, [KS11] used this to argue
+that the Wilson lines whose charges diﬀer by a multiple of k are isomorphic,
+18
+
+thus showing that the ﬁnite spectrum of Wilson lines in a CS theory is a non-
+perturbative eﬀect manifested via the monopoles. Note that all the statements
+we referred to here are about the non-SUSY CS theory, but they extend verbatim
+to the half-BPS lines in the N = 2 case.
+To apply these observations to the interval theory, it is convenient to assume
+that the interval is long enough, such that we ﬂow to the CS ﬁrst and only then
+to 2D. At least for k > h∨ this appears to be a harmless assumption, since
+SUSY suggests that the BPS sector is not sensitive to the interval length, and
+the IR physics is also more straightforward in this case [Bas+18].
+It is therefore natural to conjecture that the non-perturbative eﬀects on the
+interval with Dirichlet boundaries are captured by the monopoles. In the bulk
+they ensure that there are only ﬁnitely many inequivalent Wilson lines, and the
+boundary monopoles modify the boundary VAs. The details depend strongly
+on whether G is abelian or non-abelian.
+In the non-abelian case, the monopoles at the level k−h∨ boundary, accord-
+ing to the conjecture in [CDG20], turn the perturbative aﬃne VOA Vk−h∨(g)
+into its simple quotient Lk−h∨(g). The ﬁnitely many bulk Wilson lines cor-
+respond to the integrable representations of the latter.
+As for the bound-
+ary monopoles at the opposite end, it seems unlikely that they can modify
+V−k−h∨(g), which is already simple. The total nonperturbative interval alge-
+bra in this case appears to be some modiﬁcation of the CDO that contains
+Lk−h∨(g) rather than Vk−h∨(g). We do not know its structure yet, and will
+explore it elsewhere.
+The abelian case will be studied in the next sections, where we consider
+G = U (1) in detail. It is a little bit diﬀerent as there is no abelian WZ term
+in 2D, and the level is encoded in the periodicity of the compact boson. The
+monopole corrections extend the boundary aﬃne u(1) to the lattice VOA, also
+known as the abelian WZW. The non-isomorphic Wilson lines correspond to
+the ﬁnite set of modules of the lattice VOA.
+3.3
+U(1)
+We restrict k to lie in 2Z and consider the k ̸= 0 case ﬁrst. Bℓ, Br,
+�
+At dt
+are the possible candidates for elements of the perturbative interval VOA. The
+analysis of the OPE of B’s still holds, and B’s on the left and right boundaries
+commute with each other (have the regular OPE). So, the full set of OPEs again:
+Br (z) Br (w) ∼
+k
+(z − w)2 ,
+Bℓ (z) Bℓ (w) ∼
+−k
+(z − w)2 ,
+Br(z)Bℓ(w) ∼ 0.
+(3.16)
+Surprisingly, there is also one relation connecting the left and right B’s to the
+Wilson line, which follows from the following transformation:
+Q
+�
+A∗
+t dt = Br − Bℓ − k
+2π ∂z
+�
+At dt .
+(3.17)
+Thus, we see that the derivatives of
+�
+A are not independent operators in the
+Q-cohomology. We will encounter similar phenomena when we consider sin-
+gular vectors for aﬃne algebras on the boundaries later. We can choose any
+19
+
+two operators out of {Br, Bℓ, ∂
+�
+A} as the independent generators, and to be
+consistent with the previous section, we take {Bℓ,r}. The stress-energy tensor
+is also included, but for k ̸= 0 it should be expressed in terms of Bℓ,r as we
+discussed before. In this particular case, T =
+1
+2kB2
+r −
+1
+2kB2
+ℓ .
+We also expect that this perturbative algebra is extended by the boundary
+monopole operators [DGP18]. The boundary monopole Mp is obtained from
+the usual monopole by cutting it in half and restricting to a half-space in such
+a way that the integral over the half sphere is
+�
+HS2 F = 2πp,
+p ∈ Z.
+(3.18)
+Due to the CS term, the monopole operator Mp develops an electric charge,
+as was mentioned before. Hence this operator can only exist by itself on the
+boundary where its electric charge is global. Under the global boundary U(1)
+action by eiα it transforms as:
+Mp → e−ipkαMp.
+(3.19)
+To insert it in the bulk, we need to consider a composite operator with a Wilson
+line attached to a monopole eikp
+� t
+0 AMp(t) to cancel the anomalous transfor-
+mation. In fact, we can pull a boundary monopole Mp oﬀ the boundary while
+extending a Wilson line of charge kp between the monopole and the boundary
+to respect gauge invariance, as shown in Fig. 4. Recall that the twisted the-
+Mp
+Wkp
+⇐⇒
+Mp
+Figure 4: The bulk monopole Mp is connected by a Wilson line of charge kp to
+the Dirichlet boundary. In the Q-cohomology, due to the topological invariance
+in the t direction, this is equivalent to the boundary monopole Mp.
+ory is topological in the t direction [CDG20], meaning that the t translations
+are Q-exact. Thus the length of the Wilson line in Fig. 4 is irrelevant, and
+pulling a monopole oﬀ the boundary is an identity operation in the cohomology.
+Using this observation, we can easily ﬁnd the OPE of a boundary monopole
+Mp with the boundary current B. Pulling Mp far away from the boundary, it
+can no longer contribute to such an OPE. Essentially, for the purpose of com-
+puting OPEs with the boundary operators (in the cohomology), the boundary
+monopole Mp is equivalent to the Wilson line of charge kp ending at the bound-
+ary. We have already computed the OPE of B with the Wilson line in earlier
+sections, so the answer follows immediately. Recalling that there is also a sec-
+ond boundary (and operators on the opposite boundaries have regular OPE),
+we thus ﬁnd:
+Bℓ(z)Mp(0) ∼ kp
+z Mp(0),
+Br(z)Mp(0) ∼ 0,
+(3.20)
+20
+
+for the monopole on the left boundary. A more semi-classical way to derive this
+is by computing the on-shell value of B in the twisted formalism. One can easily
+check that the monopole singularity implies B ∼ kp
+z (where the factors of 2π
+were scaled away). Similar results hold for monopoles on the right boundary.
+In fact, repeating our argument, a monopole on the right boundary is equiva-
+lent to the same monopole on the left boundary connected by the Wilson line to
+the right boundary (and vice versa), see Fig. 5. Thus, one can express the right
+monopoles as M ′
+p(t = L) = e−ikp
+� L
+0 AMp(t = 0), and they are not independent
+generators.
