Dual Boundary Conditions in 3d SCFT's
JJanuary 31, 2018
Prepared for submission to JHEP
Dual Boundary Conditions in 3d SCFT’s
Tudor Dimofte Davide Gaiotto Natalie M. Paquette Department of Mathematics and Center for Quantum Mathematics and Physics (QMAP), Universityof California, Davis, CA 95616, USA Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, ON N2L 2Y5, Canada Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, CA91125, USA
Abstract:
We propose matching pairs of half-BPS boundary conditions related by IR dual-ities of 3d N = 2 gauge theories. From these matching pairs we construct duality interfaces.We test our proposals by anomaly matching and the computation of supersymmetric indices.Examples include basic abelian dualities, level-rank dualities, and Aharony dualities. CALT-TH-2017-065 a r X i v : . [ h e p - t h ] J a n ontents N = (0 , boundary conditions 6 N b.c. 303.4 3d gauge symmetry: D b.c. and boundary monopoles 323.5 Summary 363.6 Line operators 363.7 Difference equations 38 U (1) + a chiral 424.3 U (1) − + a chiral 464.4 Left vs. right boundary conditions, and CS levels 48 U ( N ) SQCD with N f = N U ( N ) k + N ↔ U ( k ) − k − N level-rank duality 799.2 U ( N ) k + N − Nf ↔ U ( k ) − k − N + Nf with N f fundamentals 819.3 SU ( N ) k + N − Nf ↔ U ( k ) − k − N + Nf , − k + Nf with N f fundamentals 839.4 U ( N ) k + N − nf + na ↔ U ( k ) − k − N + nf + na with n f fundamentals, n a anti-fundamentals 859.5 Other Seiberg-like dualities 86
10 Adjoint matter 86A Fermionic T-duality 88B Character of the Vacuum Module 90
Three-dimensional N = 2 supersymmetric gauge theories have a vast, intricate network ofinfrared dualities. Some of the first dualities were discovered in [1–7], motivated in partby 3d N = 4 mirror symmetry [8–10], and later extended to included level-rank-like orSeiberg-like dualities of 3d N = 2 theories, with “chiral” matter content and nontrivialChern-Simons couplings [11–16]. Dualities of 3d N = 2 theories played a central role inthe 3d-3d correspondence [17–20], i.e. in compactifications of M5 branes on 3-manifolds,and are intimately connected to three-dimensional geometry and topology. They have alsorecently been used [21–25] to motivate non-supersymmetric dualities of 3d Chern-Simons-matter theories, notably 3d “bosonization” dualities and their cousins [26–46], includingclassic particle-vortex duality [47–49].Our goal in this paper is to advance the study of boundary conditions for 3d N = 2 gaugetheories (with various Chern-Simons levels, matter content, and superpotential couplings)and their network of dualities. Specifically, we focus on half-BPS boundary conditions thatpreserve a 2d N = (0 ,
2) subalgebra of the bulk N = 2 supersymmetry as well as a non-anomalous U (1) R R-symmetry. A systematic study of N = (0 ,
2) boundary conditions using These references still only represent a small sample of the recent literature in this extremely fruitful area. – 1 –nomalies and gauge dynamics was pioneered in [50, 51], particularly in the context of the 4d-2d correspondence (M5 branes on four-manifolds), and has led to many new and surprisingresults, including the N = (0 ,
2) trialities of [52]. Related works on N = (0 ,
2) boundaryconditions include [53–55]; and analogous discussions of 2d N = (1 ,
1) boundary conditionsappeared in [53, 56]. As in [50, 51, 53], we will define boundary conditions in the UV of thebulk 3d theory, by making some elementary choices of boundary conditions for the bulk fields,and then potentially coupling to an additional 2d N = (0 ,
2) theory on the boundary.Suppose one has a pair of dual 3d N = 2 theories, i.e. a pair of UV gauge theories T , T ∨ that flow to the same superconformal infrared theory T IR . Given a UV boundary condition(b.c.) B for T that flows to a superconformal b.c. B IR , it may be possible to find a UVboundary condition B ∨ for T ∨ that flows to the same B IR . This is what we mean by dualboundary conditions. There is no a priori reason why such a B ∨ should exist for any B ,unless T ∨ is free. Nevertheless, we will see in numerous examples that dual pairs ( B , B ∨ ) canbe explicitly (and fairly simply) constructed.One way to facilitate the existence of a dual UV boundary condition B ∨ for any B is toconstruct a duality interface I , along the lines of [20, 50, 57–60]. By definition, this is aninterface between the UV theories T and T ∨ that flows to the trivial/identity interface inthe infrared (between T IR and itself). Such a duality interface can be used to generate dualboundary conditions, by defining (say) B ∨ to be the collision of I and B — assuming that theRG flow implicit in the collision commutes with the RG flows defining the IR theories andboundary conditions, as in Figure 1. We will construct duality interfaces for all the examplesin this paper. B B _ B IR I id TT _ T IR T _ T IR T IR B IR collidecollide=flow flow = collision of I , B Figure 1 . Using collision with a duality interface to generate dual boundary conditions. Note thatboth collisions and the bulk flows to the IR are RG flows that hold different parameters fixed. If thediagram commutes, then the collision of I and B defines a dual to B . – 2 –ost of the dual 3d N = 2 gauge theories that we consider fit into a large family ofSeiberg-like dualities, obtained from a “parent” duality of Aharony [4] by a series of flows[14, 15]. Specifically, we will propose dual boundary conditions and duality interfaces for: • The basic Aharony duality [4] relating U ( N ) and U ( N f − N ) Yang-Mills theories with N f fundamental and N f antifundamental chirals, together with additional singlets and asuperpotential on one side. • Its precursor [3], corresponding to the case N f = N , which relates U ( N ) Yang-Millswith N fundamental and N antifundamental chirals to a Landau-Ginzburg model. For N f = N = 1, this becomes the basic duality between SQED and the XYZ model, whichplayed a prominent role in the 3d-3d correspondence. • Supersymmetric level-rank duality, relating U ( N ) k + N and U ( k ) − k − N , as well as U ( N ) k + N,k and SU ( k ) − k − N pure Yang-Mills-Chern-Simons theories [11, 14], a supersymmetric gen-eralization of classic level-rank duality [61–65]. • Level-rank duality with fundamental and/or antifundamental matter, relating U ( N ) k + N − Nf + Na theory with N f ( N a ) fundamental (antifundamental) chirals and U ( k ) − k − N + Nf + Na theorywith N a ( N f ) fundamental (antifundamental) chirals [11, 14]. • The basic supersymmetric particle-vortex “triality” relating a free chiral, U (1) theorywith a charged chiral, and U (1) − theory with a charged chiral. This may be considered aspecial case of level-rank duality with matter, or the simplest 3d N = 2 “mirror symmetry”[1], or obtained from SQED ↔ XYZ duality by a real mass deformation.We also consider SU (2) theory with adjoint matter, which is dual to a free chiral [66].In all these cases, we find that a Neumann-like b.c. for one 3d theory T is dual to aDirichlet-like b.c. for T ∨ . Making this precise requires a longer discussion of multiplets andsupersymmetric boundary conditions. Such a discussion was initiated in [50, 51, 53], and wewill summarize and slightly extend it in Section 2. The basic construction of an N = (0 , N = (0 ,
2) SUSY to the entire gauge multiplet).2) Choosing Neumann or Dirichlet b.c. for the scalars in each chiral multiplet (the b.c. isthen extended by N = (0 ,
2) SUSY to the entire chiral multiplet).3) Choosing additional 2d N = (0 ,
2) boundary matter (or a boundary CFT) with appropriateboundary couplings.4) In the case of Neumann b.c. for the 3d gauge fields, ensuring that all boundary gaugeanomalies vanish. – 3 –) If there is a bulk superpotential, ensuring that it is properly “factorized” at the boundary.Together with matching of boundary flavor symmetries and boundary ’t Hooft anomalies, theconstraints of anomaly cancellation and superpotential factorization turn out to be surpris-ingly restrictive. Just as in the initial examples of [50, 51], most of the dual pairs of boundaryconditions that we propose are actually the simplest possible choices that are consistent withall these constraints.We note that some basic dualities of boundary conditions in abelian theories, in particularparticle-vortex duality and SQED ↔ XYZ, were already proposed in [50] and [53], based onholomorphic block identities of [67] and a moduli space analysis. An extended discussion ofNeumann-like b.c. for pure Yang-Mills-Chern-Simons theory, their IR behavior, and theirconnection to level-rank duality appeared in [51] and led to N = (0 ,
2) trialities [52]. Relatedconstructions of WZW and coset models on boundaries in supersymmetric Chern-Simonstheory appeared in [68, 69].Our main tool in the analysis of boundary conditions is the 3d half-index. Indices, half-indices, and more general partition functions have been used to spectacular effect in verifyingand even predicting dualities of supersymmetric theories in recent years (see the review [70]and references therein). Examples involving dual boundary conditions and interfaces for 4d N = 2 theories, closely analogous to our current topic, appeared in [20, 71–74]. Early checksof bulk 3d dualities based on the 3d index and S partition function include [75–80].The 3d half-index “counts” boundary operators in the cohomology of one of the su-percharges in the N = (0 ,
2) superalgebra preserved by the boundary condition. The 3dhalf-index may equivalently be identified with the partition function of a 3d N = 2 theory ona hemisphere HS times S , with a chosen boundary condition on ∂ ( HS × S ) (cid:39) S × S .From this perspective, it is clear that if the 3d bulk is trivial the partition function shouldsimply reduce to an S × S partition function, which turns out to be the 2d elliptic genus.A UV formula for the 3d half-index of a gauge theory with Neumann b.c. for the gaugefields was given in [50, 51], analogous to UV formulae for the 2d elliptic genus [50, 87–89].By “UV formula,” we mean a prescription for computing the half-index from the UV fieldcontent of a theory as in [90] — essentially obtained by counting operators constructed fromthe classical fields and (in the case of Neumann b.c.) using a contour integral to project togauge invariants. Intuitively, this is possible because the index is insensitive to RG flow, andlargely insensitive to UV superpotential couplings. Such UV formulae for indices or half-indices are often reproduced by localization computations of partition functions. In particular, The twist of 3d N = 2 theory with respect to this supercharge was recently studied by [81]. The twistedtheory is topological in the direction perpendicular to the boundary, and holomorphic in the plane parallel tothe boundary. Thus boundary operators counted by the index have the structure of a chiral algebra. Indeed,if the 3d bulk theory were empty, the boundary chiral algebra would precisely coincide with that appearing inthe half-twist of 2d N = (0 ,
2) theories [82, 83], and the half-index would coincide with the 2d elliptic genus[84–86]. UV formulae for the index are only available if the IR U (1) R coincides with some (possibly unknown)linear combination of the UV U (1) R symmetry and other U (1) flavour symmetry generators which are presentin the UV. This is the main reason we focus on boundary conditions which preserve some UV U (1) R symmetry. – 4 –ocalization of the full 3d index was performed in [79, 91] (further analyzed, extended, andrelated to three-sphere partition functions [75, 78, 92] in [67, 80, 93–96]); and localization ofthe 3d half-index with Neumann b.c. appeared in [55]. In Section 3 we propose an extensionof the half-index to Dirichlet b.c., which instead of a projection to gauge invariants involvesa sum over boundary monopole sectors.The half-indices that we consider in this paper are closely related to the holomorphicblocks of [67, 98] (see also [99–103]). As discussed in [67], holomorphic blocks are defined inthe infrared of a 3d N = 2 theory. They are defined as partition functions of a mass-deformedtheory on C × S , with a boundary condition near infinity on C labelled by the choice of anIR vacuum. The most general N = (0 ,
2) boundary condition in the IR may be describedas a superposition (direct sum) of boundary conditions labelled by vacua, each optionallydecorated with a boundary N = (0 ,
2) theory. Correspondingly, the C × S partition functionof a general IR boundary condition will be a linear combination of holomorphic blocks, withcoefficients valued in elliptic genera.Since any (SUSY-preserving) UV boundary condition B flows to some IR boundarycondition, one should expect that the UV half-index of B will be equal to such a linearcombination of holomorphic blocks. We indeed find this to be the case, though we do notanalyze the phenomenon in detail here. We show that half-indices are solutions to the samedifference equations that characterize the holomorphic blocks (up to a systematic modificationthat we explain), which implies that the former is a linear combination of the latter. Theseare also the same difference equations that are obeyed by the S × S index and by the S partition function of a 3d theory [18, 73]; in all cases, the difference equations capturerelations in the category of bulk BPS line operators. The results of this paper can be extended in numerous orthogonal directions, some of whichoriginally motivated this work.1. The analysis of boundary gauge and ’t Hooft anomalies and the matching of symmetrieson the two sides of the dualities is not specific to supersymmetric gauge theories. Thesame anomaly matching considerations allow one to formulate plausible pairs of dualboundary conditions for non-supersymmetric “bosonization” dualities. Preliminary re-sults were announced in [104]. While this draft was completed, further work appearedon the subject [105].2. In addition, we expect that one can recover dual pairs of boundary conditions in [105]associated with bosonization and particle-vortex duality by judiciously breaking super-symmetry in our dual pairs of boundary conditions (analogous to the ‘bulk’ study of[24, 25]). We have performed a similar breaking of the 3d N = 4 abelian mirror symme-try interface obtained in [60] to the 3d N = 2 interface. We derive the latter in Section These localization computations were all largely inspired by the computation of [97] of the 4d S partitionfunction. We again refer readers to [70] for a review of localization techniques, with many further references. – 5 –.2 by independent means. In completely breaking SUSY, of course, one is subject tothe usual caveats associated with uncontrolled RG flows.3. The tools we employed in our analysis apply equally well to boundary conditions for3d N = 4 gauge theories, preserving N = (2 ,
2) and N = (0 ,
4) SUSY. The corre-sponding dualities of boundary conditions may also be derivable through brane con-structions [106].4. The N = (2 ,