+Via the semiclassical analysis of the monopole operator, one can show that
+its OPE with the Wilson line is
+Mp(z)eiq
+� L
+0 A(0,t) ∼ zqp : Mp(z)eiq
+� L
+0 A(0,t) :,
+(3.21)
+where :: means the normal ordering. From this equation we see that the normal
+ordering for e−ikp
+� L
+0 AMp(t = 0) is not required: The leading OPE term scales
+as zp2. To compute the missing Mp1(z)Mp2(0) OPE, we again use the trick of
+replacing the left boundary monopole Mp2 by the Wilson line attached to the
+right monopole eikp2
+� L
+0 AtdtM ′
+p2,
+Mp1(z, 0)Mp2(0, 0) ∼ Mp1(z, 0)eikp2
+� L
+0 At(0,t)dtM ′
+p2(0, L).
+(3.22)
+In this equation, the monopole M ′
+p2 is separated in the t direction from
+the rest of the operators. Hence the singular terms only come from the OPE
+between the Wilson line and the monopole Mp1. This results in the following:
+Mp1(z, 0)Mp2(0, 0) ∼ zkp1p2 : Mp1(z, 0)eikp2
+� L
+0 At(0,t)dt : M ′
+p2(0, L).
+(3.23)
+It remains to bring M ′
+p2 back to the left boundary, colliding with it along the
+t direction, which produces no further singularities due to the topological in-
+variance. Finally, we observe that the magnetic charges are additive under the
+collision, and conclude:
+Mp1(z)Mp2(0) ∼ zkp1p2 : Mp1(z)Mp2(0) := zkp1p2Mp1+p2(0) + . . . .
+(3.24)
+The full set of strong generators can be chosen to be
+�
+Bℓ,r, eip
+�
+A, Mp|p ∈ Z
+�
+.
+(3.25)
+Looking closer at (3.16), (3.20) and (3.24), we recognize the rank one lattice
+VOA setting, with the compact boson of radius
+√
+k. We can extend the U (1)k
+current B by the monopoles (vertex operators) Mp to the Z[
+√
+k] lattice VOA,
+also called the U(1)k WZW. This can be done individually on the left and right
+boundaries, giving the abelian WZWk and WZW−k, respectively.
+Then the
+number of Wilson lines is rendered ﬁnite, and the algebra is generated as an
+extension of
+WZWk ⊗ WZW−k
+(3.26)
+by the bimodules corresponding to the Wilson lines e−i k−2
+2
+�
+A, . . . , ei k
+2
+�
+A. In-
+deed, it is consistent with the limit of the large interval. The theory in this limit
+reduces to the Chern-Simons at level k in the IR, which only admits a ﬁnite
+21
+
+number (k, to be more precise) of Wilson lines in the bulk, and supports WZW
+at the boundaries.
+Now, let us turn to the k = 0 case. One can easily see that we no longer
+have two separate operators Bℓ,r, as B is in the bulk cohomology. Thus, we can
+pull it oﬀ the boundary and bring it all the way to the opposite one without
+changing the cohomology class. Technically, this follows from the relation (3.17)
+involving the descendant line of B. Monopole operators no longer have electric
+charge, so they can be freely moved into the bulk, and they commute (have
+regular OPE) with all local operators. In other words, monopoles are naturally
+elements of the bulk VA V. The stress-energy tensor is no longer a sum of the
+two boundary terms but an independent operator. So, the algebra ceases to
+be generated as a bimodule of the left and right algebras, and we propose the
+following set of strong generators:
+�
+B, T, ein
+�
+A, Mn|n ∈ Z
+�
+.
+(3.27)
+4
+2D Perspective or βγ System
+The IR limit of a 3D N = 2 YM-CS theory on a slab is in general controlled
+by the dimensionless parameter γ = Le2
+3d.
+When L ≫
+1
+e2
+3d or γ ≫ 1, the
+bulk ﬁrst ﬂows as a 3D theory to the Chern-Simons TFT with keﬀ = k − h∨
+(for k ≥ h∨). The boundaries ﬂow to some 2D relative CFTs matching the CS
+boundary anomalies. The interval VOA in this limit was studied in the previous
+section. We do expect, however, that the answer does not depend on γ, at least
+for k > h∨ (when the SUSY is unbroken).
+In the opposite limit γ ≪ 1, the system behaves as a 2D sigma model. It was
+shown in [DL22] that in this regime, the theory is described by an N = (0, 2)
+NLSM into the complexiﬁed group GC ≈ T ∗G at level k. The 2D coupling is
+related to the 3D coupling as
+L
+e2
+3d =
+1
+e2
+2d . The gauge degrees of freedom are
+integrated out, except for the complex Wilson line along the interval, which is
+left as an eﬀective degree of freedom. Its compact part is valued in G, and
+the vector multiplet scalars lie in the cotangent space. The Chern-Simons term
+reduces to a non-trivial B-ﬁeld, or the Wess-Zumino term, necessary for anomaly
+matching. In this section, we explore our problem from such a 2D viewpoint,
+as well as study connections between the 3D and 2D setups.
+As reviewed in Section 2, the perturbative chiral algebra of 2D N = (0, 2)
+NLSM into the target X is captured by the βγ system into X [Wit07; Nek05].
+More precisely, it is given by the cohomology of a sheaf of βγ systems on X,
+also called the sheaf of chiral diﬀerential operators (CDO) [GMS99; GMS01;
+GMS04].
+It is conveniently computed using the Čech cohomology, and the
+resulting vector space with the VA structure on it is denoted Dk[X], where
+k is the B-ﬁeld.
+In our case, the target is a simple complex group GC, so
+k ∈ C = H3(GC, C). This parameter is the well-known modulus of the CDO
+valued in H1(X, Ω2,cl(X)), which in general is not quantized, however, in our
+models k is an integer originating as the CS level in 3D. Since GC has a trivial
+tangent bundle, c1(GC) = 0 and p1(GC) = 0, so the sigma model anomalies
+[MN85] vanish. As reviewed before, c1(GC) = 0 implies that Dk[GC] is a VOA,
+i.e., it has a well-deﬁned Virasoro element.
+22
+
+Notably, Dk[GC] containes the aﬃne sub-VOAs Vk−h∨(g) and V−k−h∨(g)
+corresponding to the left and right G-actions on GC [GMS01]. These clearly
+originate as the perturbative baoudnary VOAs in 3D. Additionally, Dk[GC]
+contains holomorphic functions on the group (functions of γ in the βγ lan-
+guage).
+These originate from the Wilson lines stretched across the interval.
+For generic k ∈ C \ Q, functions on the group together with the pair of aﬃne
+VOAs V±k−h∨(g) generate the whole Dk[GC]. For k ∈ Q this is no longer true,
+and Dk[GC] as a Vk−h∨(g) ⊗ V−k−h∨(g)–bimodule has a very intricate structure
+[Zhu08]. Nonetheless, Dk[GC] remains a simple VOA for all values of k.