2) boundary conditions played an important role in the physical explana-tion of symplectic duality [60]. They should also be useful in interpreting the resultsof [102].5. On the other hand, N = (0 ,
4) boundary conditions for 3d N = 4 gauge theories play animportant role in past [107] and upcoming [108] work on gauge theory constructions ofvertex operator algebras with applications to the Geometric Langlands program. Ourhalf-index calculations provide important checks in the form of characters of the relevantVOAs.6. Our abelian examples are the basic building blocks for an analysis of boundary condi-tions and duality interfaces for 3d N = 2 theories of class R [18, 109], which correspondgeometrically to ideal triangulations of 3-manifolds. They are also the building blocksfor an IR description of codimension-two defects in 4d N = 2 theories of class S, alongthe lines of [110] (and related to [111, 112]).7. Some of our proposed dual pairs of boundary conditions are mutually consistent thanksto some known dualities of N = (0 ,
2) gauge theories [52]. It would be interesting toexplore the relationship further.8. Many 3d dualities descend from Seiberg duality for 4 d N = 1 gauge theories, cf. [15].The basic N = (0 ,
2) boundary conditions in the 3d theories would arise naturally fromcigar configurations with a N = (0 ,
2) surface defect at the tip of the cigar. It maybe possible to lift some dualities of boundary conditions to Seiberg dualities of surfacedefects (see e.g. [113]).9. In the opposite direction, it would be interesting to reduce some of the 3d dualities ofboundary conditions considered here to two dimensions, along the lines of of [114, 115],and relate to the analysis of UV boundary conditions in 2d N = (2 ,
2) GLSM’s carriedout in [54, 116, 117]. N = (0 , boundary conditions We describe several very general families of boundary conditions for 3d N = 2 theories thatpreserve two-dimensional N = (0 ,
2) supersymmetry, and their associated anomalies. We alsointroduce some compact notation that we’ll use throughout the paper.– 6 –ost aspects of boundary conditions presented below appeared in a slightly more con-densed form in [50, 51, 53]. New features of our discussion include a refined computation ofboundary anomalies, showing that they are independent of the signs of bulk real masses orthe signs of bulk Chern-Simons levels — in contrast to the well-known shifts of bulk Chern-Simons levels induced by massive fermions. We also initiate a study of singular boundaryconditions, analogous to the Nahm-pole b.c. in 4d N = 4 super-Yang-Mills [58, 59] andgeneralizing the 3d N = 4 analysis of [106]. Basic conventions:
We identify the local neighborhood of a boundary with half of 3d Minkowski space R , × R ≤ . We use coordinates x µ ( µ = 0 ,
1) on R , , and a spatial coordinate x ⊥ ≤ z, ¯ z in the plane of the boundary.The 3d N = 2 SUSY algebra has four real odd generators that can be regrouped intocomplex spinors Q α , Q α ( α = − , +), satisfying { Q + , Q + } = − P + , { Q − , Q − } = 2 P − , { Q + , Q − } = − i ( P ⊥ − iZ ) , { Q − , Q + } = 2 i ( P ⊥ + iZ ) , (2.1)where P ± = ± P + P are the left and right translation operators in R , and Z is a realcentral charge. The algebra admits a U (1) R R-symmetry with charges Q + Q + Q − Q − U (1) R − − . (2.2)A given boundary condition can (potentially) preserve any subalgebra of 3d N = 2 thatdoes not contain P ⊥ . We are interested in the N = (0 ,
2) subalgebra that is generated by Q + and Q + , and inherits the U (1) R symmetry of 3d N = 2. We focus our attention onboundary conditions that preserve this (0 ,
2) subalgebra and leave U (1) R unbroken. Notethat unbroken U (1) R is necessary in order to define a half-index. Superconformal boundaryconditions always preserve some U (1) R , though it may be emergent in the IR. We begin with the simplest 3d theory: that of a free chiral multiplet. Off-shell, it contains acomplex boson φ , a complex fermion ψ α and its conjugate ¯ ψ α , and a complex auxiliary field F , which can be grouped in a superfieldΦ d = φ + θ + ψ + + θ − ψ − + θ + θ − F + ... , (2.3)with remaining components determined by the chirality condition D + Φ d = D − Φ d = 0.The 3d theory has an action (cid:82) d xd θ Φ † d Φ d , and the equations of motion set F = 0. Ourconventions for 3d and 2d SUSY coincide with [3, 83] and with a dimensional reduction of 4d– 7 – = 1 following [118], up to numerical factors that we do not carefully keep track of, as theydo not affect the structure of the results.The multiplet Φ d can be decomposed into two multiplets under the (0 ,
2) subalgebra, a(0 ,
2) chiral Φ = Φ d (cid:12)(cid:12) θ − =¯ θ − =0 = φ + θ + ψ + − iθ + ¯ θ + ∂ + φ (2.4)that satisfies D + Φ = 0, and a (0 ,
2) Fermi multipletΨ = ¯ ψ − + θ + f − iθ + ¯ θ + ∂ + ψ − ( = D − Φ † d (cid:12)(cid:12) θ − =¯ θ − =0 on shell) (2.5)that also satisfies D + Ψ = 0. The auxiliary field f in the Fermi multiplet is equal on-shell to f = ∂ ⊥ ¯ φ , as can be seen from f ∼ D + Ψ (cid:12)(cid:12) θ + =¯ θ + =0 = D + D − Φ † d (cid:12)(cid:12) all θ = 0 ∼ ∂ ⊥ Φ † d (cid:12)(cid:12) all θ = 0 = ∂ ⊥ ¯ φ In the language of [83], one would say that Ψ has a “J-term” J ∼ ∂ ⊥ Φ . (2.6)Written in (0,2) superspace, the bulk action takes the form (cid:90) d x dx ⊥ (cid:104) (cid:90) d ¯ θ + dθ + (cid:0) Φ † ∂ − Φ + Ψ † Ψ (cid:1)(cid:124) (cid:123)(cid:122) (cid:125) standard 2d (0 ,
2) kin. term + (cid:90) dθ + Ψ ∂ ⊥ Φ (cid:124)(cid:123)(cid:122)(cid:125) (0,2) J-term + c.c. (cid:105) (2.7)with a fermionic superpotential Ψ ∂ ⊥ Φ that sets f ∼ ∂ ⊥ ¯ φ and ultimately gives rise to the ∂ ⊥ derivatives in the 3d kinetic term.There are two basic boundary conditions for a free 3d chiral. They follow most simply byobserving that the action (2.7) contains a term ¯ ψ − ∂ ⊥ ψ + , which forces one or the other of thefermions to be zero at the boundary in order to avoid a boundary term in the equations ofmotion (at least in the absence of any additional boundary matter). The SUSY completionof setting ψ − to zero, which we’ll call Neumann and denote as ‘N’, sets to zero the entire(0,2) multiplet Ψ: N b.c. : Ψ (cid:12)(cid:12) ∂ = 0 ⇒ ∂ ⊥ φ (cid:12)(cid:12) ∂ = 0 , ψ − (cid:12)(cid:12) ∂ = 0 . (2.8)(We call this Neumann due to the induced Neumann b.c. on φ .) One SUSY completion ofsetting ψ + to zero, which we’ll call Dirichlet and denote ‘D’, sets to zero the entire multiplet Φ:D b.c. : Φ (cid:12)(cid:12) ∂ = 0 ⇒ φ (cid:12)(cid:12) ∂ = 0 , ψ + (cid:12)(cid:12) ∂ = 0 . (2.9)Alternatively, we can set Φ equal to a constant (background) chiral superfield at theboundary, i.e. to a complex number c , leading to a deformation of the Dirichlet b.c. that wedenote ‘D c ’ D c b.c. : Φ (cid:12)(cid:12) ∂ = c ⇒ φ (cid:12)(cid:12) ∂ = c , ψ + (cid:12)(cid:12) ∂ = 0 . (2.10)For the D c b.c., we always assume that c (cid:54) = 0.Clearly, a D c boundary condition for a free chiral will not flow in the IR to a superconfor-mal boundary condition. More generally, in order for D c to preserve U (1) R , the R-charge of– 8 – must be zero. This precludes any straightforward RG flow to a superconformal boundarycondition even in the presence of interactions, since the R-charge is below the unitarity bound.There is, however, a useful loophole: in a gauge theory, if φ is not gauge invariant, it mayhave R-charge zero as long as all gauge-invariant chiral operators have an R-charge above theunitarity bound. Thus, giving charged chirals D c b.c. in a gauge theory may still be part ofthe UV definition of superconformal boundary conditions. Note that in order to include D c b.c. in a gauge theory, the gauge group will have to be partly broken at the boundary. Any quantum field theory, even a free one, will generally admit a very large variety of “bound-ary conditions,” some of which may not even admit a weakly-coupled description. A generalclassification of such boundary conditions would be at least as intricate as the classificationof quantum field theories in one dimension lower.Even semi-classically, there may be a large variety of elliptic boundary conditions on thebulk fields. We will ignore for now the possibility of boundary conditions defined by a singularbehaviour of the bulk fields near the boundary — we will come back to that in Section 2.6.A fairly general strategy to define boundary conditions is to start from a simple “refer-ence” boundary condition and deform it by a boundary action, possibly involving auxiliaryboundary degrees of freedom. Intuitively, one picks conjugate variables ( u, v ) in phase spaceand a boundary action S ∂ ( u ) to deform the boundary conditions as v | ∂ = 0 → (cid:18) v + ∂S ∂ ( u ) ∂u (cid:19) (cid:12)(cid:12)(cid:12) ∂ = 0 (2.11)As a shortcut, we can think about the original and modified boundary condition as arisingas boundary equations of motions for half-space actions (cid:90) x ⊥ ≤ v∂ ⊥ u (cid:90) x ⊥ ≤ v∂ ⊥ u + S ∂ ( u ) (2.12)The shortcut is correct as long as we forbid the boundary action from depending on v .In general, a boundary condition is supersymmetric if the perpendicular component of thesuper-current is a total derivative along the boundary. A simple way to obtain supersymmetricboundary conditions is to start from a supersymmetric reference boundary condition and addsupersymmetry-preserving boundary couplings.We illustrate this perspective by reformulating the basic N, D, and D c b.c. for a free3d chiral in terms of half-space actions. Effectively, Φ and Ψ are conjugate variables. Thereference N and D boundary conditions are associated to bulk actions S N = (cid:82) d x (cid:82) x ⊥ ≤ dx ⊥ (cid:104) (cid:82) d ¯ θ + dθ + (cid:0) Φ † ∂ − Φ + Ψ † Ψ (cid:1) + (cid:82) dθ + Ψ ∂ ⊥ Φ + c.c. (cid:105) S D = (cid:82) d x (cid:82) x ⊥ ≤ dx ⊥ (cid:104) (cid:82) d ¯ θ + dθ + (cid:0) Φ † ∂ − Φ + Ψ † Ψ (cid:1) − (cid:82) dθ + ( ∂ ⊥ Ψ)Φ + c.c. (cid:105) (2.13)– 9 –arying S N leads to a boundary term δS N = (bulk terms) + (cid:90) d x (cid:90) dθ + Ψ δ Φ (cid:12)(cid:12) ∂ + c.c. (2.14)whose associated equation of motion sets Ψ (cid:12)(cid:12) ∂ = 0, effectively imposing N b.c. Similarly, thevariation of S D has a boundary term (cid:82) dθ + δ ΨΦ (cid:12)(cid:12) ∂ that imposes D b.c.: δS N = 0 ⇒ N b.c.; δS D = 0 ⇒ D b.c. . (2.15)The modification D c of the Dirichlet boundary condition can be achieved by adding aboundary superpotential c Ψ. Consider S D c := S D + (cid:90) d x (cid:90) dθ + c Ψ (cid:12)(cid:12) ∂ + c.c. . (2.16)Then δS D c = (bulk) − (cid:82) d x (cid:82) dθ + δ Ψ(Φ − c ) (cid:12)(cid:12) ∂ , so that δS D c = 0 ⇒ D c b.c. (2.17) In this section we discuss boundary conditions for free chirals that involve boundary degreesof freedom. In order to obtain N = (0 ,
2) boundary conditions, we add (0 ,
2) chiral and/orFermi multiplets on the boundary, coupled to the bulk with a fermionic superpotential. Wedescribe some general consequences of such a modification.Consider a 2d N = (0 ,
2) theory with chiral multiplets C α and Fermi multiplets Γ i . Recallthat interactions in this theory may be introduced via a choice of ‘E’ and ‘J’ terms for theFermi multiplets. In superspace E i = D + Γ i , J i = D + Γ † i (on shell) . (2.18)Both E i and J i are necessarily chiral (since D = 0), and we assume they are holomorphicfunctions of the C α . The 2d action may be written as S d = (cid:82) d xd ¯ θ + dθ + ( C † α ∂ − C α + Γ i Γ † i ) + (cid:82) d x dθ + J i Γ i + c.c. . (We use canonical kinetic terms.) The action is supersymmetric providedthat E · J := (cid:88) i E i J i = 0 . (2.19)In this theory, the equations for a supersymmetric vacuum include E i ( C ) = J i ( C ) = 0 ∀ i ,∂ α J · Γ + ∂ α E · Γ † = 0 ∀ α , (2.20) This standard constraint arises from the variation of the fermionic superpotential Q + (cid:82) dθ + J · Γ ∼ (cid:82) dθ + D + ( J · Γ) = (cid:82) dθ + E · J . This will vanish as desired if E · J is constant. Further requiring that thetheory admits a supersymmetric vacuum, as in (2.20), forces E · J = 0. Meaning: a combination of the BPS equations for the full (0 ,
2) algebra together with the equations ofmotion in the low-energy limit, so that derivatives in the x , x directions drop out. – 10 –here ∂ α J := ∂J ( C ) /∂C α , etc.Note that there is an inherent symmetry under exchange Γ i ↔ Γ † i and E i ↔ J i . This well-known symmetry may actually be viewed as fermionic analogue of T-duality. Some details ofthis perspective are presented in Appendix A.Now suppose we have a 3d chiral multiplet Φ d = (Φ , Ψ) with N b.c., encoded in the action S N from (2.13), and use a boundary N = (0 ,
2) theory as above to modify it. We introduceboundary chiral and Fermi multiplets C α , Γ i , and couple them to the bulk by allowing theE and J terms to contain a holomorphic dependence on the boundary values of Φ . Themodified action takes the form S mod N = S N + i (cid:90) d x (cid:90) dθ + (cid:80) i J i ( C, Φ | ∂ )Γ i + c.c. + (bdy kinetic terms) (2.21)The variation of S mod N with respect to the boundary fields C, Γ gives rise, at low energy, tothe usual 2d constraints (2.20). In addition, the variation of S mod N with respect to the bulkfield Φ takes the form δS mod N = (bulk) + i (cid:90) d x (cid:90) dθ + (cid:2) Ψ δ Φ | ∂ + ∂ Φ J · Γ δ Φ | ∂ (cid:3) + c.c. + δ (bdy kinetic) . (2.22)Setting this variation to zero leads toΨ (cid:12)(cid:12) ∂ = − ∂ Φ J · Γ − ∂ Φ E · Γ † . (2.23)Thus, in the infrared, we expect the modified boundary condition to flow to one combining(2.23) and E = J = ∂ α J · Γ + ∂ α E · Γ † = 0 from (2.20).We may modify the D b.c. on a 3d chiral in a similar way. We begin with the action S D from (2.13). We add 2d multiplets C α , Γ i with E and J terms that depend only on C .Then we introduce a boundary J-term J Ψ ( C ) for the restriction to the boundary of the bulkmultiplet Ψ. The modified action takes the form S mod D = S D + i (cid:90) d x (cid:90) dθ + (cid:2) J Ψ ( C )Ψ | ∂ + (cid:80) i J i ( C )Γ i ( C ) (cid:3) + c.c. + (bdy kinetic) . (2.24)Setting to zero the boundary variation of this action leads at low energy toΦ (cid:12)(cid:12) ∂ = J Ψ ( C ) , E i ( C ) = J i ( C ) = 0 ,∂ α J Ψ Ψ (cid:12)(cid:12) ∂ + (cid:80) i ( ∂ α J i Γ i + ∂ α E i Γ † i ) = 0 . (2.25)The first equation is the deformed boundary condition, the other equations are required fora low energy supersymmetric vacuum. There are two simple, important special cases of modified boundary conditions for a 3d chiral.Starting with N b.c., we may introduce a single boundary Fermi multiplet Γ with J = Φ.The modified action is S N, Γ = S N + i (cid:90) d x (cid:90) dθ + Φ | ∂ Γ + c.c. + (Γ kinetic) , (2.26)– 11 –hose boundary variation (at low energy) setsΨ (cid:12)(cid:12) ∂ = − Γ , J = Φ (cid:12)(cid:12) ∂ = 0 . (2.27)This both relieves the constraint on the boundary value of Ψ (imposed by pure N b.c.) andconstrains Ψ. Indeed, (2.27) looks just like D b.c.Conversely, starting with D b.c., we may introduce a single boundary chiral multiplet C and a J-term J Ψ = C for Ψ. The modified action is S D,C = S D + i (cid:90) d x (cid:90) dθ + Ψ (cid:12)(cid:12) ∂ C + c.c. + ( C kinetic) , (2.28)whose boundary variation (at low energy) setsΦ (cid:12)(cid:12) ∂ = C , ∂ C J Ψ = Ψ (cid:12)(cid:12) ∂ = 0 . (2.29)This frees up the boundary value of Φ while constraining Ψ, so that we effectively end upwith N b.c. (Note also that the definition of D c b.c. in (2.16) is a special case of (2.28), with C = c a constant rather than a dynamical boundary field.)We say that coupling to a boundary Fermi multiplet (and flowing to the IR) “flips” N toD, while coupling to a boundary chiral multiplet (and flowing to the IR) “flips” D to N:flips : D[C] (cid:32) NN[Γ] (cid:32) D . (2.30)These “flips” are analogous to modifications of boundary conditions of 4d N = 2 theoriesthat were discussed in [18], inspired by the field-theoretic Fourier transform of [119, 120]. If we have multiple 3d free chiral multiplets (cid:8) Φ a d (cid:9) N f a =1 , we can define a basic boundary con-dition by choosing N, D, or D c b.c. for each Φ a d independently. This basic combination ofboundary conditions can be further modified by introducing boundary fields and couplings,in a straightforward generalization of the above analysis.For example, we could use a modification to engineer a boundary condition that restrictsthe bosons φ a to lie on an arbitrary holomorphic submanifold S ⊂ C N f , and the fermions ψ a − to take values in the (parity-reversed) conormal bundle of S :( φ, ψ − ) ∈ N ∗ S ⊂ T ∗ [1] C N f . (2.31)To do so, we begin with N b.c. for all the Φ a d , and introduce boundary Fermi multiplets { Γ i } di =1 with J-tems J i (Φ | ∂ , ..., Φ N f | ∂ ) and E i = 0. This results in a modified boundarycondition J (Φ) (cid:12)(cid:12) ∂ = ... = J d (Φ) (cid:12)(cid:12) ∂ = 0 , Ψ a − (cid:12)(cid:12) ∂ ∼ ∂J i (Φ) ∂ Φ a Γ i ∀ a , (2.32)which is precisely of the form (2.31) for S = { J = ... = J d = 0 } ⊂ C N f . Such boundaryconditions are familiar in the 2d B-model, where they given rise to coherent sheaves.– 12 – .3 Bulk superpotentials A standard bulk deformation of the theory of N f free chirals Φ a d corresponds to adding asuperpotential (cid:90) d x (cid:90) d θ W (Φ d ) + c.c. (2.33)to the bulk action. In the presence of a superpotential, Neumann-type boundary conditionsrequire a modification (often a severe modification, which can make them partially Dirichlet)in order to preserve 2d (0 ,
2) supersymmetry [121]. The structure of the possible modificationsis encapsulated by the same “matrix factorizations” that show up in 2d Landau-Ginzburgmodels. We review a few relevant details here, following [51, Sec. 4.1].First, let’s recall why Neumann b.c. are incompatible with superpotentials. When W (cid:54) = 0,each 3d chiral multiplet may still be decomposed into a 2d chiral Φ a and a 2d Fermi multipletΨ a , as in (2.4), (2.5). However, the Fermi multiplets no longer satisfy D + Ψ = 0; rather, onefinds D + Ψ a ∼ ∂W (Φ) ∂ Φ a . (2.34)In N = (0 ,
2) terminology, Ψ a has both a J-term J a ∼ ∂ ⊥ Φ a and an E-term E a ∼ ∂ a W . Wemay now try to engineer a Neumann boundary condition by writing down an appropriatehalf-space action and setting the boundary variation to zero — just as in Section 2.1.1. Theaction S N in (2.13) is a natural choice, and certainly induces ∂ ⊥ φ a (cid:12)(cid:12) ∂ = 0. However, thisaction no longer preserves (0 ,
2) SUSY, because it violates the standard E · J = 0 constraint(2.19). In the present case, we have“ E · J ” = (cid:90) x ⊥ ≤ dx ⊥ (cid:80) a E a · J a = (cid:90) x ⊥ ≤ dx ⊥ (cid:80) a ∂ a W ∂ ⊥ Φ a = (cid:90) x ⊥ ≤ dx ⊥ W = W (Φ) (cid:12)(cid:12) ∂ , (2.35)whence the constraint is W (Φ) (cid:12)(cid:12) ∂ = 0. This clearly cannot be satisfied if W is nontrivial andthe φ a are left unconstrained on the boundary, as for standard N b.c. Fortunately, the computation (2.35) tells us how to remedy the situation. We can intro-duce additional boundary Fermi multiplets Γ i ( i = 1 , ...d ) with their own E and J terms E i Γ , J i Γ (which are holomorphic functions of Φ | ∂ , and perhaps other boundary chirals) such that (cid:80) i E i Γ J i Γ = − W (Φ) (cid:12)(cid:12) ∂ + const . (2.36)The factorization (up to a constant shift) of W in this manner is a basic example of a matrixfactorization. The constant will have to vanish for boundary conditions that preserve U (1) R .Note that many choices of E Γ and J Γ are possible. The relation (2.36) ensures that the total E · J vanishes, so that the modified half-space action S N + i (cid:90) d x (cid:90) dθ + (cid:88) i Γ i J i Γ (Φ | ∂ , ... ) + c.c. + (bdy kinetic) (2.37) From a more fundamental perspective, the incompatibility of N b.c. with bulk superpotentials may beseen as a consequence of the fact that N b.c. do not ensure that the bulk BPS equations dW = 0 are satisfiedat the boundary. – 13 –s once more supersymmetric. The actual boundary condition resulting from the variation of(2.37) now becomes ( cf. (2.23))MF J,E : Ψ a (cid:12)(cid:12) ∂ ∼ (cid:80) i (cid:2) ∂ a J i Γ (Φ | ∂ )Γ i + ∂ a E i Γ (Φ | ∂ )Γ i † (cid:3) ; J i Γ (Φ | ∂ ) = E i Γ (Φ | ∂ ) = 0 ∀ i . (2.38)The “matrix factorization” (2.38) is the most general sort of boundary condition we willneed in the presence of a superpotential. It may be a mild or severe modification of standardN b.c., depending on the precise choice of J Γ and E Γ . Notice that J Γ (cid:12)(cid:12) ∂ = E Γ (cid:12)(cid:12) ∂ = 0 imposeDirichlet-like b.c. on some of the φ a . The additional relations Ψ (cid:12)(cid:12) ∂ ∼ dJ · Γ constrain some ofthe Ψ’s (those in the cokernel of the map dJ ), leading to Neumann b.c. for others of the φ a .We should also discuss the effect of a bulk superpotential on Dirichlet D c boundaryconditions. Generic Dirichlet boundary conditions do not appear to break SUSY explicitlywhen the bulk superpotential is turned on, but they break it spontaneously if the boundaryvalue of the fields is not a critical value of W . That means we need to impose ∂W ( c ) = 0 inorder to have (classically) low energy supersymmetry.It is easy to reproduce this statement as a special case of (2.38). Adding auxiliaryboundary Fermi multiplets Γ a with J a = Φ a − c a to go from Neumann to general Dirichletb.c. we have a constraint (Φ a − c a ) E a (Φ , c ) = W (Φ) − W ( c ) . (2.39)that can be solved in a straightforward manner. Low energy SUSY requires both Φ a = c a and E a (Φ , c ) = E a ( c, c ) = ∂W ( c ) = 0 (2.40)In a similar way, we can argue that if we give Dirichlet b.c. to a subset Φ of the bulkfields, Neumann for the remaining Φ (cid:48) , then we will need a matrix factorization of W ( c, Φ (cid:48) ).Compatibility with the bulk SUSY vacua will further require at low energy ∂ c W ( c, Φ (cid:48) ) = 0.Although Dirichlet boundary conditions do not need extra conditions to preserve SUSYin the UV, there are constraints on boundary couplings of the form J Φ ( C α )Ψ | ∂ , in the sensethat the E.J = const constraint is deformed to
E.J − W ( J Φ ) = const. A simple example of a 3d N = 2 theory with a superpotential is the “XYZ model,” i.e. threechiral fields X d , Y d , Z d coupled by a superpotential W = X d Y d Z d . What are the basic(0,2) boundary conditions for this theory?The preceding analysis suggests that we cannot choose N b.c. for all three chirals withoutadding some extra degrees of freedom to factorize XY Z . Setting D b.c. for a single chiral,say X = 0, is already enough to preserve SUSY in the UV. On the other hand, if we deformthe boundary condition for X to D c , we will still have to deal with the restriction cY Z of thebulk superpotential.Altogether, we can consider the following simple boundary conditions:– 14 – (D,D,D): D b.c. for all three fields. This can be deformed to a (D x ,D y ,D z ), but thedeformation will break SUSY spontaneously unless xy = xz = yz = 0. In other words,we should really only consider (D c ,D,D), (D,D c ,D) or (D,D,D c ) deformations. • (D,D,N): Dirichlet for X and Y , Neumann for Z (or permutations thereof). We canconsider deformations (D c ,D,N) or (D,D c ,N). • (D,N,N): Dirichlet for one field, Neumann for the other two.In Section 6, we will identify the duals of most of these boundary conditions in 3d N = 2SQED. Boundary conditions that preserve N = (0 ,
2) supersymmetry typically have ’t Hooft anoma-lies for global symmetries. We would like to determine what they are.We use the following conventions: • A purely two-dimensional left-handed chiral fermion (such as the leading component γ − in a Fermi multiplet Γ = γ − + ... ) contributes +1 to the anomaly for the U (1) symmetryit is charged under. Letting f denote the field strength for the symmetry, the anomalypolynomial is I ( f ) = f . • A purely two-dimensional right-handed chiral fermion (such as the fermion χ + in a chiralmultiplet C = c + θ + χ − + ... ) contributes − U (1) symmetry,with anomaly polynomial − f . • Three-dimensional U (1) Chern-Simons theory (purely bosonic) has a level k ∈ Z natu-rally quantized to be an integer; and it induces an anomaly + k f on the boundary.Now, a three-dimensional fermion ( ψ + , ψ − ) has both left-handed and right-handed com-ponents with respect to the 2d Lorentz group. Typical boundary conditions set either ψ + or ψ − to zero at the boundary. Let f be the field strength for the U (1) symmetry rotating ψ .We claim that • A three-dimensional fermion with b.c. ψ + (cid:12)(cid:12) ∂ = 0 (so that ψ − survives at the boundary)contributes f to the anomaly polynomial. • A three-dimensional fermion with b.c. ψ − (cid:12)(cid:12) ∂ = 0 (so that ψ + survives at the boundary)contributes − f to the anomaly polynomial.To verify the claim, we introduce a real mass for the 3d fermion ψ α , flow to the IR, andmatch UV and IR anomalies. A real mass term im(cid:15) αβ ψ α ¯ ψ β has two effects at energies below | m | . First, the fermion becomes fully massive in the bulk, and integrating it out at one-loop generates a (background) Chern-Simons term for its U (1) symmetry, at level sign( m )– 15 –122–124] ( cf. [3] in the supersymmetric case). Second, since the Dirac equations take theform ( ∂ ⊥ + m ) ψ − + ( ∂ ± of other fermions) = 0 , ( ∂ ⊥ − m ) ψ + + ( ∂ ± of other fermions) = 0 , (2.41)normalizable edge modes ( i.e. purely 2d fermions) may survive, with profiles (cid:40) ψ − = a − ( x µ ) e − mx ⊥ if m < ψ + = a + ( x µ ) e mx ⊥ if m > x ⊥ ≤ . (2.42)If the boundary condition is ψ + (cid:12)(cid:12) ∂ = 0, then an edge mode of ψ − exists when m <
0, so thetotal IR anomaly is 12 sign( m ) + (cid:40) m >
01 if m < (cid:41) = 12 (2.43)On the other hand, if the boundary condition is ψ − (cid:12)(cid:12) ∂ = 0, then an edge mode of ψ + existswhen m >
0, so the total IR anomaly is12 sign( m ) + (cid:40) − m >
00 if m < (cid:41) = −
12 (2.44)This substantiates our claim .In the case of a free chiral multiplet Φ d , there are two U (1) symmetries around: a flavorsymmetry U (1) f under which Φ d has charge 1 (say), and the R-symmetry U (1) R under whichΦ d has some charge ρ . The component fields have chargesΦ , φ ψ + Ψ , ¯ ψ − U (1) f − U (1) R ρ ρ − − ρ (2.45)(Note that the superpotential Φ ∂ ⊥ Ψ has U (1) f charge 0 and U (1) R charge 1, as requiredfor unbroken flavor and R-symmetry.) Therefore, our basic boundary conditions come withanomalies N b.c. : − ( f + ( ρ − r ) ; D b.c. : ( f + ( ρ − r ) , (2.46)encoded as polynomials I ( f , r ) in the U (1) f and U (1) R curvatures.This prescription is perfectly consistent with the “flips” that modify D to N or vice versa.For example, in order to flip D to N we must introduce a boundary chiral multiplet C . Theboundary superpotential Ψ C fixes the U (1) f , U (1) R charges of C to be (1 , ρ ), so that theright-handed chiral fermion in C contributes an anomaly − ( f + ( ρ − r ) , which is exactlyright to modify the D anomaly to the N anomaly. The non-supersymmetric analysis of [105] proceeds in a similar spirit; we expect their results are recoverablefrom our analysis. – 16 –t is somewhat insightful to repeat the analysis of edge modes for a free chiral in super-space. If we introduce a real mass m as a background value of the scalar field in the U (1) f gauge multiplet, the real mass will enter the Lagrangian through complexified covariant ∂ ⊥ derivatives. In other words, the superpotential Ψ ∂ ⊥ Φ in (2.7) gets modified toΨ ∂ ⊥ Φ → Ψ( ∂ ⊥ − m )Φ . (2.47)Recalling that the real mass m plays the role of a central charge Z , we may view (2.47) as aconsequence of the fact that Z complexifies P ⊥ in the SUSY algebra (2.1). The N = (0 , ∂ ⊥ − m )Φ = 0 , ( ∂ ⊥ + m )Ψ = 0 , (2.48)which lead directly to the edge mode profiles in (2.42). The anomaly for a nonabelian symmetry group G may be fixed by ensuring that the aboverules/conventions are consistent with the maximal abelian torus of G . As might be expected,we find that 3d fermions that survive as chiral fields on the boundary contribute half of thestandard 2d anomaly.We briefly recall the standard 2d result. Let G be a simple compact group and R anirreducible unitary representation of G . Let T R denote the quadratic index of R , definedso that the ratio of traces in any pair of representations obeys Tr R Tr R (cid:48) = T R T R (cid:48) , and normalizedso that T adjoint = 2 h where h is the dual Coxeter number. For example, this means thatif G = SU ( N ) we have T fundamental = 1 and T adjoint = 2 N = 2 h . In general, the index sonormalized may be computed as a sum of lengths-squared of weights T R = 1rank G (cid:88) λ ∈ R || λ || (2.49)where long roots α have || α || = 2. When G is U ( N ) or SU ( N ), we simply write ‘Tr’ theusual trace (on the enveloping algebra) in the fundamental representation; and for general G define Tr := h Tr adjoint . Let f denote the field strength of a G connection. Then • A 2d left-handed (resp., right-handed) complex fermion in representation R of G con-tributes Tr R ( f ) = T R Tr( f ) (resp., − T R Tr( f )) to the anomaly polynomial. • Correspondingly, the left-handed (resp., right-handed) component of a 3d fermion in rep-resentation R of G that is unconstrained by a boundary condition contributes T R Tr( f )(resp., − T R Tr( f )) to the anomaly polynomial. • In this same normalization, 3d Chern-Simons term for G at level k contributes k Tr( f ).– 17 –e may extend this analysis to a 3d chiral multiplet Φ d in representation ( R , ρ ) of G × U (1) R , where G is simple and U (1) R is the R-symmetry. We find that Neumann andDirichlet b.c. have boundary anomaliesN b.c.: − T R Tr( f ) − (dim R )( ρ − r , D b.c.: T R Tr( f ) + (dim R )( ρ − r . (2.50)If G is not simple, the anomaly can be computed by directly ensuring agreement withthe abelian anomaly for the maximal torus. We list a few special cases for G = U ( N ). Let f denote the usual N × N field strength. A 3d chiral of R-charge ρ in the fundamental,anti-fundamental, and adjoint representations of G = U ( N ) has a boundary anomalyfund: ± (cid:2) Tr( f ) + ( ρ − r Tr f + N ( ρ − r (cid:3) , anti-fund: ± (cid:2) Tr( f ) − ( ρ − r Tr f + N ( ρ − r (cid:3) , adj: ± [ N Tr( f ) − (Tr f ) + N ( ρ − r ] , (2.51)with a ‘-’ sign for Neumann b.c. and a ‘+’ sign for Dirichlet. We next extend the analysis of N = (0 ,
2) boundary conditions to gauge fields.