+Below we will ﬁnd a full non-perturbative answer in the abelian case and
+comment on what is known in the non-abelian case. By comparing the 3D and
+2D answers when possible, we will veryfy that the interpolation between γ ≫ 1
+and γ ≪ 1 determines an isomorphism of the respective chiral algebras:
+H• �
+Ops3d, Q+
+� ∼
+−→ H• �
+Ops2d, Q
+2d
++
+�
+.
+(4.1)
+4.1
+U(1)
+For G = U(1), the target space of our model is U (1)C ∼= C∗, which is a cylinder.
+The N = (0, 2) sigma model into C∗ was considered in [DGP17], and we will
+make contact with it later. For now, let us follow the βγ approach ﬁrst. The
+CDO only have zeroth cohomology in this case, which are the global sections
+of this sheaf forming the perturbative VOA. Then we will identify its non-
+perturbative extension. We work with the even Chern-Simons level k in what
+follows.
+We introduce two cooridnate systems on the cylinder: One is just γ ∈ C∗,
+and another has �γ as a periodic complex boson. Its periodicity is what encodes
+the “level” in the abelian case, descending from the CS level in 3D:
+�γ ∼ �γ + iπ
+√
+2k,
+(4.2)
+where the conventions are adjusted to match [Wit07]. The relation between the
+two is, naturally,
+γ = e
+√
+2
+k �γ.
+(4.3)
+The quantum transformation between β and �β is
+�β =
+�
+2
+k
+�
+βγ − 1
+2γ−1∂γ
+�
+,
+β =
+�
+k
+2
+�
+�βe−√
+2
+k �γ − 1
+k e−√
+2
+k �γ∂�γ
+�
+.
+(4.4)
+Let us deﬁne two currents:
+Jℓ =
+1
+√
+2
+�
+�β + ∂�γ
+�
+,
+Jr =
+1
+√
+2
+�
+�β − ∂�γ
+�
+,
+(4.5)
+where �γ ∝
+�
+t A is a (log of a) Wilson line from the 3D perspective, and the
+diﬀerence is
+√
+2∂�γ, which is proportional to ∂
+�
+t A as in 3.17. One can easily
+see that these operators commute and have the following OPEs:
+Jℓ(z)Jℓ(0) ∼ −1
+z2 ,
+Jr(z)Jr(0) ∼ 1
+z2 .
+(4.6)
+23
+
+We observe that they only diﬀer from the 3D boundary currents Bℓ,r by the
+normalization factor
+√
+k. We ﬁnd such conventions useful for this section. The
+stress-energy tensor is −�β∂�γ and the central charge is equal to 2. The stress-
+energy tensor does not require any modiﬁcations. The single-valued operator
+corresponding to the charge p Wilson line is Wp = γp. The conformal dimensions
+of Wp can be easily found to be equal to zero.
+The operators β and γp, p ∈ Z, are already global sections of the CDO, and
+they generate the full perturbative VOA, which is most concisely described as
+the C∗-valued βγ system. Note that this implies that γ can be inverted. In
+practice, it is convenient to do so by inverting the zero mode of γ only:
+γ−1(z) =
+1
+γ0 + ∆γ = 1
+γ0
+∞
+�
+n=1
+(−1)n �
+γ−1
+0 ∆γ
+�n ,
+(4.7)
+where
+∆γ =
+�
+n∈Z\0
+γn
+zn .
+(4.8)
+So, what are the non-perturbative eﬀects that we are missing here?
+3D
+physics suggests that they are boundary monopoles. Imagine inserting an im-
+properly quantized monopole on the boundary with
+�
+HS2 F = 2πα at the origin
+z = 0. We know that γ is related to the connection γ ∝ ei
+�
+t A. Now we can
+consider moving γ around the insertion of this monopole, as in γ
+�
+eiφz
+�
+, and let
+us track the phase that is acquired in the process. It is clear that the Wilson
+line sweeps the entire magnetic ﬂux of the monopole in the end, and we obtain:
+γ
+�
+e2πiz
+�
+= ei
+�
+M F γ (z) = e2iπαγ (z) ,
+(4.9)
+where M is a tube stretched between the two boundaries.
+Thus, γ in the
+presence of the improperly quantized monopole behaves as:
+γ =
+�
+n
+γn
+zn−α .
+(4.10)
+Now let us go back to the actual monopole that has α = p ∈ Z. Equation (4.9)
+shows that γ winds p times around the origin as we go once around z = 0, and
+we expect the same shift as in (4.10) with α = p. The actual answer is a little
+bit trickier, but this gives us a good starting point.
+An operator that “shifts the vacuum” in the βγ system by p units will be
+called Mp. The module that it creates is known as the spectrally ���owed module.6
+We can look at the Hilbert space interpretation of these modules. If we place
+an Mp operator at the origin, then the mode expansions of β and γ take the
+following form:
+β =
+�
+n
+βn
+zn+p+1 ,
+γ =
+�
+n
+γn
+zn−p .
+(4.11)
+Here the modes obey the usual commutation relations:
+[βn, γm] = δn+m,0,
+(4.12)
+6In superstrings such operators are often called the picture changing operators.
+24
+
+and the lowest state |p⟩ corresponding to Mp via the state-operator map is
+deﬁned by
+γn+1|p⟩ = βn|p⟩ = 0,
+for n ≥ 0.
+(4.13)
+Essentially, we shifted the modes of the vacuum module by p positions and then
+relabeled them.
+The inverse of γ in the presence of Mp is deﬁned in the similar manner:
+γ−1 (z) =
+1
+γ0zp + ∆γ = z−p
+γ0
+∞
+�
+n=0
+(−1)n �
+zpγ−1
+0 ∆γ
+�n .
+(4.14)
+Let us compute charges and the conformal dimension of Mp. The currents
+Jℓ, Jr in the γ coordinate system are
+�
+Jℓ
+Jr
+�
+=
+�
+�
+�
+1
+√
+k
+�
+βγ + (k−1)
+2
+γ−1∂γ
+�
+,
+1
+√
+k
+�
+βγ + (−k−1)
+2
+γ−1∂γ
+�
+.
+(4.15)
+The following product can be computed using the mode expansions:
+β (z) γ (w) |p⟩ = −
+�w
+z
+�p
+1
+z − w |p⟩ .
+(4.16)
+Subtracting the singularity
+1
+z−w, we are getting:
+: βγ : |p⟩ = p
+z |p⟩ .
+(4.17)
+Let us denote : βγ : as J0 in what follows. The J0 charge of Mp is p, and the
+charges of β and γ are 1 and −1 respectively. The diﬀerence of Jℓ and Jr is
+proportional to γ−1∂γ, which is itself a current measuring the winding number.
+In the presence of Mp, the expression γ−1∂γ can be computed directly from the
+deﬁnition:
+γ−1∂γ = p
+z + . . .