For gauge group G , the 3d N = 2 gauge multiplet contains the connection A m , a real scalar σ , a complex fermion λ α , and (off shell) a real auxiliary field D d , all valued in the real Liealgebra g . These fields may be grouped in a Hermitian vector superfield V d = θσ m ¯ θA m + iθ ¯ θσ − iθ ¯ θ ¯ λ − i ¯ θ θλ + θ ¯ θ D d (WZ gauge) (2.52)or a gauge-covariant linear superfieldΣ d = − i (cid:15) αβ D α D β V d = σ + ¯ θσ mn θF mn + iθ ¯ θD − θ ¯ λ + ¯ θλ + ... (2.53)that contains the field strength and satisfies D α D α Σ d = D α D α Σ d = 0, cf. [3]. On shell,the auxiliary field D d is set equal to the moment map for G acting on the matter scalars(which necessarily parameterize a K¨ahler manifold) : e D d = µ matter . (2.54)We typically set the gauge coupling e = 1, as it can be restored by dimensional analysis.In the presence of an FI term t and/or a supersymmetric Chern-Simons term at level k , theD-term is modified to e D d = µ matter − t − kσ . (2.55) The relative sign of the moment map and Chern-Simons term in (2.55) is somewhat important (in contrastto most numerical factors in the SUSY analysis). It enters in the (0,2) BPS equations (Section 2.6), whichin turn control edge modes that can contribute to anomalies. The sign in (2.55) was carefully computed, andagrees with our sign conventions for anomalies. – 18 –he 3d gauge multiplet may be decomposed into an N = (0 ,
2) gauge multiplet andan N = (0 ,
2) chiral multiplet. The chiral multiplet contains the complexified connection σ + iA ⊥ in the x ⊥ direction, together with the right-handed fermion λ + . As an N = (0 , S = σ + iA ⊥ − θ + ¯ λ + + ... , (2.56)with its remaining θ + ¯ θ + component determined by the (gauge-covariant) chirality constraint D + S = 0. In the presence of chiral matter, the superfield S appears in complexified covariantderivatives. For example, the fermionic superpotential Ψ ∂ ⊥ Φ from (2.7) that we encounteredwhen writing an N = (0 ,
2) superspace action for a 3d chiral gets modified to (cid:90) dθ + Ψ( ∂ ⊥ − S )Φ . (2.57)Note how this is compatible with the BPS equations (2.48) in the presence of a backgroundgauge multiplet (containing m rather than σ ), and the fact that Z complexifies P ⊥ in theSUSY algebra (2.1).The remaining fields of the 3d gauge multiplet go into an N = (0 ,
2) gauge multiplet,which comprises a pair of superfields A = θ + ¯ θ + A + , V − = A − − iθ + ¯ λ − − i ¯ θ + λ − + 2 θ + ¯ θ + D (in WZ gauge) , (2.58)where A ± = A ± A . Here ‘ D ’ is a new auxiliary field that is equal on shell to D = D d − ∂ ⊥ σ . (2.59)(The RHS of (2.59) will contain covariant ∂ ⊥ derivatives when the gauge group is nonabelian.)From the gauge multiplet one can also construct the fermionic field-strength superfieldΥ = λ − + θ + ( F + iD ) + ... , (2.60)which is covariantly chiral D + Υ = 0.The kinetic terms of the 3d gauge theory action may be written more or less as ordinary N = (0 ,
2) kinetic terms for a vector multiplet (
A, V − ) coupled to a chiral S . Some care mustbe taken to account for the fact that S has an exotic gauge transformation , but this is notimportant for our subsequent analysis. Additionally, an FI term for the abelian part of thegauge group appears in a superpotential (cid:90) d xdx ⊥ (cid:90) dθ + t Tr Υ + c.c. ; (2.61) The relative sign of ∂ ⊥ σ appearing here is important, just like the sign of k was in (2.55). It can ultimatelyfixed by observing that the N = (0 ,
2) BPS equations take the form of a gradient flow, as discussed inSection 2.6. If Λ is the chiral parameter of a super-gauge transformation, then the gauge transformation takes the from S → e − i Λ Se i Λ + e − i Λ D ⊥ e i Λ , mirroring the gauge transformation of A ⊥ . For abelian gauge group, this lookslike S → S + i∂ ⊥ Λ. – 19 –hile a 3d Chern-Simons term at level k manifests as k (cid:90) d xdx ⊥ (cid:90) d ¯ θ + dθ + Tr( A∂ ⊥ V − V ∂ ⊥ A ) − k (cid:90) d xdx ⊥ (cid:90) dθ + Tr(Υ S ) + c.c. , (2.62)which also includes a superpotential piece.We note that the U (1) R charges of fields in the 3d gauge multiplet are uniquely determinedfrom the fact that A µ , A ⊥ , σ have R-charge zero. Explicitly, the R-charges are S Υ λ ± ¯ λ ± U (1) R − . (2.63) N , D boundary conditions In pure, non-supersymmetric gauge theory, there always exist two basic boundary conditionscompatible with the Maxwell/Yang-Mills equations of motion. A Neumann boundary condi-tion sets F ⊥ µ (cid:12)(cid:12) ∂ = 0, and preserves gauge symmetry on the boundary. A Dirichlet boundarycondition sets A µ (cid:12)(cid:12) ∂ = 0, and breaks gauge symmetry to a global symmetry at the bound-ary (since only gauge transformations that are constant along the boundary preserve theconstraint A µ (cid:12)(cid:12) ∂ = 0). In 3d N = 2 gauge theory, there exist natural supersymmetric com-pletions of these basic Neumann and Dirichlet b.c., which we denote N and D , respectively.The N = (0 ,
2) supersymmetric completion of Dirichlet simply sets the superfields
A, V − from (2.58) to zero on the boundary: D b.c. : A (cid:12)(cid:12) ∂ = V − (cid:12)(cid:12) ∂ = 0 ⇒ A ± (cid:12)(cid:12) ∂ = 0 , D (cid:12)(cid:12) ∂ = 0 , λ − (cid:12)(cid:12) ∂ = 0 . (2.64)The constraint on the D-term translates on-shell to ∂ ⊥ σ (cid:12)(cid:12) ∂ = (cid:2) µ matter + t + kσ (cid:3) ∂ , (2.65)which is a modified Neumann b.c. for the scalar σ . (As usual, if the gauge group is nonabelian, ∂ ⊥ should be promoted to a covariant derivative.) In a quantum theory, the usual one-loopcorrections from massive fermions will modify the Chern-Simons term on the RHS of (2.65).The N = (0 ,
2) supersymmetric completion of Neumann may be understood as setting tozero the complexified (chiral) covariant derivative ∂ ⊥ − S (as in (2.57)) in a gauge-invariantway. This results in N b.c. : F ⊥± (cid:12)(cid:12) ∂ = 0 , σ (cid:12)(cid:12) ∂ = 0 , λ + (cid:12)(cid:12) ∂ = 0 , (2.66)which includes a Dirichlet b.c. for the scalar σ . In the presence of additional boundary mattercharged under the bulk gauge group G , the constraint on the field strength is modified to F ⊥± (cid:12)(cid:12) ∂ = J ∂ ± , where J ∂ ± is the current for the boundary G symmetry. Moreover, for abelianfactors in the gauge group, one may introduce boundary FI parameters t ∂ , which modify theNeumann b.c. to σ (cid:12)(cid:12) ∂ = t ∂ . (2.67)– 20 –he FI parameters enter supersymmetrically via a boundary superpotential (cid:82) dθ + t ∂ Υ | ∂ + c.c. A standard generalization of the basic N , D b.c. in gauge theory involves choosing asubgroup H ⊆ G of the gauge group to remain unbroken on the boundary. Let h ⊆ g denotethe Lie algebra of H , and h ⊥ its orthogonal complement with respect to the Cartan-Killingform; and let π : g → h and π ⊥ : g → h ⊥ denote the corresponding orthogonal projections.Then we may define a hybrid boundary condition N H b.c. : π ( A ) (cid:12)(cid:12) ∂ = π ( V − ) (cid:12)(cid:12) ∂ = 0 , π ⊥ ( F ⊥± ) (cid:12)(cid:12) ∂ = π ⊥ ( σ ) (cid:12)(cid:12) ∂ = π ⊥ ( λ + ) (cid:12)(cid:12) ∂ = 0 , (2.68)which is compatible with H gauge symmetry at the boundary. In addition, there is a boundaryflavor symmetry N G ( H ) /H , where N G ( H ) is the normalizer of H in G . Note that for twoextreme choices of H we have N H = G = N , N H = { } = D . (2.69)In the presence of bulk chiral matter (and superpotentials, etc.), the basic N and D b.c.for the gauge multiplet may be combined with N, D, or D c , or any of the more complicatedchoices of boundary conditions for matter fields discussed in Sections 2.1–2.3. We will seehow this works in many examples. When the bulk gauge group is abelian (or has an abelian factor) one may dualize the gaugefield to a circle-valued scalar, the “dual photon” γ , which obeys dγ = ∗ F . Supersymmetrically,the complex combination σ + iγ is the leading component of a 3d N = 2 chiral multiplet [3].Just as in Section 2.1, this 3d chiral may be decomposed into • a N = (0 ,
2) chiral multiplet S ∨ containing σ + iγ and λ + , ¯ λ + , which may be identifiedas the two-dimensional T-dual of S • a N = (0 ,
2) Fermi multiplet containing λ − , ¯ λ − and the normal derivative ∂ ⊥ S ∨ = ∂ ⊥ ( σ + iγ )+ ... , which may be identified with Υ (note that on-shell, Υ contains F + iD ∼ i∂ ⊥ ( σ + iγ ))Our basic N and D boundary conditions for abelian gauge theory now look just like basic Dand N b.c. (respectively) for the 3d dual-photon multiplet! In particular, the D b.c. (2.64)is compatible with setting Υ (cid:12)(cid:12) ∂ = 0, while leaving the boundary value of S ∨ unconstrained.Conversely, the N b.c. (2.66) is compatible with setting S ∨ (cid:12)(cid:12) ∂ = 0. This may be deformed to S ∨ (cid:12)(cid:12) ∂ = t ∂ (2.70)by introducing a complex boundary FI parameter (including a real FI parameter and a theta-angle). Thus N b.c. sets the dual photon γ (cid:12)(cid:12) ∂ equal to the boundary theta-angle. Thedeformation (2.70) is exactly of the same “D c ” type discussed in Section 2.1, correspondingto the addition of the boundary superpotential (cid:82) dθ + t ∂ Υ | ∂ + c.c. .– 21 – .5.4 Anomalies The gauginos λ + , λ − in the 3d gauge multiplet necessarily have R-charge +1, as in (2.63),and also transform in the adjoint representation of the gauge group G . These gauginos maytherefore contribute to boundary anomalies.A Neumann b.c. for the 3d gauge multiplet sets λ + | ∂ = 0 but does not constrain λ − .Thus, our standard UV computation (Section 2.4) would suggest a contribution from λ − tothe boundary anomaly N b.c. : h Tr( f ) + r (2.71)if G is simple. If G is not simple, than a computation such as (2.51), for the adjoint repre-sentation, gives the anomaly. If G is abelian, then the anomaly is simply r .The prediction (2.71) may again be compared with an IR calculation. We focus on the G symmetry, and assume that G is simple. We introduce a supersymmetric Chern-Simonsterm at level k , which includes a real mass for the gauginos. Indeed, in the presence of theChern-Simons term, the Dirac equations for gauginos take the form( D ⊥ + k ) λ + = 0 , ( D ⊥ − k ) λ − = 0 (2.72)(assuming that λ ± are constant in the x µ directions parallel to the boundary), from whichwe recognize − k as the real mass. In the bulk, integrating out massive gauginos shifts theChern-Simons coupling k → k − h sign( k ). On the boundary, λ − has an edge mode when k >
0, which contributes a standard 2d anomaly 2 h Tr( f ). Thus, the total IR anomaly for G is k − h sign( k ) + (cid:40) h Tr( f ) k > k < (cid:41) = k + h Tr( f ) . (2.73)This agrees with the UV prediction, which gives k from the CS term, plus h Tr( f ) from(2.71).Note that, unlike ’t Hooft anomalies for flavor symmetries, the anomaly for a G gaugesymmetry on a Neumann b.c. must somehow be cancelled! Either bulk our boundary mattermay be added to effect the cancellation.A Dirichlet b.c. for the 3d gauge multiplet behaves in the opposite way to Neumann.Since D sets λ − | ∂ = 0 while leaving λ + unconstrained, it comes with an anomaly D b.c. : − h Tr( f ) − r (2.74)for (say) a simple gauge group G . Again, this may be computed either in the UV or the IR.Note that (2.74) contains the anomaly for the boundary flavor symmetry G on a Dirichletb.c., rather than the bulk gauge symmetry. It is an ordinary ’t Hooft anomaly, and does notneed to be cancelled. – 22 – .5.5 Relations between gauge theory boundary conditions When describing boundary conditions for a free 3d chiral, we observed in Section 2.2.1 that D(resp. N) can be “flipped” to N (resp. D) by coupling to boundary matter. In gauge theory,we can look for similar relations between N and D .Modifying D to N , possibly enriched by any boundary matter, is relatively easy. Let G denote the bulk gauge group. Since D has a global boundary G symmetry, we can “gauge”the boundary symmetry by adding 2d N = (0 ,
2) gauge fields with gauge group G , togetherwith any other 2d N = (0 ,
2) degrees of freedom charged under the new G gauge symmetry.(In doing so, we should carefully ensure that all G anomalies cancel.) The resulting boundarycondition should be equivalent to N , as the 2d G gauge fields are eaten up by the bulk gaugefields.In the case of abelian G = U (1) gauge theory, we saw in the previous section that thebulk gauge multiplet may be dualized to a chiral dual-photon multiplet, grouped into two N = (0 ,
2) superfields ˆ S and Υ. The D b.c. sets Υ | ∂ = 0, and the flip to N introducesa boundary U (1) gauge symmetry with field strength Υ ∂ , coupled to ˆ S by a superpotential (cid:82) dθ + ˆ S (cid:12)(cid:12) ∂ Υ ∂ . Just as in Section 2.2.1, this coupling modifies the b.c. toˆ S (cid:12)(cid:12) ∂ = 0 , Υ (cid:12)(cid:12) ∂ = Υ ∂ , (2.75)which is equivalent to N .For general G , there is no systematic operation that maps a generic N boundary conditionto a D boundary condition. In the case of abelian G = U (1) gauge theory, it is possible tomodify N to D by promoting the boundary FI parameter to a dynamical chiral superfield.Comparing to (2.70) we see that this is essentially a D → N flip for the dual photon.
So far, we have focused on UV boundary conditions defined by fixing the values or normalderivatives of various fields at the boundary. There is another possibility that we brieflydiscuss, though it will not play a major role in this paper: we may require the bulk fields tohave a singular profile in the neighborhood of the boundary that is compatible with N = (0 , N = 4 super Yang-Mills theory,and we follow the basic logic of [58, 59] here.One inspiration for considering singular boundary conditions arises from the IR behaviorof D c b.c. Recall from Section 2.1 that D c boundary conditions for chiral multiplets mayflow to useful superconformal boundary conditions in the presence of gauge fields. In fact, innon-supersymmetric theories, D c -type b.c. may lead to conformal boundary conditions evenin the absence of gauge fields! A notable example is a real 3d scalar with a φ potential, whichflows to the 3d Ising CFT. This theory admits a conformal boundary condition that flowsfrom φ | ∂ = c in the UV, known as the “exceptional transition” boundary condition [127]. Inthe IR, the operator φ simply diverges at the boundary, as the RG flow sends the scale c toinfinity. – 23 –e thus find it natural to ask: in a given 3d N = 2 gauge theory, can we define singularboundary conditions directly in the UV that flow to the same IR superconformal boundarycondition as some D c ? The answer, in certain cases, seems to be affirmative.In general, in a Lagrangian QFT, a disorder operator supported on a submanifold S maybe defined by finding a singular solution to the equations of motion in a neighborhood of S , andrequiring fields to asymptotically approach this solution when performing the path integral.One may further require that the operator preserves certain symmetries ( e.g. flavor symmetry,supersymmetry, conformal symmetry...). These symmetries must then leave invariant thesingular solution to the equations of motion. In the case at hand, we want to define a singularboundary condition that preserves 2d N = (0 ,
2) supersymmetry, so we look for solutions tothe equations of motion that are fixed by the N = (0 ,
2) subalgebra of 3d N = 2; in otherwords, we look for solutions of the N = (0 ,
2) BPS equations.In a 3d gauge theory with chiral matter, most of the BPS equations for scalar fields canbe deduced by rewriting the theory in 2d N = (0 ,
2) superspace (as in Sections 2.1, 2.2, 2.5),and requiring E , J , and D terms to vanish. Let’s describe these in turn. • The vanishing of E terms requires fields to sit at a critical point of the 3d superpotential, dW = 0. • The vanishing of J terms sets ( D ⊥ − σ ) φ = 0 (2.76)for the bosonic component φ of each 3d chiral, where D ⊥ − σ is the complexified covariantderivative perpendicular to the boundary. Here σ ∈ g implicitly “acts” on φ in theappropriate representation; for example, if the gauge group is G = U (1) and φ hascharge q , the equation (2.76) would read ( ∂ ⊥ − iqA ⊥ − qσ ) φ = 0. • The vanishing of D terms as in (2.59) schematically sets D ⊥ σ = µ matter − t − kσ . (2.77)We can now specialize to the definition of UV boundary conditions. In the UV, we expectCS terms to be subleading compared to YM gauge kinetic terms. Indeed, they will providesubleading corrections to the solutions we describe.It is natural to look for solutions where the fields have a singular power law behaviournear the boundary. In an axial gauge, (2.76) immediately tells us that σ ∼ − sx ⊥ (2.78)with s some constant element of the Lie algebra that determines the scaling behavior of allchiral fields near the boundary. On the other hand, (2.77) then requires the matter momentmap µ matter to diverge as s ( x ⊥ ) near the boundary.– 24 –or simplicity, we can restrict ourselves to solutions where the chiral fields have theminimal divergence compatible with the behaviour of the moment map, i.e. φ ∼ ux ⊥ (2.79)Restoring dimensionful units, that would become uex ⊥ , with u dimensionless and φ and e ofdimension . The differential equations thus collapse to ordinary equations: s · u = u s = µ matter ( u ) (2.80)These solutions are a close analogue of the Nahm pole boundary conditions. Indeed, theNahm pole boundary condition occurs precisely in this setting for 3d N = 4 gauge theory[106], where only the adjoint chiral in the N = 4 gauge multiplet acquires a singularity. Then s and u are generators of an su (2) embedding in the gauge group. The resulting boundaryconditions preserve N = (0 ,
4) SUSY in that case.Surprisingly, the equations (2.80) have solutions even for simple abelian gauge theories.For example, if G = U (1) and φ consists of a single chiral of charge 1, we can simply take s = 1 and u = 1. The resulting boundary condition preserves the same symmetries as a( D ,D c ) boundary condition and presumably flows to the same boundary condition in the IR.In general, these singular boundary conditions require the boundary gauge symmetry tobe broken to gauge transformations that leave s and u fixed. The residual gauge group maythen be broken further and/or coupled to extra boundary degrees of freedom.We leave a full discussion of such boundary conditions to future work. In this section, we give the basic rules for computing the half-index of various boundaryconditions. Formulae for the half-index with Neumann ( N ) b.c. for gauge multiplets (andvarious b.c. for matter) were derived in [50, 51, 55]. We review these results, and propose ageneralization to Dirichlet ( D ) b.c. for gauge multiplets that involves a sum over monopolesectors.The half-index may be defined as the character of the vector space of local operatorson the boundary : for a 3d N = 2 theory T with a N = (0 ,
2) boundary condition B , thehalf-index is a trace II T , B ( x ; q ) = Tr Ops B ( − R q J + R x e , (3.1)where R is the R-charge operator, J generates Spin (2) (cid:39) U (1) J rotations in the plane ofthe boundary, and e is a flavor charge operator, measuring charges under a maximal torusof the flavor symmetry group on the boundary. The character (3.1) counts operators in the Often one finds the expression Tr( − F q J + R x instead of (3.1), where F = 2 J is fermion number. Aslong as all R-charges can be chosen to have integral values, the two formulae are simply related by substituting q → − q . All our examples will allow integral R-charges. – 25 –ohomology of the supercharge Q + , which is part of the N = (0 ,
2) algebra preserved by ahalf-BPS boundary condition. For a superconformal boundary condition, with the superconformal assignment of R-charges, the half-index should begin with ‘1 + ... ’ and thereafter only contain positive powersof q . More generally, the same properties should hold a 3d N = 2 gauge theory and a 2d N = (0 ,
2) boundary condition B that flows to a superconformal boundary condition in theinfrared, assuming an appropriate assignment of R-charges. In the gauge theory there istypically some range of R-charge assignments that preserve the positivity of J + R . If one canchoose integral R-charge assignments within this range (which will be true for the theoriesconsidered in this paper), then we expect II T , B ∈ Z [[ q ]][ x, x − ] , (3.2) i.e. the half-index is a formal Taylor series in q , whose coefficients are Laurent polynomialsin x , themselves with integer coefficients.An equivalent definition of the half-index is as a hemisphere ‘ HS × S ’ partition function,with the boundary condition B placed at the equatorial boundary of the hemisphere HS andwrapping S . This perspective is amenable to a localization computation [55]. In this section,however, we take the perspective of counting boundary local operators seriously, and employphysical intuition to derive practical formulae for the half-index.The general formula for the half-index of a Lagrangian 3d gauge theory, summarized inSection 3.5, takes the schematic form II gauge multiplets × II
3d matter multiplets × I
2d boundary index . (3.3)This is further projected to gauge-invariant operators (given a N b.c. for gauge fields) orsummed over monopole sectors (given a D b.c. for gauge fields). We describe the variousingredients, from right to left. Notation:
In formulae for the half-index, we use the q-Pochhammer symbols( x ; q ) k := k − (cid:89) n =0 (1 − q n x ) , ( x ; q ) ∞ := (cid:81) n ≥ (1 − q n x ) , ( q ) k := ( q ; q ) k = k (cid:89) n =1 (1 − q n ) , ( q ) ∞ := ( q ; q ) ∞ = (cid:81) n ≥ (1 − q n ) . (3.4)Also, for a general symmetry group G we work with a vector of fugacities x ∈ T C valued in themaximal torus of G . If R is a (unitary) representation of G , and λ ∈ wt( R ) (cid:39) Hom( T C , C ∗ ) aweight of R , then the fugacity corresponding to the λ weight space is λ ( x ), which we denotewith the more common notation x λ . As mentioned in the introduction, the cohomology of Q + actually has the structure of a chiral algebra A ∂ ,and from this perspective the half-index is the character of a vacuum module for A ∂ . The characters of othermodules may be obtained by inserting line operators in the index. – 26 –n this section, we will work in Euclidean signature and use complex coordinates z, ¯ z inthe plane parallel to the boundary, so that the ( ∂ − , ∂ + ) derivatives become ( ∂ z , ∂ ¯ z ). A purely two-dimensional N = (0 ,
2) theory may be considered a boundary condition fora trivial three-dimensional bulk. In this case, the half-index is simply the elliptic genus ofthe 2d N = (0 ,
2) theory [84–86] — or, more accurately, the “flavored” generalization of theelliptic genus discussed in [50, 87–89]. We refer to these latter references for details, andsimply summarize the main features here.For a 2d N = (0 ,
2) gauge theory, the elliptic genus may be constructed as a UV index,by starting with a product of elliptic genera for matter multiplets, then projecting to gauge-invariants by doing an integration along the torus of the gauge group. Superpotentials (Eand J terms) only affect the index insofar as they constrain flavor symmetries.
The index of a 2d chiral multiplet Φ = φ + θ + ψ + + ... of charge +1 under a U (1) x flavorsymmetry (with fugacity ‘ x ’) and R-charge 0 isC( x ; q ) = (cid:89) n ≥ − q n x )(1 − q n +1 x − ) = 1( x ; q ) ∞ ( qx − ; q ) ∞ . (3.5)On the RHS, we use q-Pochhammer notation ( x ; q ) ∞ := (cid:81) n ≥ (1 − q n x ). This is the inverse ofa Jacobi theta function, up to a 1 / ( q ; q ) ∞ prefactor. The operators in Q + -cohomology countedby this index are φ and its ∂ z derivatives, ∂ z ¯ φ and its further ∂ z derivatives, and normal-ordered monomials in these basic operators. Indeed, the supersymmetry transformations takethe form Q + φ = 0 , Q + ψ + ∼ ∂ ¯ z ¯ φ , Q + ¯ φ ∼ ¯ ψ + , Q + ¯ ψ + = 0 (3.6)The operators ∂ nz φ ( n ≥
0) are Q + -closed on the nose; while the operators ∂ n +1 z ¯ φ ( n ≥ Q + ∂ n +1 z ¯ φ ∼ ∂ n +1 z ¯ ψ + and are closed modulo the Dirac equation, which sets ∂ z ¯ ψ + = 0.Since the flavor charges, R-charges, and spins are ∂ nz φ ∂ n +1 z ¯ φU (1) x − U (1) R U (1) J n n + 1fugacity in index: q n x q n +1 x − (3.7)and these operators are all bosonic, the index (3.5) results.Similarly, the index of a 2d Fermi multiplet Γ = γ − + ... counts the operators ∂ nz γ − and ∂ nz ¯ γ − ( n ≥
0) and normal-ordered polynomials thereof. If Γ has U (1) x flavor charge − ∂ nz γ − ∂ nz ¯ γ − U (1) x − U (1) R − U (1) J n + n + fugacity in index: − q n +1 x − − q n x (3.8)and remembering that they are fermionic leads to the indexF( x ; q ) = ( x ; q ) ∞ ( qx − ; q ) ∞ = C( x ; q ) − . (3.9)The fact that F( x ; q ) = C( x ; q ) − is not coincidental. It reflects the fact that a chiralmultiplet and a Fermi multiplet of the charges given above can be coupled via a superpotential (cid:82) dθ + ΦΓ that preserves flavor and R-symmetry. The superpotential makes all the fieldsmassive, triggering a flow to a trivial theory in the infrared, whose index is trivial due to theidentity F( x ; q )C( x ; q ) = 1.We also see that F( x ; q ) = F( qx − ; q ). This reflects the fact that a free Fermi multipletmay equivalently be encoded in a superfield Γ = γ − + ... or ˜Γ = ¯ γ − + ... . In the presence ofgauge fields and superpotentials, the fundamental ambiguity in how one treats the on-shellFermi multiplet persists, though the relation between superfields Γ and ˜Γ is somewhat moreinteresting: it can be interpreted as a fermionic remnant of 2d N = (2 ,
2) T-duality that wedescribe in Appendix A.
In pure 2d N = (0 ,
2) gauge theory with compact gauge group G , the G gauge multiplettakes the same form as (2.58). The gauginos and their derivatives are in Q + -cohomology.In particular, Q + λ − = 0 and Q + D z ¯ λ − ∼ D z F z ¯ z = 0 by the equations of motion, so theoperators contributing to the index are D nz λ − D n +1 z ¯ λ − G adj adj U (1) R − U (1) J n + n + fugacity in index: − q n +1 s α − q n +1 s − α ( n ≥
0) (3.10)where s ∈ T C is the fugacity for the G symmetry, and α ∈ wt(adj) are the weights of theadjoint representation (including the zero weights), counted with multiplicity. These operatorsprovide a contribution (cid:81) α ∈ wt(adj) ( qs α ; q ) ∞ ( qs − α ; q ) ∞ in the index, which can usefully berewritten as( q ) G ) ∞ (cid:89) α ∈ roots( G ) ( qs α ; q ) ∞ ( qs − α ; q ) ∞ = ( q ) G ) ∞ (cid:89) α ∈ roots( G ) F( s α ; q )1 − s α . (3.11)– 28 –rojecting to G -invariants by integrating over the torus of G with a Vandermonde determinant | Weyl( G ) | (cid:81) α ∈ roots( G ) (1 − s α ) then leads to an index( q ) G ) ∞ | Weyl( G ) | (cid:73) ds πis (cid:89) α ∈ roots( G ) F( s α ; q ) . (3.12)For 2d gauge theory coupled to chiral or Fermi matter multiplets, the index is obtainedby combining the gauge and matter contributions: I d = ( q ) G ) ∞ | Weyl( G ) | (cid:73) ds πis (cid:89) α ∈ roots( G ) F( s α ; q ) × (matter index( s ; q )) . (3.13)For example, given a chiral multiplet (of R-charge zero) in representation R of G , and aFermi multiplets (of R-charge 1) in representation R (cid:48) of G , the matter index takes the form (cid:81) λ ∈ wt( R ) C( s λ ; q ) (cid:81) µ ∈ wt( R (cid:48) ) F( s µ ; q ). The formula (3.13) combines operators from the chiraland Fermi multiplets with gauginos, and projects to G -invariants. We perform an analysis analogous to that of Section 3.1.1 to find the half-index for 3d chiralmultiplets.We begin with a single free 3d N = 2 chiral multiplet Φ , with charge +1 under a U (1) x flavor symmetry (with fugacity x ) and charge zero for U (1) R . Recall from Section 2.1 thatΦ decomposes into a N = (0 ,