+(4.18)
+Thus the winding charge is equal to p for Mp and 0 for β and γ. The U (1)ℓ,r
+charges for Mp are then computed from (4.15) and are
+p
+√
+k( 1+k
+2 , 1−k
+2 ).
+The stress energy tensor written in terms of βγ is −β∂γ + 1
+2 (log γ)′′ as the
+holomorphic top form in this case is w ∝ dγ
+γ . Moreover, it can be written as
+Tβγ = 1
+2J2
+r − 1
+2J2
+ℓ , which is indeed what was expected. Computing the conformal
+dimensions of Mp requires again the normal ordering prescription:
+lim
+z→w
+�
+−β (z) ∂γ (w) |p⟩ −
+1
+(z − w)2 |p⟩
+�
+=
+�p − p2
+2w2
++ β−1γ0
+w
++ . . .
+�
+|p⟩ ,
+1
+2(log γ)′′ |p⟩ =
+�
+− 1
+w2
+p
+2 + . . .
+�
+|p⟩ ,
+(4.19)
+where we again subtracted all singular terms. It follows that the dimension of
+the operator Mp is − p2
+2 .
+25
+
+≈
+Figure 5: Moving a monopole of magnetic charge p from one boundary to an-
+other creates a Wilson line of electric charge kp.
+Now that we understand the operators Mp fairly well, we need to iden-
+tify precisely those that should be added to the βγ system to obtain our non-
+perturbative VOA. They correspond to the boundary monopoles in 3D. For that
+we need to ﬁnd operators that are only charged under the left or right aﬃne
+currents Jℓ,r. If we had k = 1 for a moment, the left monopole is just the ﬂowed
+module Mp, as can be seen from its charges (Jℓ, Jr) = (p, 0). The mathematical
+perspective on such modules was given in [RW14; AW22], and in their notations,
+Mp generates σpW+
+0 , where σ denotes the spectral ﬂow automorphism.
+To tackle the general case, we just need to take a composite operator that
+has the correct charge. The Mp by itself is charged as
+p
+√
+k
+� k+1
+2 , −k+1
+2
+�
+. So, if
+we take M ′
+p = (γ
+p(k−1)
+2
+0
+|p⟩)(z), we get an operator with the charges p(
+√
+k, 0),
+as required, and the conformal dimension ∆M ′p = p2 k
+2.
+This M ′
+p generates
+the spectrally ﬂowed module σpW p(k−1)
+2
+in the notations of [AW22], which is
+just σpW+
+0 for p(k − 1) even and σpW 1
+2 for p(k − 1) odd.7
+In order to get
+monopoles on the other boundary, we need to consider the composite operator
+with the Wilson line WkM ′
+1 (Fig.
+5), which has charges p(0, −
+√
+k) and the
+same conformal dimension. Thus we do not need a separate generator for that
+monopole. The generator content of our algebra matches the eqn. (3.25).
+Let us summarize the result that we have obtained so far for the full non-
+perturbative VOA. It is given by the C∗-valued βγ VOA (i.e., γ is invertible,)
+extended8 by its modules σ2pW+
+0 and σ2p+1W 1
+2 for all p ∈ Z, in the notations
+of [RW14; AW22].
+One can also easily compute a character over the full chiral algebra:
+Z = TrH(qL0− c
+24 xJℓyJr) =
+1
+η(q)2
+�
+n,m∈Z
+qnmx
+n
+√
+k +m
+√
+k
+2 y
+n
+√
+k −m
+√
+k
+2 ,
+(4.20)
+which is clearly not a meromorphic function and can only be understood as a
+formal power series. The obvious non-convergence of the character is expected,
+as characters on the boundary with the positive level are convergent when |q| < 1
+7Since we consider even k, this is determined by the parity of p only.
+8In fact, it is slightly redundant to say that γ is invertible. The module W+
+0 of the usual βγ
+system coincides with the vacuum module of such a C∗-valued βγ system. Thus the extension
+by W+
+0 automatically inverts γ.
+26
+
+[DGP18], and on the opposite boundary the convergence is at |q| > 1. This
+trace, of course, can be reinterpreted from the 3D perspective as an index on
+the interval (see also [SY20]):
+ZI×T 2 = TrH(−1)F e−2πRH �
+e2πiJizi,
+(4.21)
+where Ji are generators of the maximal tori of the boundary symmetries.
+The k = 0 situation is diﬀerent, and does not appear to be particularly
+interesting and well behaved from the 2D viewpoint, so we skip it.
+Dual boson
+One can also calculate the same algebra directly in the C∗ sigma model, without
+going to the βγ-description. The calculation was ﬁrst done in [DGP17]. Let us
+connect it with our formulas for completeness. Let σ be the radial coordinate
+and X = XL(z) + XR(z) be an angular coordinate on C∗. Then �β and �γ are
+related to the free boson as in Sec. 2.4:
+∂�γ = ∂σ + i∂X,
+�β = ∂σ − i∂X.
+(4.22)
+In the γ coordinate system one gets from (4.4):
+γ = e
+√
+2
+√
+k (σ+i(XL+XR)),
+β =
+√
+k
+√
+2
+�
+∂(σ − iX)e−
+√
+2
+√
+k (σ+iX) − 1
+k e−
+√
+2
+√
+k (σ+iX)∂(σ + iX)
+�
+= −
+√
+k
+√
+2
+�
+(1 − 1
+k )∂σ + i(1 + 1
+k )∂X
+�
+e−
+√
+2
+√
+k (σ+iX).
+(4.23)
+Redeﬁne both X and σ by
+√
+2 to match the notations of [DGP17], so we ﬁnd:
+γ = e
+1
+√
+k (σ+i(XL+XR)),
+β = −
+√
+k
+2
+�
+(1 − 1
+k )∂σ + i(1 + 1
+k )∂X
+�
+e−
+1
+√
+k (σ+iX).
+(4.24)
+The BPS vertex operators in this description take the form:
+eikℓXL+kr(σ+iXR),
+(4.25)
+with
+(kℓ, kr) =
+� n
+R + wR
+2 , n
+R − wR
+2
+�
+,
+n, w ∈ Z.
+(4.26)
+The radius R is related to the Chern-Simons level as R2 = k.
+We can see
+that these operators form the same lattice as we found in the βγ system with
+the special shifted modules included.
+In particular, γn here is the vertex
+operator with kℓ = kr =
+n
+R.
+At the same time the left monopoles have
+(kℓ, kr) = p(
+√
+k, 0), which means n = wk/2 = pk/2, and the right monopoles
+have (kℓ, kr) = p(0, −
+√
+k), which corresponds to n = −wk/2 = −pk/2.