2) chiral multiplet Φ = φ + θ + ψ + + ... and a Fermi multipletΨ = ¯ ψ − + ... . In the 3d bulk, the operators ∂ nz φ and ∂ nz ¯ ψ − are in Q + -cohomology. However,in contrast to Section 3.1.1, the operators ∂ z ¯ φ and ψ − (and their further derivatives) are nolonger Q + -closed: we have Q + ( ∂ z ¯ φ ) ∼ ∂ z ¯ ψ + and Q + ( ψ − ) ∼ ∂ ⊥ φ , neither of which are set tozero by the 3d equations of motion. Thus, the bulk operators potentially contributing to anindex are ∂ nz φ ∂ nz ¯ ψ − U (1) x − U (1) R U (1) J n n + fugacity in index : q n x − q n +1 x − ( n ≥ . (3.14)On a Neumann (N) boundary condition, all the ∂ nz ¯ ψ − operators are killed. The bosonic ∂ nz φ operators survive, and lead to the half-index II N ( x ; q ) = (cid:89) n ≥ − q n x = 1( x ; q ) ∞ . (3.15)On a Dirichlet (D) b.c., the ∂ nz φ operators are killed while the fermionic ∂ nz ¯ ψ − operatorssurvive, giving a half-index II D ( x ; q ) = (cid:89) n ≥ (1 − q n +1 x − ) = ( qx − ; q ) ∞ . (3.16)– 29 –n a similar way, a 3d chiral multiplet of R-charge ρ has half-indices II N (( − q ) ρ x ; q ) and II D (( − q ) ρ x ; q ), obtained by a standard shift in fugacities x → − q x . We can use these basic half-indices to test the operations that “flip” between N and D b.c.,as in 2.2.1. We expect that a 3d chiral with N b.c. coupled to a 2d boundary Fermi multipletof opposite flavor charge is equivalent to D b.c.; and conversely that a 3d chiral with D b.c.coupled to a boundary chiral multiplet of the same flavor charge is equivalent to N b.c. Theseoperations are reflected in the obvious identities II N ( x ; q )F( x ; q ) = II D ( x ; q ) , II D ( x ; q )C( x ; q ) = II N ( x ; q ) . (3.17) c and singular b.c. A D c boundary condition, where Φ (cid:12)(cid:12) ∂ = c is set to a nonzero value at the boundary, is verysimilar to D b.c. The same ∂ nz ¯ ψ − operators survive. However, U (1) x flavor symmetry isbroken, and U (1) R symmetry is preserved precisely if Φ has R-charge zero. Correspondingly,the half-index is II D c ( q ) = II D ( x ; q ) (cid:12)(cid:12) x =1 = ( q ) ∞ . (3.18)This analysis indicates how we should treat D c boundary conditions in more interestinggauge theories with chiral matter, superpotentials, etc.: the index may first be written downfor D b.c., and then “deformed” to D c by specializing the flavor fugacities of the relevantchiral multiplets according to the broken flavor symmetry, i.e. x →
1. The specialization isdone in the entire index. N b.c. The 3d gauge multiplet decomposes into a 2d gauge multiplet with fermionic field strengthΥ and a chiral multiplet S , as discussed in Section 2.5.1. Given a Neumann ( N ) boundarycondition, only the operators formed out of the fields in Υ survive at the boundary. Ignoringgauge invariance for the moment, the operators in Q + -cohomology are as follows.In a pure gauge theory, the elementary bulk operators in Q + -cohomology are the gaugino λ − and its D z -derivatives. In contrast to a purely 2d gauge theory, D z λ − is no longer closed,since Q + D z λ − ∼ D z F z ¯ z is no longer set to zero by the 3d equations of motion. Thus, theoperators contributing to the half-index for gauge group G are D nz λ − G adj U (1) R U (1) J n + fugacity in index: − q n +1 s α ( n ≥ . (3.19)The full contribution to the index from polynomials in the D nz λ − is (cid:89) α ∈ wt(adj) ( qs α ; q ) ∞ = ( q ) rank( G ) ∞ (cid:89) α ∈ roots( G ) ( qs α ; q ) ∞ . (3.20)– 30 –s in 2d, the fugacity s is valued in the complexified torus T C of G . Projecting to gauge-invariants with a contour integral and a Vandermonde determinant leads to( q ) rank( G ) ∞ (cid:73) ds πis (cid:89) α ∈ roots( G ) (1 − s α )( qs α ; q ) ∞ = ( q ) rank( G ) ∞ (cid:73) ds πis (cid:89) α ∈ roots( G ) ( s α ; q ) ∞ (3.21)The half-index for a N b.c. is not directly sensitive to a bulk Chern-Simons level. How-ever, since gauge symmetry is preserved by a N b.c., the boundary gauge anomaly must becancelled. The computations of anomalies in Section 2.5.4 show that unless additional matteris present, a N b.c. for pure gauge theory only makes sense at Chern-Simons level k = − h .Thus it is only for k = − h that (3.21) computes an actual half-index.In the presence of bulk or boundary matter, other Chern-Simons levels are possible.Matter is incorporated into the half-index in a standard way: • The 2d index of boundary matter (or a boundary gauge theory, or a boundary CFT)should be inserted directly into the integrand of (3.21). The 2d theory will typicallyhave a G flavor symmetry that is gauged in coupling to the bulk, thus the 2d index willdepend on the fugacity s . • For 3d chiral matter with N b.c., D b.c., D c b.c., or some combination thereof, the half-index is computed as if G were a flavor symmetry, and then inserted into the integrandof (3.21). (Note that D c b.c. are not possible for 3d chirals charged under G , if gaugesymmetry is to be preserved at the boundary.)For example, a G gauge theory with a chiral multiplet in representation ( R , ρ ) of G × U (1) R has a half-index( N , N) : ( q ) rank( G ) ∞ (cid:73) ds πis (cid:89) α ∈ roots( G ) ( s α ; q ) ∞ (cid:89) λ ∈ wt( R ) II N (( − q ) ρ s λ ; q ) (3.22)( N , D) : ( q ) rank( G ) ∞ (cid:73) ds πis (cid:89) α ∈ roots( G ) ( s α ; q ) ∞ (cid:89) λ ∈ wt( R ) II D (( − q ) ρ s λ ; q ) (3.23)for N and D b.c. on the chiral, respectively. Additional fugacities may be added for flavorsymmetries. Again, cancellation of the gauge anomaly constrains the bulk CS level. Here( N , N) b.c. requires a CS level k = − h + T R , while ( N , D) requires k = − h − T R .As observed in many analogous index computations, and in particular in (0 ,
2) ellipticgenus calculations [52, 88, 89], doing the integration carefully may require careful contoursprescriptions. As long as the 2d index of boundary matter does not contribute poles to theintegrand for 3d gauge fields, our naive operator-counting approach works in a straightforwardway. This will be the case in many of our examples, where the boundary matter is a collectionof 2d Fermi multiplets.If the 2d index of boundary matter does contribute poles, then our naive approach wouldneed to be re-considered. There is a simple strategy which should work: replace N b.c. witha combination of D b.c. and 2d gauge fields and then apply the known contour prescriptionsfor contour integrals associated to 2d gauge fields.– 31 – .4 3d gauge symmetry: D b.c. and boundary monopoles Now consider pure 3d gauge theory in the presence of a Dirichlet ( D ) boundary condition.The λ − gaugino is killed, but there are other operators in Q + -cohomology formed fromthe leading component σ + iA ⊥ of S and its D z derivatives. Recall that D b.c. breaks G gaugesymmetry to a global G ∂ symmetry at the boundary, allowing only gauge transformationsthat are constant along the boundary. Nevertheless, one may still perform “residual” gaugetransformations that depend on x ⊥ , given by elements g ( x ⊥ ) ∈ G that restrict to the identityat the boundary. The boundary operator σ + iA ⊥ is not quite invariant under these residualgauge transformations, since ( σ + iA ⊥ ) (cid:12)(cid:12) ∂ → ( σ + iA ⊥ ) (cid:12)(cid:12) ∂ + ig − ∂ ⊥ g (cid:12)(cid:12) ∂ . However, the “fieldstrength” [ D z , ∂ ⊥ + σ + iA ⊥ ] = D z σ + iF z ⊥ and its further D z derivatives are invariant, andgenerate the Q + -cohomology. They have charges D n +1 z σ + iD nz F z ⊥ G ∂ adj U (1) R U (1) J n + 1fugacity in index: q n +1 s α ( n ≥ , (3.24)leading to a half-index (cid:89) α ∈ wt(adj) qs α ; q ) ∞ = 1( q ) rank( G ) ∞ (cid:89) α ∈ roots( G ) qs α ; q ) ∞ . (3.25)Here s is a fugacity for the boundary G ∂ symmetry, and there is no need for any projectionto G -invariants.Note that F z ⊥ is the chiral current for the G ∂ flavor symmetry on the boundary. Thus,the operators in (3.24) may be interpreted as positive modes of a complexified version of thiscurrent. Alternatively, in an abelian gauge theory, the operators in (3.24) are just modes ofthe dual photon.In a 3d gauge theory with D b.c., coupled to 3d and/or 2d matter, the index (3.25) issimply multipled by the matter index. As usual, the matter index may depend on the fugacity s . For example, a 3d gauge theory with a chiral multiplet of R-charge zero in representation R of G , with D b.c. for the gauge multiplet and (say) D b.c. for the chiral, has boundaryoperators counted by the index1( q ) rank( G ) ∞ (cid:89) α ∈ roots( G ) qs α ; q ) ∞ (cid:89) λ ∈ wt( R ) II D ( s λ ; q ) . (3.26)So far, the index is completely insensitive to a bulk Chern-Simons coupling. There is no needto cancel the ’t Hooft anomaly for G ∂ . In truth, (3.25) and (3.26) must be supplemented by nonperturbative contributions frommonopole operators. The D b.c. for gauge fields are just right to support boundary monopoles– 32 –ith a conserved topological charge, which enhance the space of local operators. The basicanalysis of these operators appeared in [60], and we review the main ideas here.It is well known ( cf. [3, 128]) that one can define a BPS monopole operator (that is, inparticular, Q + -closed) in the bulk of a 3d N = 2 theory as a disorder operator, by specifyinga singular solution to the BPS equations F = ∗ Dσ , D ∗ σ = 0 . (3.27)For gauge group G = U (1), the basic solutions are Dirac monopoles: they have σ = m r , where r is the radial distance from the singularity and m is a constant. The condition that theflux through a 2-sphere surrounding the singularity be quantized, (cid:72) S F ∈ π Z , constrains m ∈ Z . For general gauge group G , the singular solutions to (3.27) come from embeddingsof the basic abelian monopole σ = r into G , and are thus labelled by cocharacters m ∈ Hom( U (1) , T ) , (3.28)where T ⊂ G is the maximal torus of G . (More accurately, the embeddings are labelled bycocharacters modulo the action of the Weyl group.)In the present case, we want to consider a monopole operator on the boundary, defined(say) by a singular solution to (3.27) on a half-space. It is easy to see that in abelian G = U (1)gauge theory the basic solution for the scalar field σ = 1 (cid:112) ( x ⊥ ) + | z | , F = ∗ Dσ on x ⊥ ≤ D b.c. (2.64)-(2.65), which do not constrain σ (cid:12)(cid:12) ∂ of F z ⊥ (cid:12)(cid:12) ∂ ,but set F z ¯ z (cid:12)(cid:12) ∂ = 0. One can also find a gauge transformation near the boundary so that theconnection A corresponding to (3.29) satisfies A (cid:12)(cid:12) ∂ = 0 away from z = ¯ z = 0.More generally, we can use a cocharacter m as in (3.28) to embed the basic abelian solution(3.29) into any gauge group G , thus defining a boundary monopole operator of “charge” m .Geometrically, one may think of boundary monopole operators as follows. Take G = U (1).Consider surrounding an operator O at x ⊥ = z = ¯ z = 0 with a hemisphere HS . The D b.c. trivializes the principal G bundle at the boundary and sets A z (cid:12)(cid:12) ∂ = A ¯ z (cid:12)(cid:12) ∂ = 0 there. Inparticular, D b.c. trivializes the G -bundle on ∂ ( HS ) = S . In the interior of HS , thebundle may be topologically nontrivial, and its topological type is precisely measured by thecurvature integral (cid:90) HS F π = m ∈ Z . (3.30)This is the monopole charge of O . The curvature integral gives a well-defined integer preciselybecause of the trivialization at the boundary, and the boundary condition A z (cid:12)(cid:12) ∂ = A ¯ z (cid:12)(cid:12) ∂ = 0. In general, given a U (1) bundle with connection on HS that is trivialized at the boundary, Stokes’ theoremreads (cid:82) HS F = 2 πm + (cid:72) S A . – 33 –n the case of the full 3d index (or S × S partition function), it is well known howbulk monopole operators contribute. A localization computation [79, 91] expresses the full3d index as a sum over flux sectors on S , i.e. over cocharacters of G . Notably, this is asum over abelian flux sectors, i.e. over all cocharacters rather than their Weyl-orbits. Thelocalization computation for the half-index with D b.c. proceeds a similar way, summing overabelian fluxes on the hemisphere.We are led to a complete, nonperturbative formula for the half-index of the form1( q ) rank( G ) ∞ (cid:88) m ∈ cochar( G ) (cid:20) (cid:89) α ∈ roots( G ) q m · α s α ; q ) ∞ (cid:21) q k eff Tr( m ) s k eff m × [matter index]( q m s ) . (3.31)Here all the fugacities s for the boundary G ∂ symmetry have been shifted s → q m s , reflectingthe fact that electrically charged states acquire spin in the presence of magnetic flux. In termsof operators, one would expect that a monopole of charge m dressed by operators of electriccharge λ ∈ wt( G ) acquires spin m · λ . In addition, in the presence of a Chern-Simons couplingat level k , even a bare monopole operator is induced to have nontrivial electric charge km and spin k Tr( m ). This leads to the extra weight q k Tr( m ) s km in the monopole sum.We propose that for boundary monopole operators the correct Chern-Simons level toinclude in the index (3.31) is not the bare UV Chern-Simons level k from the bulk, but ratheran effective k eff := CS level that captures the total boundary G ∂ anomaly,including shifts from gauginos and bulk or boundary matter (3.32)In other words, k eff is such that the boundary anomaly I is the exterior derivative of a Chern-Simons form at level k eff . A strong argument for using this effective level comes from observingthat bulk Chern-Simons terms in the presence of D b.c. are equivalent in the IR to boundarychiral matter, and the effective level k eff (or more accurately, the boundary anomaly) is theonly quantity that consistently captures the effect of both. We will thoroughly test theproposed use of k eff in many examples. It would be satisfying to reproduce this proposalusing localization, as in the case of Neumann boundary conditions [55].As a simple test, consider an abelian G = U (1) bulk gauge theory at level k − witha 3d chiral of U (1) ∂ charge +1 and R-charge +1 (for convenience). By our proposal, thehalf-index of D b.c. on the gauge multiplet and N b.c. on the chiral is computed by II D , N ( s ; q ) = 1( q ) ∞ (cid:88) m ∈ Z q ( k − m s ( k − m − q + m s ; q ) ∞ , (3.33) Just as in Section 2.4.1, ‘Tr( m )’ denotes the Cartan-Killing form, the same one appearing in the Chern-Simons action k Tr[
AdA + ... ], normalized to be the usual trace in the fundamental representation of u ( N ).Similarly, the Cartan-Killing form is implicitly being used to transform a magnetic charge m ∈ cochar( G ) toan electric charge km ∈ wt( G ), so that s km makes sense. – 34 –hich reflects the boundary anomaly I = ( k − ) f − f = ( k − f in the presence of Nb.c., so k eff = k −
1. Similarly, the half-index with D b.c. on the chiral is computed by II D , D ( s ; q ) = 1( q ) ∞ (cid:88) m ∈ Z q km s km ( − q − m s − ; q ) ∞ , (3.34)where now I = ( k − ) f + f = k f , so k eff = k . These two boundary conditions shouldbe related by a flip, e.g. ( D ,N) should be equivalent to ( D ,D) coupled to a boundary Fermimultiplet Γ of U (1) ∂ charge − II D , D ( s ; q ) = F( − q s − ; q ) II D , N ( s ; q ) (3.35)due to the theta-function identity( − q − m s − ; q ) ∞ = F( − q − m s − ; q )( − q + m s ; q ) ∞ = q − m s − m F( − q s − ; q )( − q + m s ; q ) ∞ (3.36)that can be applied to the summand of II D , D ( s ; q ).Formula (3.31) has a natural generalization to a product of gauge groups, or gauge andflavor groups. Let us denote G entire gauge group (possible a product of groups) F entire global symmetry group, including flavor and R-symmetry˜ G = G × F full symmetry group g , f , ˜ g the gauge, global, and gauge × global Lie algebras s, x, − q gauge, flavor, and R-symmetry fugacities˜ s = ( s, x, − q ) joint fugacity for ˜ G (3.37)Then, given D b.c. for all of G , one sums over cocharacters m ∈ cochar( G ) ⊂ g , whichby the embedding G ⊂ ˜ G may also be thought of as cocharacters m ∈ cochar( ˜ G ) ⊂ ˜ g .In the monopole sum, all fugacities are shifted ˜ s → q m ˜ s (meaning explicitly ( s, x, − q ) → ( q m s, x, − q )), and there is an extra factor q k eff [ m,m ] ˜ s k eff [ m, − ] (3.38)where k eff : ˜ g × ˜ g → R (3.39)is the bilinear form defined by the full boundary ’t Hooft anomaly polynomial, and s k eff [ m, − ] means exp (cid:0) k eff [ m, log ˜ s ] (cid:1) .For example, in pure 3d U (1) gauge theory at level k , with a D b.c., the boundary anomalypolynomial is k f + 2 ff x , where f x is a field strength for the topological U (1) x flavor symmetry(the R-symmetry does not enter here). The corresponding bilinear form is (cid:0) k
11 0 (cid:1) . Letting s, x denote the fugacities for the boundary U (1) ∂ symmetry and for U (1) x , the half-indexbecomes 1( q ) ∞ (cid:88) m ∈ Z q ( m (cid:16) k
11 0 (cid:17) ( m ) e ( m (cid:16) k
11 0 (cid:17)(cid:16) log s log x (cid:17) = 1( q ) ∞ (cid:88) m ∈ Z q km s km x m . (3.40)– 35 – .4.2 Comparison between D and N We now have prescriptions for D and N boundary conditions. when the D and N boundaryconditions are both well-defined (with the same boundary conditions on the rest of the othersuper-multiplets), it should be also possible to obtain the half-index of N boundary conditionsby 2d gauging the half-index of the corresponding D boundary conditions.If the boundary matter does not contribute poles to the index, so that we do have areliable prescription for both sides, it should be possible to match the answers of our twoprescriptions.We will not do so in detail, but the match is intuitively clear: in the correct circumstancesthe N contour integral can be executed on C ∗ , picking a semi-infinite sequence of poles. Onthe other hand, the 2d contour integral will only pick the poles in (3.31) which lie in thefundamental region | q | < | s | <
1. These come again in semi-infinite sequences labeled by themagnetic charge m , as the q m shifts push the poles out of the fundamental region when m issufficiently large with the appropriate sign.On the other hand, as we mentioned before, when the boundary matter does contributepoles to the index, then we expect the 2d prescription applied to (3.31) to produce the correct,unknown prescription for the half-index of N boundary conditions. Altogether, the computation of a half-index involves of a 3d N = 2 gauge theory T with a2d N = (0 ,
2) boundary condition B involves:1. Computing a 2d index I d , as in Section 3.1, for any 2d N = (0 ,
2) theory that is coupledto the bulk in defining B .2. Multiplying by II N or II D half-indices for all 3d chiral multiplets as in Section 3.2, de-pending on whether these multiplets are given N or D b.c., prior to coupling to any 2dmatter.3. Integrating over fugacities ( i.e. projecting to invariants) for the part of the bulk gaugegroup given N b.c., using the measure (3.21) from Section 3.3.4. Summing over monopole sectors for the part of the bulk gauge group given D b.c., withthe “measure” (3.31) from Section 3.4, along along with shifts of fugacities and theeffective Chern-Simons contribution from (3.31).5. For any boundary flavor symmetries broken by D c b.c. or singularities such as Nahmpoles, setting the corresponding fugacities to 1, as in Section 3.2.2. A useful modification of the half-index comes from including a half-BPS line operator thatpreserves the Q + supercharge. Here we envision a line operator L supported on a ray perpen-dicular to the boundary, as in Figure 2, hitting the boundary at (say) the origin z = ¯ z = 0.– 36 –uch a line operator can preserve a full 1d N = 2 subalgebra of 3d N = 2 generated by Q + and Q − . The half-index can be defined to count local operators at the intersection of L andthe boundary. Alternatively, under a state-operator correspondence, the half-index with a line operator L can be interpreted as a D × S partition function with L inserted along { } × S . L B M L Figure 2 . Boundary local operators at the end of a line operator L form a module M L for theboundary chiral algebra. There are two simple types of line operators that we will consider here. The first is asupersymmetric Wilson line W R in representation R of the bulk gauge group, defined as W R = P exp i (cid:90) x ⊥≤ z =¯ z =0 ( A ⊥ − iσ ) dx ⊥ . (3.41)If the gauge multiplet is given Neumann ( N ) b.c., the boundary operators at the end of W R are no longer gauge-invariant, but rather must be in representation R . The half-index isaccordingly modified by inserting a character N : (cid:73) ds πis II matter ( s ) W R (cid:32) (cid:73) ds πis Tr R ( s ) II matter ( s ) , (3.42)which precisely projects to operators in representation R . Alternatively, if the bulk gaugemultiplet is given Dirichlet ( D ) b.c., the boundary operators at the end of W R are simplytensored with the representation R of the G ∂ boundary flavor symmetry. In the presence of W R and monopole flux, this can modify the spin of boundary operators by the mechanismof Section 3.4; accordingly, the half-index takes the form D : (cid:88) m ∈ cochar( G ) II matter , CS W R (cid:32) (cid:88) m ∈ cochar( G ) Tr R ( q m s ) II matter , CS . (3.43) These form a module M L for the usual chiral algebra of boundary local operators (since there is an OPEbetween generic boundary local operators and boundary local operators stuck to the endpoint of L ). Thus,the half-index in the presence of L can be interpreted as a character of the module M L . This is analogous tothe behavior of the 4d N = 2 index in the presence of a surface operator [129, 130]. – 37 –ore trivially, we can insert a bulk Wilson line for a flavor symmetry rather than a gaugesymmetry. This again just tensors the boundary operators by representation R for the flavorsymmetry, and multiplies the entire half-index by Tr R ( x ), where x is the flavor fugacity.A second simple type of line operator is a vortex line V n for an abelian gauge or flavorsymmetry, which can be understood as an insertion of n units of flux F z ¯ z ∼ πn δ (2) ( z, ¯ z ) (3.44)through the z, ¯ z plane. More precisely, V n is a disorder operator that requires the connectionto attain the singular profile A ∼ n dθ near z = ¯ z = 0. Working in holomorphic gauge A ¯ z = 0,this profile looks like A z ∼ nz , A ¯ z = 0 , (3.45)and can reached by applying a singular complex gauge transformation g ( z ) = z n to a smoothconfiguration. This forces charged matter fields to have a zero or pole of order ∼ n atthe location of the vortex. The insertion of a V n vortex line for an abelian gauge or flavorsymmetry G shifts the spin of all charged operators by − n units (times the operator’s charge),and thus acts on the index by shifting the corresponding fugacity s V n −→ q − n s or x V n −→ q − n x . (3.46)Vortex lines for a dynamical abelian gauge symmetry are relatively boring, as they canbe screened in the bulk (in other words, the gauge transformation g ( z ) = z n can be undone).With N b.c. for the gauge symmetry, they can also be screened on the boundary, so the N half-index is insensitive to a V n insertion. This is manifest in (3.46), since the shift s → q − n s is invisible after one integrates over s (projects to gauge invariants). With D b.c., the vortexline does have a mild effect, as it leaves behind a singularity on the boundary. The shift s → q − n s could be removed from the monopole sum (3.31) by redefining m → + n , were itnot for the additional prefactor q k eff Tr( m ) s k eff m , controlled by effective Chern-Simons terms.Due to this extra prefactor, the D half-index obeys II D ( q − n s, x ) = q − k eff n x n s k eff n II D ( s, x ) , (3.47)where x is the fugacity for the topological U (1) symmetry dual to the bulk G gauge symmetry.This identity reflects the classic phenomenon that, in the presence of Chern-Simons terms, avortex line for an abelian gauge symmetry is equivalent to a Wilson line — here, a Wilsonline both for G and the topological U (1). The bulk 3d index, S partition, and holomorphic blocks of a 3d N = 2 theory all satisfy acommon set of difference equations [18, 67, 73]. On the holomorphic blocks B ( x ; q ), whichdepend on fugacities x i for each bulk flavor symmetry, the difference equations take the Here we will assume that the flavor symmetry is abelian. Given a nonabelian symmetry, we work with amaximal torus. – 38 –orm f a ( p i , x i ; q ) · B ( x, q ) = 0 , (3.48)where the f a are a finite set of polynomials in the q -commuting operators p i , x i , which obey p i x j = q δ ij x j p i . (3.49)The x i operators act on B ( x ; q ) by multiplication, while the p i operators act by a q -shift,sending x i → qx i . The 3d index and S partition functions obey two sets of equations of theform (3.48), involving two mutually commuting copies of the algebra (3.49) and exactly the same polynomials f a .Physically, all these difference equations are a consequence of identities in the algebra ofhalf-BPS line operators of the 3d theory — in particular, identities among abelian vortex linesand flavor Wilson lines [18, 131]. We therefore expect that 3d half-indices would also satisfy(3.48). This turns out to be true, up to a small modification controlled by the boundary ’tHooft anomalies.To understand the modification, we consider a 3d N = 2 theory with U (1) x flavor sym-metry, and a boundary condition that preserves this flavor symmetry. Suppose that thehalf-index obeys some difference equation f ( p, x ; q ) · II ( x ; q ) = 0 , (3.50)where x acts by multiplication and p acts by shifting x → qx ( i.e. p , x represent the insertionsof flavor vortex and Wilson lines, respectively). Consider what happens if we were to changethe boundary ’t Hooft anomaly for U (1) x by +1 units. The shift can be achieved by addinga boundary Fermi multiplet of U (1) x charge +1 and U (1) R charge zero, whose 2d index isF( − q x ; q ) = ( − q x ; q ) ∞ ( − q /x ; q ) ∞ . The new half-index is II (cid:48) ( x ; q ) = II ( x ; q )F( − q x ; q ),and is annihilated by the operator f (cid:48) ( p, x ; q ) = F( − q x ; q ) f ( p, x ; q )F( − q x ; q ) − = f ( q xp, x ; q ) . (3.51)Thus, shifting the boundary ’t Hooft anomaly modifies the difference operator by replacing p → q xp . More symmetrically, we might write the modification as a normal-ordered product p → : xp : = q xp = q − px . (3.52)The redefinition (3.52) is an automorphism of the Weyl algebra (3.49).This argument is easily generalized to determine how any shift in the boundary ’t Hooftanomaly modifies operators acting on the half-index. To express the result concretely, supposethat the difference in boundary ’t Hooft anomalies is encoded in a quadratic polynomial The actual algebra (3.49) has a nice interpretation as arising from ’t Hooft and Wilson lines in an Omega-deformed abelian 4d N = 2 theory, cf. [132, 133], coupled to a 3d N = 2 theory on its boundary. See [18, 67]for further discussion. – 39 – I ( x i , r ). Let q = e β , so that we may represent p i = exp (cid:0) βx i ∂∂x i (cid:1) . Then the shift inanomalies modifies difference operators by conjugation, f ( p i , x i ; q ) → exp (cid:104) − β δ I (log( x i ) , iπ + β ) (cid:105) f ( p i , x i ; q ) exp (cid:104) β δ I (log( x i ) , iπ + β ) (cid:105) (3.53)or simply p j → exp (cid:104) − β δ I (log( x i ) , iπ + β ) (cid:105) p j exp (cid:104) β δ I (log( x i ) , iπ + β ) (cid:105) . (3.54)Despite the complicated-looking expression, (3.54) is simply a redefinition of p i by amonomial in the x ’s. For example, if there is a single U (1) flavor symmetry and we shift theanomaly by δ I ( x , r ) = x as above, then p → e − β (log x ) pe β (log x ) = e − β (log x ) e β (log x + β ) p = q xp , (3.55)just as in (3.52). The appearance of ‘ iπ + β ’ in (3.54) reflects the fact that the U (1) R fugacityin the half-index is exp (cid:0) iπ + β (cid:1) = − q .We can now understand which operators will actually annihilate the half-index of a UVboundary condition in a 3d N = 2 theory. Suppose that no flavor symmetry is broken by theboundary condition: so the chirals all have N or D b.c. (not D c ); and in the case of N b.c.,there is no mixed gauge-flavor anomaly. If the boundary ’t Hooft anomaly were exactly zero,we would expect the very same difference operators (3.48) to annihilate the half-index f a ( p i , x i ; q ) · II ( x ; q ) = 0 . (3.56)This is consistent with the fact that the holomorphic blocks of [67] were carefully engineeredto have zero boundary anomaly. In addition, given D b.c. for the gauge multiplets, there areadditional equations expressing independence under q -shifts of the G ∂ fugacities ‘ s ’, namely( p s − · II ( x, s ; q ) = 0 . (3.57)When the boundary anomaly is not zero, we use (3.54) to modify the difference operatorsaccordingly. Explicitly, given a boundary anomaly I ( x i , f , r ) (where f is a G ∂ field strengthfor D b.c.), we conjugate the p ’s and p s in (3.56)-(3.57) byexp (cid:104) β δ I (log( x i ) , log( s ) , iπ + β ) (cid:105) . (3.58)For example, after conjugation, we recover from (3.57) the simple difference equation (3.47)relating G ∂ vortices and Wilson lines.If a symmetry is broken by the boundary condition, its fugacity (say ‘ y ’) should besubsequently removed from the difference operators by first eliminating the dual operator p y from (the conjugated versions of) (3.56), (3.57), and then setting y → same f a (up to monomial redefinitions) annihilate UV half-indices and holomorphic blocks (and– 40 –ull indices and S partition functions) can actually be seen explicitly from the formulaicdefinitions of all these objects. In each case, simple chiral-matter partition functions are mul-tiplied together, then gauge fugacities are integrated over (and/or monopole sectors summedover). The difference equations can correspondingly be constructed step by step, startingfrom elementary equations satisfied by free chirals. The steps are identical, up to simplemonomial redefinitions, no matter which object is being considered.
The simplest dual pair of 3d N = 2 theories is actually part of a mirror “triality,” involvingtheories T free chiral ↔ T (cid:48) U (1) + a chiral ↔ T (cid:48)(cid:48) U (1) − + a chiral (4.1)This is the simplest example of 3d N = 2 “mirror symmetry” [1], and also follows fromdeforming the SQED ↔ XYZ duality of [3] by a large real mass [18]. The triality played afundamental role in the 3d-3d correspondence, where it encoded a Z rotation symmetry of atetrahredron [18]. This triality is a supersymmetric version of classic particle-vortex duality,and can be used to derive both bosonic and fermionic versions of non-supersymmetric particle-vortex duality [24, 25, 34]. Let’s first focus on theory T . We define T to contain a free 3d N = 2 chiral multiplet(Φ , Ψ) of charge ρ = 0 for U (1) R R-symmetry and charge +1 for a U (1) x flavor symmetry.As in [18], we add − units of background Chern-Simons coupling for U (1) x to cancel aparity anomaly. Specifically, the anomaly polynomial encoding the bulk UV (background)Chern-Simons couplings is −
12 ( f x − r ) . (4.2)The two basic N = (0 ,
2) boundary conditions for the free chiral are Neumann (N, Ψ | ∂ = 0)and Dirichlet (D, Φ | ∂ = 0), as in Section 2.1. The corresponding boundary ’t Hooft anomalies(Section 2.4) and half-indices (Section 3.2) areanomaly half-indexN : I N = − ( f x − r ) II N ( x ; q ) = ( x ; q ) − ∞ , D : I D = 0 II D ( x ; q ) = ( qx − ; q ) ∞ . (4.3)Following Section 3.7, we expect that these half-indices satisfy certain difference equa-tions. The difference operator that annihilates the full 3d index of a free chiral is p + x − − In 3d theories associated to 3-manifolds via the 3d-3d correspondence, the difference equations are versionsof the “quantum A-polynomial” on the 3-manifold side [134, 135], whose analogous step-by-step constructionwas given by [136]. – 41 –ince the boundary anomaly for D b.c. is zero, this should also annihilate II D ( x ; q ), and indeed p · II D ( x ; q ) = II D ( qx ; q ) = ( x − ; q ) ∞ = (1 − x − )( qx − ; q ) ∞ = (1 − x − ) II D ( x ; q ) , (4.4)so ( p + x − − · II D ( x ; q ) = 0. For N b.c., we must conjugate the difference operator byexp (cid:2) − β (log( x ) − iπ − β ) (cid:3) , corresponding to the boundary anomaly. This modifies p → e β (cid:0) log( x ) − iπ − β (cid:1) e − β (cid:0) log( x ) − iπ + β (cid:1) p = − x − p , (4.5)and changes the difference operator to − x − p + x − −