+27
+
+No Mercy
+This ﬁnal presentation allows us to identify the nonperturbative VOA even
+more explicitly in terms of the known VOAs. Namely, let us denote the vertex
+operator representing the Q-cohomology class with the momentum and winding
+charges (n, w) ∈ Z2 by Vn,w(z). Then computing the OPE of vertex operators
+deﬁned in (4.25), we easily ﬁnd the following:
+Vn1,w1(z)Vn2,w2(0) ∼ zn1w2+n2w1 : Vn1,w1(z)Vn2,w2(0) : ,
+(4.27)
+which identiﬁes our VOA as a lattice VOA for the smallest Narain lattice [Nar86;
+NSW87], namely Z2 ⊂ R2 with the scalar product:
+(n1, w1) ◦ (n2, w2) = n1w2 + n2w1.
+(4.28)
+Note that the two U(1) currents can be obtained as V−1,0∂V1,0 and V0,−1∂V0,1:
+J1 = 1
+R (i∂XL + i∂XR + ∂σ) ,
+J2 = R
+2 (i∂XL − i∂XR − ∂σ) .
+(4.29)
+Also notice a curious fact: While many of the steps in our analysis involved
+the CS level, the ﬁnal answer does not depend on it. This in fact serves as a
+consistency check for the following reason. From the N = (0, 2) point of view,
+the compact boson radius
+√
+k only enters the Kähler potential, thus it cannot
+aﬀect the chiral algebra structure.
+Together with the other two results in the earlier sections, we thus ﬁnd three
+presentations for the nonperturbative VOA in the abelian case:
+Narain lattice VOA
+of rank two
+∼=
+βγ extended by
+σ2pW+
+0 and σ2p+1W 1
+2
+∼=
+WZWk ⊗ WZW−k
+extended by bimodules
+(4.30)
+4.2
+SU(2)
+Let us now turn to a less-trivial example and discuss G = SU(2), that is
+GC =SL(2, C).
+The computation is more involved in this case, as we need
+to deﬁne everything on patches and consider a non-trivial gluing. We will ﬁrst
+ﬁnd the global theory and discuss the moduli space, and then will turn to the
+non-trivial modules for boundary VOAs.
+SL(2, C) can be covered by two patches, the coordinates on which will be
+denoted as γi and �γi :
+�a
+b
+c
+d
+�
+, ad − bc = 1
+a̸=0
+�−−→
+�γ1
+γ2
+γ3
+�
+=
+�a
+b
+c
+�
+∈ C3 \ {γ1 = 0},
+�
+a
+b
+c
+d
+�
+, ad − bc = 1
+b̸=0
+�−−→
+�
+�γ1
+�γ2
+�γ3
+�
+=
+�
+a
+b
+d
+�
+∈ C3 \ {�γ2 = 0}.
+28
+
+Thus, the coordinate transformations have the following form:
+γ1 = �γ1, γ2 = �γ2, γ3 = �γ1�γ3 − 1
+�γ2
+or
+�γ1 = γ1, �γ2 = γ2, �γ3 = 1 + γ2γ3
+γ1
+.
+(4.31)
+The Jacobian matrix and its inverse can be computed to be
+gi
+j ≡ ∂γi
+∂�γj =
+�
+�
+1
+0
+0
+0
+1
+0
+1+γ2γ3
+γ1γ2
+− γ3
+γ2
+γ1
+γ2
+�
+� ,
+∂�γi
+∂γj =
+�
+�
+1
+0
+0
+0
+1
+0
+− 1+γ2γ3
+(γ1)2
+γ3
+γ1
+γ2
+γ1
+�
+� .
+∂jgi
+a∂igj
+b = ∂3g3
+a∂3g3
+b =
+�
+�
+1
+(γ1)2
+−
+1
+γ1γ2
+0
+−
+1
+γ1γ2
+1
+(γ2)2
+0
+0
+0
+0
+�
+� .
+As we mentioned earlier, for each left- and right-invariant vector ﬁelds there
+exists a corresponding VOA sub-algebra [GMS01] that saturates boundary anoma-
+lies. The boundary with the negative anomaly usually corresponds to a rela-
+tive CFT with the anti-holomorphic dependence on the coordinates, and it
+transforms into a chiral algebra with the negative level after passing to the
+Q-cohomology: kcoh = kℓ − kr [Wit07].
+Let us ﬁnd these algebras and the CDO sections explicitly. Note that there is
+nothing left except global sections as the manifold is Stein and does not support
+geometric objects that can form a higher degree cohomology. So, the only ﬁelds
+that can contribute are in H0(SL(2), �A).
+Let us ﬁrst write out all classical vector ﬁelds. To do this, we will use the
+following well-known form of basis at the identity point of the group:
+e =
+�0
+1
+0
+0
+�
+f =
+�0
+0
+1
+0
+�
+h =
+�1
+0
+0
+−1
+�
+(4.32)
+and carry it over the whole manifold by L∗
+gV µ∂µ|1 = (gV )µ∂µ|g.
+Local sections, corresponding to the left-invariant vector ﬁelds, then have
+the following form in both patches:
+eℓ = γ1β2
+�eℓ = �γ1 �β2 + �γ−1
+2 (�γ1�γ3 − 1)�β3
+fℓ = γ2β1 + γ−1
+1 (1 + γ2γ3)β3
+�fℓ = �γ2 �β1
+hℓ = γ1β1 − γ2β2 + γ3β3
+�hℓ = �γ1 �β1 − �γ2 �β2 − �γ3 �β3.
+(4.33)
+Local sections corresponding to the right-invariant vector ﬁelds in both patches
+are
+er = γ1β3
+�er = �γ2 �β3
+fr = γ3β1 + γ−1
+1 (1 + γ2γ3)β2
+�fr = �γ−1
+2 (�γ1�γ3 − 1)�β1 + �γ3 �β2
+hr = γ1β1 + γ2β2 − γ3β3
+�hr = �γ1 �β1 + �γ2 �β2 − �γ3 �β3.
+(4.34)
+The normal ordering for these ﬁelds is chosen exactly in the way they are written,
+abc =
+def: a : bc ::, and will be omitted from this point on to unclutter notations.
+29
+
+By using (2.12), one can easily obtain the following transformation formulas:
+�β1 = β1 + β3
+�γ3
+γ1 +
+1
+γ1γ2
+�
+− 1
+2
+�
+1
+(γ1)2 ∂γ1 −
+1
+γ1γ2 ∂γ2
+�
+,
+�β2 = β2 − β3
+γ3
+γ2 − 1
+2
+�
+−
+1
+γ1γ2 ∂γ1 +
+1
+(γ2)2 ∂γ2
+�
+,
+�β3 = β3
+γ1
+γ2 .
+(4.35)
+Note that here we choose to set the moduli parameter µab to zero. Now we will
+combine (4.31) and (4.35) to ﬁnd the corrected version of these ﬁelds. After
+either doing a tedious calculation or applying the Mathematica tool attached,
+one can obtain the corrected version of the left- and right-invariant vector ﬁelds,
+respectively:
+−Hℓ ≡ hℓ = �hℓ
+−Hr ≡ hr + ∂γ1
+γ1
+= �hr + ∂γ2
+γ2
+Eℓ ≡ eℓ = �eℓ + 1
+2∂
+��γ1
+�γ2
+�
+Er ≡ er = �er
+Fℓ ≡ fℓ + 1
+2∂
+�γ2
+γ1
+�
+= �fℓ
+Fr ≡ fr + 1
+2∂
+�γ3
+γ1
+�
++ ∂γ3
+γ1
+= �fr + 1
+2∂
+��γ3
+�γ2
+�
++ ∂�γ3
+�γ2
+.
+(4.36)
+As one can see, we regrouped terms in the expression, so the sections are actually
+smooth and well-deﬁned on the whole patch. For example, the term ∂γ1
+γ1 would
+have a pole on the second patch, where γ1 can be equal to zero. Thus, it is only
+deﬁned smoothly on the ﬁrst patch.
+The OPEs, as expected, constitute the aﬃne Kac-Moody vertex algebra.
+For example, the left OPEs are:
+Hℓ(z)Eℓ(w) ∼ 2Eℓ(w)
+z − w ,
+Hℓ(z)Hℓ(w) ∼
+−3
+(z − w)2 ,
+Hℓ(z)Fℓ(w) ∼ −2Fℓ(w)
+z − w
+,
+Eℓ(z)Fℓ(w) ∼
+−3/2
+(z − w)2 + Hℓ(w)
+z − w ,
+which make it into V−3/2 (sl(2, C)). From the anomaly inﬂow argument we know
+that the total level should be kℓ + kr = −2h∨ = −4. Thus, the right algebra
+is the aﬃne algebra V−5/2 (sl (2, C)), which we could again check by a direct
+computation.
+Operator products between all the left and right global sections are non-
+singular. The level is deﬁned with respect to the standard bilinear form (, ) =
+(2h∨)−1(, )K, where (, )K is the Killing form. In the case of sl(2, C) it is given
+by (h, h) = 2, (e, f) = (f, e) = 1.
+General k
+To ﬁnd the most general form of this algebra, we need to ﬁnd the moduli
+space of this CDO. By section 2 we know that it is equivalent to ﬁnding
+30
+
+H1 �
+Ω2,cl, SL (2, C)
+�
+. We want to show that
+µ = 2tdγ1 ∧ dγ2
+γ1γ2
+,
+t ∈ C,
+(4.37)
+is the only generator of that cohomology. We show it in three steps. First, we
+use that H1 �
+SL (2, C) , Ω2,cl�
+→ H3
+dR (SL (2, C) , C) is injective (see Appendix
+A). Second, it is known that the 3rd de Rham cohomology for simple Lie groups
+is generated by Tr
+�
+g−1 dg
+�3, i.e., H3
+dR (SL (2, C)) ∼= C. Third, the form above
+is well-deﬁned on U12 = U1 ∩ U2 ∼= C∗ × C∗ × C and has a non-trivial pe-
+riod over a non-contractible cycle on the intersection.
+Thus, it means that
+it represents a non-trivial class in H1(SL(2, C), Ω2,cl).
+So, we showed that
+H1 �
+Ω2,cl, SL (2, C)
+� ∼= H3
+dR (SL (2, C) , C) and 4.37 is the non-trivial element.
+The coeﬃcient t there is thus the only CDO modulus.
+After all these preparations, we can ﬁnally redo the calculations with the
+transformation law shifted by µ (2.12). One gets the following sections:
+−Hℓ ≡ hℓ + tγ′
+1
+γ1
+= �hℓ − tγ′
+2
+γ2
+−Hr ≡ hr + (1 − t)∂γ1
+γ1
+= �hr + (1 − t) ∂γ2
+γ2
+Eℓ ≡ eℓ = tγ′
+1
+γ2
++ �eℓ + 1
+2∂
+�γ1
+γ2
+�
+Er ≡ er = �er
+Fℓ ≡ fℓ + 1
+2∂
+�γ2
+γ1
+�
++ tγ′
+2
+γ1
+= �fℓ
+Fr ≡ fr + 1
+2∂
+�γ3
+γ1
+�
++ (1 − t)∂γ3
+γ1
+= �fr + 1
+2∂
+��γ3
+�γ2
+�
++ (1 − t)∂�γ3
+�γ2
+.
+(4.38)
+Eﬀectively, we observe that introducing the form (4.37) leads to a shift of
+levels kℓ → kℓ − t and kr → kr + t. We will set t = 1
+2 + k for convenience, where
+k now is the actual Chern-Simons level. The levels of the boundary algebras
+are then −2−k and −2+k. The quantization condition is not necessary in this
+approach, but is necessary from the 3D perspective. In this context it means
+that k ∈ Z.
+Thus, the aﬃne algebras of the global left and right G-action
+sections are V−2±k (sl (2, C)). The explicit form of these sections is one of the
+key technical results of this chapter.
+Module Structure
+To reveal the module structure of Dk[SL(2, C)] with respect to V−2±k (sl (2, C)),
+let us consider:
+Eℓ(z)γ1(w) ∼ 0
+Eℓ(z)(−γ2)(w) ∼ γ1(w)
+z − w
+Hℓ(z)γ1(w) ∼ γ1(w)
+z − w
+Hℓ(z)(−γ2)(w) ∼ γ2(w)
+z − w
+Fℓ(z)γ1(w) ∼ −γ2(w)
+z − w
+Fℓ(z)(−γ2)(w) ∼ 0.
+(4.39)
+Thus, as before, γ’s generate modules for our boundary algebras and are iden-
+tiﬁed with the Wilson lines in the 3D description. One ﬁnds that we quotient
+out the singular vector of the underlying sl2 algebra, i.e. (F 0
+ℓ )2γ1 = 0.
+31
+
+One also ﬁnds that the vectors (γ1)0 |0⟩, −(γ2)0 |0⟩, and all vectors obtained
+from them by the action of the negative modes of Eℓ, Hℓ, and Fℓ span a module
+over the left current algebra, with the vector (γ1)0 |0⟩ being the highest weight
+vector of weight 19. There is an isomorphic module over this subalgebra, “gener-
+ated” by γ3 and γ−1
+1 (1+γ2γ3), with the ﬁrst ﬁeld giving the highest vector. One
+also ﬁnds analogous modules over the right current algebra, “generated” by pairs
+of global sections γ1, γ3 and γ2, γ−1
+1 (1+γ2γ3), where again the ﬁrst ﬁeld in each
+pair deﬁnes the highest weight vector of weight 1. Note that these expressions
+indeed deﬁne global sections due to (4.31). All global functions depend only on
+(γ1, γ2, γ3, γ−1
+1 (1 + γ2γ3)) and are modules for zero modes of our currents.
+Let us put these building blocks together and consider the vector space:
+(γ1)0 |0⟩
+(γ2)0 |0⟩
+(γ3)0 |0⟩
+�
+1+γ2γ3
+γ1
+�
+0 |0⟩
+This is (1, 1) representation for sl(2)ℓ ⊗ sl(2)r. We can act on this vector space
+by negative modes of Ja
+ℓ and Ja
+r . The vector (γ1)0 |0⟩ is the highest weight
+vector of weight (1, 1) in the representations of the corresponding �gℓ ⊗ �gr aﬃne
+Kac-Moody algebras. In order to obtain other representation one can act with
+higher powers of γn
+1 on the vacuum and this yields representation (n, n). So,
+the answer at the generic point is
+Dk[SU(2)C] =
+�
+λ∈Z+
+Vλ,−2+k (g) ⊗ Vλ,−2−k (g) ,
+(4.40)
+where again Vλ,−2±k are Weyl modules.
+Two points require important clariﬁcations. First, what happens with the
+stress-energy tensor of the βγ system −βi∂γi?
+It is guaranteed to exist by
+[GMS99] as the canonical bundle on a Lie group is trivial. The holomorphic
+top form can be written as w = dγ1dγ2dγ3
+γ1
+on the ﬁrst patch. Thus, the stress-
+energy tensor gets corrected to
+T (z) = −
+�
+βi∂γi + 1
+2 (log γ1)′′ ,
+(4.41)
+where the correction is a derivative of the coeﬃcient of the holomorphic top
+form. One can show by a direct computation that outside of the critical levels
+of the boundary algebras,
+Tβγ = Tℓ + Tr,
+(4.42)
+where Tℓ,r =
+1
+2(kℓ,r+h∨)
+�
+ef + fe + hh
+2
+�
+are the Sugawara stress-energy tensors.
+This result was expected from the general discussion in 3.
+9We assumed the mathematical notation, where representations of sl2 are labeled by inte-
+gers, not half-integers.
+32
+
+The second point is that the modules that we are considering are reducible
+for the physical values of k ∈ Z.
+Not only that, but those singular vectors
+are singular for both left and right algebras [Zhu08]. To see this, let us set
+the right algebra level kr to be 0. It is an obvious limiting case, but it will
+nevertheless show the important feature that is carried over to other values of
+k. The simplest singular vector for this module can be found to be
+(Er)−1 |0⟩ .
+(4.43)
+We need to ﬁnd the form of this vector in terms of the left-invariant ﬁelds.
+Classically, the vector ﬁelds are related by the following change of basis:
+�
+�
+er
+fr
+hr
+�
+� =
+�
+�
+�
+−γ2
+2
+γ2
+1
+−γ1γ2
+(1+γ2γ3)2
+γ2
+−γ2
+3
+γ3(1+γ2γ3)
+γ1
+2 γ2(1+γ2γ3)
+γ1
+−2γ2γ3
+1 + 2γ2γ3
+�
+�
+�
+�
+�
+eℓ
+fℓ
+hℓ
+�
+� = S · Vℓ
+(4.44)
+Of course, at the quantum level the relation is corrected, and for the e ﬁeld the
+correct answer is found to be
+Er = V i
+ℓ S1i + (−2 + k)(γ1∂γ2 − γ2∂γ1).
+(4.45)
+So, we see that for the special value k = 2, the correction term disappears,
+and the vector Er
+−1 can now be obtained both from the left and right algebras.
+It is actually lying inside the γ2
+1 representation for the left algebra. Thus for
+discrete values of k, diﬀerent modules start to intersect. Moreover, now there is
+no way to obtain this correction term ωf = γ1∂γ2 − γ2∂γ1 from a Wilson line
+by the action of �gℓ ⊗ �gr. Note that this form is actually dual to the f-vector
+ﬁeld < ωf, f >= 1. This phenomenon happens for all singular vectors [Zhu08].
+One could ask what happens when we include monopoles. We do not have
+a deﬁnitive answer, but as mentioned in the introduction and in Section 3.2,
+we have conjectures as to what the answer might look like. One expects to get
+some sort of truncation of the CDO that contains simple quotients L−h∨±k(g)
+rather than V−h∨±k(g). We will look into this issue elsewhere.
+4.3
+Open Questions
+We have already emphasized many times that determining the non-perturbative
+modiﬁcation of Dk[GC] is an interesting problem. It requires, perhaps, improved
+understanding of the non-compact models in 2D, of which our theory is an
+example. The usual arguments with quotienting out singular vectors based on
+the unitarity of the Hilbert space do not work in such theories. But we expect
+that monopoles on the boundary with positive k − h∨ are still required, as in
+the opposite limit γ ≫ 1 this boundary is a relative CFT with a normalizable
+vacuum. The problems lie on the other boundary which has a non-compact
+mode [DL22] that eﬀectively renders our theory non-compact.
+Another intriguing question arises from an alternative UV completion of the
+GC NLSM via the 2D Landau-Ginzburg (LG) model described in [DL22]. The
+simplest example is for G = SU (N). The UV completion is chosen to be the
+N = (0, 2) LG model with the following ﬁeld content:
+33
+
+1. M i
+j are chiral multiplets valued in complex matrices Mat(N, C); Φa
+i is
+a chiral multiplet that is bifundamental under U (k) × G, where i, j ∈
+1, . . . , N and a ∈ 1, . . . , (k = anomaly).
+2. Fermi multiplets Γ and Λj
+b.
+3. Superpotential W = Γ (det M − 1) + µΛj
+aM i
+jΦa
+i .
+This model has the same anomalies as our theory, and the superpotential is
+engineered in such a way that it ﬂows to the SL(N, C) NLSM. It is generally
+believed that LG models do not carry any non-perturbative physics. Thus one
+could hope that the perturbative chiral algebra in this model could provide some
+useful information. It is captured by the βγ systems (V j
+i , M i
+j) (here V is the
+“beta” for M), (Ri
+a, Φa
+i ), and the bc systems (Γ, Γ) and (Λ
+a
+i , Λi
+a). The chiral
+algebra is deﬁned in the cohomology of Q that acts according to:
+QΓ = det M − 1,
+QΛ = MΦ,
+QV j
+i = Γ∂ det M
+∂M i
+j
++ µΛj
+aΦa
+i ,
+QRi
+a = µΛj
+aM i
+j,
+(4.46)
+and by zeros on the rest of ﬁelds.
+All these βγ and bc systems are already
+globally deﬁned, so one simply computes the cohomology of such Q. The answer
+appears to be just Dk[SL(2, C)], which would be interesting to prove. But more
+importantly, this, supposedly exact, answer in the LG model is the same as the
+perturbative VOA we ﬁnd in the interval theory. This suggests that the exact
+non-perturbative physics in these models may depend on the UV completion.
+Other open questions involve applications to the VOA[M4], which requires
+computing the interval reductions of more complicated gauge theories, and
+which we will study in the future works.
+5
+Conclusion
+In this paper we considered the chiral algebra of a 3D N = 2 YM on R2 × [0, L]
+with the N = (0, 2) Dirichlet boundary conditions. The algebra was computed
+both from the 3D and 2D perspectives. We analyzed this protected sector using
+the holomorphic-topological twist of the 3D theory and, among other things,
+the holomorphic twist of the 2D theory.
+From the 3D perspective, the perturbative algebra was found to be an en-
+hancement of two aﬃne vertex algebras living at the boundaries by their bimod-
+ules realized via the Wilson lines. The boundary monopoles seem to modify the
+answer non-perturbatively, on which we proposed some conjectures.
+The two-dimensional system after reduction in the right regime is an N =
+(0, 2) NLSM into GC.
+The compactiﬁcation algebra is the chiral algebra of
+this 2D model, and the beta-gamma system is the main tool to compute its
+perturbative approximation. The global sections corresponding to the left and
+right actions of the group on itself were explicitly found for G = SU(2).
+In the abelian case, we ﬁnd that the spectrally ﬂowed modules of the βγ
+system are required to get the full result for the algebra. Combining the latter
+perspective with the 3D analysis and with the known results on the sigma model
+into C∗, we obtain three diﬀerent presentations of the chiral algebra (also called
+34
+
+the interval VOA) in (4.30). We also saw that the stress-energy tensor is de-
+composed in terms of the Sugawara stress tensors for the boundary symmetries,
+both in the abelian and the non-abelian cases. The answers, when available,
+fully agree between the 2D and 3D calculations. Some puzzles and speculations
+are discussed towards the end and in Section 3.2.
+Acknowledgements
+We beneﬁted from the useful discussions and/or correspondence with: A. Abanov,
+T. Creutzig, T. Dimofte, D. Gaiotto, Z. Komargodski, I. Melnikov, N. Nekrasov,
+W. Niu, M. Roček.
+A
+De Rham cohomology
+In this appendix we will show that H1 �
+SL (2, C) , Ω2,cl�
+→ H3
+dR (SL(2, C)) is in-
+jective. H1 �
+SL (2, C) , Ω2,cl�
+is isomorphic to Zd
+�
+Ω3,0 ⊕ Ω2,1�/dΩ2,0 [Wit07]. There
+is an obvious map from Zd
+�
+Ω3,0 ⊕ Ω2,1�/dΩ2,0 to the third de Rham cohomology
+group H3
+dR(SL(2, C)) given by [α] �→ [α] for any closed 2-form α ∈ Ω3,0 ⊕ Ω2,1.
+Proposition 1. For any [α] ∈ Zd
+�
+Ω3,0 ⊕ Ω2,1�/dΩ2,0 there exists β ∈ Zd
+�
+Ω3,0�
+such that
+[α] = [β],
+(A.1)
+so there is an isomorphism:
+A ≡ Zd
+�
+Ω3,0 ⊕ Ω2,1�
+⧸dΩ2,0 ∼= Zd
+�
+Ω3,0�
+⧸dΩ2,0,
+(A.2)
+where we have made use of a slight abuse of notation, and dΩ2,0 in the last
+quotient should be understood as dΩ2,0 ∩ Ω3,0.
+Proof. A general form from A has the form α+β for some α ∈ Ω3,0 and β ∈ Ω2,1.
+The closeness conditions are
+∂α + ∂β = 0,
+∂β = 0.
+(A.3)
+The second condition says that β ∈ Z∂
+�
+Ω2,1�
+, and using the fact that SL (2, C) is
+a Stein manifold with all positive degree Dolbeault cohomology groups vanishing
+H·,·≥1
+∂
+= 0, one gets that the form β is in fact exact: β = ∂γ for some γ ∈ Ω2,0.
+So, shifting α + β by −dγ, we get the desired representative in Ω3,0.
+■
+Proposition 2. The map from A to H3
+dR deﬁned above is injective.
+Proof. Suppose we have a closed (3,0)-form ω that goes under the map to zero
+in H3
+dR(SL(2, C)), i.e. ω = dα for some α ∈ Ω2,0 ⊕ Ω1,1 ⊕ Ω0,2. Let us represent
+α as α(2,0) + α(1,1) + α(0,2), where each α(p,q) ∈ Ωp,q. Thus,
+ω = dα(2,0) + dα(1,1) + dα(0,2),
+(A.4)
+and as ω ∈ Ω3,0 we ﬁnd that ∂α(0,2) = 0. Recalling that SL(2, C) has trivial
+non-zero Dolbeault cohomology groups as a Stein manifold, one obtains that
+35
+
+α(0,2) = ∂γ(0,1) for some γ(0,1) ∈ Ω0,1.
+It means, however, that dα(0,2) ≡
+∂∂γ(0,1) + ∂
+2γ(0,1) = −∂∂γ(0,1) = ∂β(1,1). Redeﬁning α(1,1),
+ω = dα(2,0) + dα(1,1).
+(A.5)
+Repeating the same argument with α(1,1), we obtain that ω = dα(2,0), meaning
+that it was trivial in A, which proves the statement.
+■
+Let us show that the only generator of H3
+dR(SL(2, C)) (which is Tr(g−1dg)3)
+after mapping to A indeed corresponds to the closed holomorphic (2,0)-form µ
+in H1 �
+SL (2, C) , Ω2,cl�
+from the Eq. (4.37):
+H ≡ Tr
+�
+g−1dg
+�3 = 2dγ1 ∧ dγ2 ∧ dγ3
+γ1
+= 2d�γ1 ∧ d�γ2 ∧ �dγ3
+�γ2
+(A.6)
+Following the general algorithm explicitly constructing the isomorphism between
+H1 �
+SL (2, C) , Ω2,cl�
+and A, we must ﬁnd two (2,0)-forms T1 and T2 in each of
+the open sets U1 and U2, such that H = dT1 in U1 and H = dT2 in U2, and take
+their diﬀerence T12 ≡ T1 − T2 in U1 ∩ U2 to produce the element of A:
+dγ1 ∧ dγ2 ∧ dγ3
+γ1
+= d
+�γ3
+γ1
+dγ1 ∧ dγ2
+�
+,
+(A.7)
+d�γ1 ∧ d�γ2 ∧ �dγ3
+�γ2
+= d
+��γ3
+�γ2
+d�γ1 ∧ d�γ2
+�
+,
+(A.8)
+γ3
+γ1
+dγ1 ∧ dγ2 − �γ3
+�γ2
+d�γ1 ∧ d�γ2 = −dγ1 ∧ dγ2
+γ1γ2
+,
+(A.9)
+which indeed coincides with the Cech cohomology class deﬁned by µ on U1∩U2.
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