1. Thus we expect that ( p − x ) · II N ( x ; q ) = 0, and we check that this is true: p · II N ( x ; q ) = II N ( qx ; q ) = ( qx ; q ) − ∞ = (1 − x )( x ; q ) − ∞ = (1 − x ) II N ( x ; q ) . (4.6)We may also modify the Dirichlet b.c. to D c (Φ | ∂ = c ). This breaks U (1) x flavorsymmetry but preserves U (1) R . As discussed in Section 3.2.2, the effect is to set x → m ∈ Z as inSection 3.6, sending x → q − m x , and subsequently impose D c b.c.; overall, this sets x = q − m in the Dirichlet index. We have: anomaly half-indexD c : 0 II D c ( q ) = ( q ) ∞ , D c + Vortex m : 0 II D c,m ( q ) = ( q m ; q ) ∞ . (4.7)Note that the vortex line effectively sets φ ∼ z m at the boundary, which breaks SUSY unless m ≥
0. Correspondingly, the half-index II D c,m ( q ) vanishes unless m ≥ U (1) + a chiral Next, consider the theory T (cid:48) . It is a U (1) gauge theory, with a chiral multiplet (Φ (cid:48) , Ψ (cid:48) ) ofcharge +1 under the gauge group and R-charge ρ = 0. There is also a topological flavorsymmetry U (1) x . The bulk UV Chern-Simons levels required for this theory to be dual to T are encoded in the anomaly polynomial I (cid:48) bulk = 12 ( f − r ) + 2 f f x − r . (4.8)In particular, the U (1) gauge symmetry (with field strength f ) has Chern-Simons level + .Recall that under the duality T ↔ T (cid:48) , the fundamental chiral of T maps to a monopoleoperator of T (cid:48) , and (correspondingly) the ordinary flavor symmetry U (1) x of T maps to thetopological flavor symmetry of T (cid:48) .We would like to find boundary conditions for T (cid:48) that are dual to N, D, D c , and D c,m in T . It is most enlightening to begin with N. Notice that the N b.c. leaves the chiral Φ of T free at the boundary. Thus we would expect that a dual boundary condition in T (cid:48) wouldleave the vev of a monopole operator unconstrained at the boundary. There is only one choice– 42 –f boundary condition for the gauge fields that has this property, namely Dirichlet ( D ) asin Sections 2.5, 3.4. Recall that D b.c. has an additional boundary U (1) ∂ flavor symmetry.We try choosing D b.c. for the chiral of T (cid:48) as well, and examine the half-index. To usethe prescription for the half-index from Section 3.4, we must compute the boundary ’t Hooftanomaly: I (cid:48)D , D = ( f − r ) + 2 f f x − r (cid:124) (cid:123)(cid:122) (cid:125) bulk CS − r (cid:124) (cid:123)(cid:122) (cid:125) D for gauge + ( f − r ) (cid:124) (cid:123)(cid:122) (cid:125) D for chiral = f + 2( f x − r ) f (4.9)where now f is the field strength for U (1) ∂ . We read off from this a matrix of effectiveChern-Simons levels. The half-index then becomes a monopole sum II (cid:48)D , D ( x, y ; q ) = 1( q ) ∞ (cid:88) m ∈ Z q m ( − q − x ) m y m II D ( q m y ; q ) (cid:124) (cid:123)(cid:122) (cid:125) ( q − m y − ; q ) ∞ , (4.10)where we have used ‘ x ’ to denote the topological U (1) x fugacity and ‘ y ’ to denote the boundary U (1) ∂ fugacity, and the monomial prefactor q m ( − q − x ) n y n corresponds to the effectiveChern-Simons levels. Due to the q m , this series converges as an element of Z [[ q ]][ x ± , y ± ],and rather beautifully sums up to II (cid:48)D , D ( x, y ; q ) = ( xy ; q ) ∞ ( qx − y − ; q ) ∞ ( x ; q ) ∞ = II N ( x ; q ) F( xy ; q ) . (4.11)This calculation strongly suggests that the ( D , D) b.c. for T (cid:48) is dual to N b.c. for the freechiral of T together with a free Fermi multiplet of charge ( − , − ,
1) for U (1) x , U (1) ∂ , U (1) R .The appearance of this extra Fermi multiplet is not a surprise. First, since the ( D ,D) b.c. for T (cid:48) had a U (1) ∂ symmetry that acted on the boundary but not in the bulk, we would expecta dual b.c. for T to have some purely two-dimensional degrees of freedom charged under thissymmetry. Second, we may compare anomaly polynomials (4.9), (4.3), whose difference I (cid:48)D , D − I N = ( f + f x − r ) (4.12)is precisely the contribution of an extra boundary Fermi multiplet Γ. Thus anomaly matchingalone requires Γ. The map of operators across the duality identifies the boundary value Ψ (cid:48) | ∂ in theory T (cid:48) (which is gauge-invariant at the boundary) with the composite operator Φ | ∂ Γin T .Several more dualities may be inferred from (4.11). The ( D , D c ) b.c. for T (cid:48) , which setsΦ (cid:48) | ∂ = c and breaks the U (1) ∂ boundary symmetry, corresponds to setting y → For a quick proof that (4.10) and (4.11) are equivalent, it suffices that the two expressions obey thesame first-order q -difference equations in x and in y , and that they agree at a particular value of ( x, y ). Infact, it suffices to use the single difference equation in y , II (cid:48)D , D ( x, qy ) = − xy II (cid:48)D , D ( x, y ; q ), which follows fromF( qxy ) = − xy F( xy ) in (4.11) and follows from simple manipulations of the sum in (4.10). Then we checkagreement at y = 1 (for any x ), which is the well-known identity in (4.13). – 43 –ndex. Since ( q − m y − ; q ) ∞ vanishes at y = 1 unless m ≤
0, we find II (cid:48)D , D c ( x ; q ) = 1( q ) ∞ (cid:88) m ≤ q m ( − q − x ) n ( q − m ; q ) ∞ = (cid:88) m ≥ q m ( − q x − ) m ( q ) m = II N ( x ; q ) F( x ; q ) = II D ( x ; q ) . (4.13)This is actually a well-known q-series identity. It suggests that the ( D , D c ) b.c. for T (cid:48) is dualto ordinary D b.c. for T . Similarly, adding an additional vortex line of charge n and thenusing D c b.c. sets y → q − n II (cid:48)D , D c,n ( x ; q ) = 1( q ) ∞ (cid:88) m ≤ n q m ( − q − x ) m ( q n − m ; q ) ∞ = ( − n q − n n x n II D ( x ; q ) , (4.14)which suggests that the ( D , D c,n ) b.c. is dual to D b.c. for T together with a flavor Wilsonline of charge n (corresponding to x n ) and an extra shift of the background spin and R-charge.More interestingly, we may compute the half-index of ( D , N) b.c. for T (cid:48) , which give N b.c.to the chiral. Now the boundary ’t Hooft anomaly is just I (cid:48)D , N = I (cid:48) bulk − r − ( f − r ) = 2 f f x ,so the monopole sum takes the form II (cid:48)D , N ( x, y ; q ) = 1( q ) ∞ (cid:88) n ∈ Z x n II N ( q n y ; q ) . (4.15)It is convenient to rewrite II N ( q n y ; q ) = II D ( q n y ; q ) / F( q n y ; q ) = ( − n q n − n y n II D ( q n y ; q ) / F( y ; q ) =( − n q n − n y n II D ( q n y ; q ) C( y ; q ), whence II (cid:48)D , N ( x, y ; q ) = C( y ; q ) 1( q ) ∞ (cid:88) n ∈ Z ( − n q n − n y n x n II D ( q n y ; q ) (4.16) (4.10) = C( y ; q ) II (cid:48)D , D ( x, y ; q ) (4.17) (4.11) = II N ( x ; q ) F( xy ; q )C( y ; q ) . (4.18)This suggests that the ( D , N) b.c. for T (cid:48) is dual to the N b.c. for T , coupled to both aboundary Fermi multiplet Γ and a boundary chiral multiplet C . The symmetries and R-charges are just right for a boundary superpotential coupling (cid:90) dθ + Φ (cid:12)(cid:12) ∂ Γ C . (4.19)Notice how the chiral C may be understood as flipping D to N b.c. for the chiral (Φ (cid:48) , Ψ (cid:48) ) oftheory T (cid:48) , by a boundary superpotential coupling Ψ (cid:48) (cid:12)(cid:12) ∂ C ; the dual of this coupling in T isprecisely (4.19). Here we used F( q − m x ) = ( − m q − m m x m F( x ). – 44 –here are also a family of boundary conditions for T (cid:48) that use Neumann ( N ) for thegauge multiplet, preserving dynamical U (1) gauge symmetry at the boundary. Some of thesewere first discussed in [50]. Suppose we combine N with N b.c. for the charged chiral. Theboundary anomaly polynomial is 2 f f x , where f is now the curvature of the dynamical gaugesymmetry at the boundary. There is no gauge anomaly, so no additional boundary matter isrequired, but we see that the topological U (1) x symmetry will be broken by a mixed anomaly.Following Section 3.3, the index is simply computed as II (cid:48)N , N ( q ) = ( q ) ∞ (cid:73) ds πis II N ( s ; q ) (4.20)where s is the gauge fugacity. This can be evaluated by residues to give II (cid:48)N , N ( q ) = ∞ (cid:88) a =0 q − ; q − ) a = ( q ) ∞ = II D c ( q ) , (4.21)suggesting that ( N , N) b.c. is dual to D c b.c. for T . This is actually a physically sensibleanswer: the Neumann b.c. for an abelian gauge multiplet sets the scalar σ field equal to aboundary FI parameter, and more generally (as explained in Section 2.5.3) sets the chiraldual-photon field equal to a complexified FI parameter t d . In the quantum gauge theory,this means that the monopole operator of T (cid:48) should be set to a nonzero constant ∼ e t d at theboundary. The dual statement in theory T is that the chiral field should be set to a nonzeroconstant, which is exactly what D c b.c. does.A generalization of (4.20) is to add a Wilson line of charge n ending on the boundary.This leads to a projection onto boundary operators of gauge charge − n . The half-indexbecomes II (cid:48)N m , N ( q ) = ( q ) ∞ (cid:73) ds πis s n II N ( s ; q ) = ∞ (cid:88) a =0 q − an ( q − ; q − ) a = ( q − n ; q ) ∞ = II D c, − n ( q ) . (4.22)In other words, we find D b.c. for T deformed by a flavor vortex of charge − n . This reflectsthe standard fact that a Wilson line in abelian gauge theory is equivalent to a vortex line forthe topological flavor symmetry — i.e. the ordinary flavor symmetry in T .It is worth observing that the ( N , N) should be obtainable from ( D , N) by gauging the U (1) ∂ boundary symmetry. This breaks U (1) x , but it is useful to keep x in the calculationuntil the very end. Then the 2d gauging prescription is to multiply II (cid:48)D , N ( x, y ; q ) by ( q ) ∞ andtake the residue at y = 1, leading to II N ( x ; q ) F( x ; q ). Setting x = 1 we recover II D c ( q ).The physical interpretation of this calculation is somewhat interesting: on the mirror sidewe have an N b.c. coupled to a boundary theory consisting of an a U (1) gauge theory coupledto a single 2d chiral and a single 2d Fermi multiplet. It appears that the 2d theory flowsto a single Fermi multiplet in the IR, the “meson” µ = Γ C , with a dynamically generatedfermionic superpotential cµ , which then flips the boundary condition to D c .We could also combine N b.c. for the gauge fields with D b.c. for the chiral, but thisboundary condition does have a gauge anomaly, since its anomaly polynomial is (4.9). In– 45 –rinciple, we could also attempt to modify the N b.c. with additional boundary matter inorder to cancel the mixed anomaly for the topological U (1) x symmetry. Because of the signof the anomaly, though, we cannot do so by boundary Fermi multiplets: we need boundarychiral multiplets. For example, ( N , D) b.c. together with a boundary chiral multiplet ofcharges (1 , ,
0) for U (1) , U (1) x , U (1) R has a total boundary anomaly polynomial ( f − r ) +2 f f x − ( f + f x − r ) = − ( f x − r ) + r , so only an ’t Hooft anomaly remains.If we couple the boundary chiral multiplet to the bulk chiral multiplet by a bi-linearfermionic superpotential we simply convert D b.c. back to N b.c. and we do not get anythingnew. If we do not add such a coupling, the naive half-index takes the form II (cid:48)N , D+chiral ( x ; q ) = ( q ) ∞ (cid:73) ds πis II D ( s ; q ) C( sx ; q ) (4.23)= ( q ) ∞ (cid:73) ds πis ( qs − ; q ) ∞ ( sx ; q ) ∞ ( qs − x − ; q ) ∞ , but there is no obvious prescription to deal with the infinite line of poles of C ( sx ; q ) at s = q n x − ( n ∈ Z ).Instead, we can go back to the ( D , D) boundary condition, add the 2d chiral field to cancelthe U (1) ∂ ’t Hooft anomalies and then gauge U (1) ∂ as a 2d gauge symmetry. The resultingindex would have an overall factor of II N ( x ; q ) but would vanish, as on the mirror side onehas a boundary theory consisting of a 2d U (1) d gauge theory coupled to a Fermi multipletof charge ( − ,
1) for U (1) d , U (1) R and a chiral multiplet of charges (1 ,
0) for U (1) d , U (1) R ,which cancel each other in the index and leave no poles to be picked.We can interpret this tentatively as a manifestation of spontaneous SUSY breaking. Thisagrees with the physical picture we discussed for ( N , N): in the absence of the coupling Φ | ∂ µ ,the dynamically generated cµ fermionic superpotential for the U (1) d gauge theory will breakSUSY.We summarize the plethora of dual boundary conditions we have found so far: B (theory T ) B (cid:48) (theory T (cid:48) )N + 2d fermi ( D , D)N + 2d fermi/chiral ( D , N)D ( D , D c )D + Wilson n ( D , D c ) + Vortex n D c ( N , N)D c + Vortex n ( N , N) + Wilson − n (4.24) U (1) − + a chiral The theory T (cid:48)(cid:48) is similar to T . However, its Neumann boundary conditions turn out to bebetter behaved, while some of its Dirichlet boundary conditions exhibit bad behavior.We define theory T (cid:48)(cid:48) as a U (1) gauge theory with a chiral of gauge charge +1 and R-chargezero, with bulk UV Chern-Simons levels encoded by the anomaly polynomial I (cid:48)(cid:48) bulk = − ( f − r ) − f − r ) f x − f x − r . (4.25)– 46 –s before, f is the field strength of the dynamical gauge field, while − f x is the topological U (1) x flavor symmetry.We start with the Neumann ( N ) boundary conditions for the gauge fields, which allbehave nicely. Giving N b.c. to the chiral as well, we find a boundary anomaly I (cid:48)(cid:48)N , N = I (cid:48)(cid:48) bulk − ( f − r ) + r = − ( f + f x − r ) , (4.26)which can be neatly cancelled by a boundary Fermi multiplet of charges (1 , , −
1) for U (1) × U (1) x × U (1) R . Note that the extra boundary Fermi cancels both the gauge and the mixedgauge- U (1) x anomalies, so this boundary condition preserves the U (1) x topological symmetry.Just as in T (cid:48) , we expect that ( N , N) is dual to a Dirichlet-like b.c. for T ; in fact, since U (1) x is preserved, we expect pure D b.c. Indeed, the half-index shows II (cid:48)(cid:48)N , N+fermi ( x ; q ) = ( q ) ∞ (cid:73) ds πis F( sx ; q ) II N ( s ; q ) (4.27)= ( q ) ∞ (cid:73) ds πis F( sx ; q )( s ; q ) ∞ residues = (cid:88) a ≥ F( q − a x ; q )( q − ) a = F( x ) (cid:88) a ≥ x a ( q ) a = F( x )( x ; q ) ∞ = ( qx − ; q ) ∞ = II D ( x ; q ) . The ( N , N) b.c. can be modified by a bulk Wilson line of charge n that ends on theboundary, which keeps boundary operators of gauge charge − n . We expect this to be dualto a bulk flavor-vortex in theory T , which shifts the spins of operators charged under U (1) x .We check: II (cid:48)(cid:48)N n , N+fermi ( x ; q ) = ( q ) ∞ (cid:73) ds πis s n F( sx ; q ) II N ( s ; q ) = II D ( q − n x ; q ) , (4.28)as expected.The ( N , D) b.c. is even simpler: the boundary anomaly is I (cid:48)(cid:48)N , D = I (cid:48)(cid:48) bulk + ( f − r ) + r = − f − r ) f x − f x , so U (1) x is anomalous, but there is no gauge anomaly to cancel. We wouldexpect this to be dual to D c b.c. for the free chiral, and indeed II (cid:48)(cid:48)N , D ( q ) = ( q ) ∞ (cid:73) ds πis II D ( s ; q ) = ( q ) ∞ (cid:73) ds πis ( qs − ; q ) ∞ = ( q ) ∞ = II D c ( q ) . (4.29)Additionally, bulk Wilson lines in T (cid:48)(cid:48) are dual to vortex lines in T : II (cid:48)(cid:48)N n , D ( q ) = ( q ) ∞ (cid:73) ds πis s n II D ( s ; q ) = ( − n q n ( n +1)2 ( q n ; q ) ∞ = ( − n q n ( n +1)2 II D c,n ( q ) . (4.30)In contrast to N b.c., the half-indices for Dirichlet ( D ) b.c. on the gauge multiplet behavevery badly due to the negative bulk Chern-Simons coupling. For example, ( D , N) and ( D , D)– 47 –.c. have boundary ’t Hooft anomalies I (cid:48)(cid:48)D , N = I (cid:48)(cid:48) bulk − ( f − r ) − r = − ( f + f x − r ) − r and I (cid:48)(cid:48)D , D = I (cid:48)(cid:48) bulk + ( f − r ) − r = − f − r ) f x − f x − r , respectively. Letting y denotethe fugacity for the boundary U (1) ∂ symmetry, we find putative half-indices II (cid:48)(cid:48)D , N ( x, y ; q ) = 1( q ) ∞ (cid:88) m ∈ Z q − m y − m ( − q x − ) m q m x ; q ) ∞ , (4.31) II (cid:48)(cid:48)D , D ( x, y ; q ) = 1( q ) ∞ (cid:88) m ∈ Z x − m ( q − m y − ; q ) ∞ . (4.32)Both sums diverge badly as series in q : for ( D , N), the n -th term in the sum begins with thelarge negative power q − n ( n − for both positive and negative n ; while for ( D , D), the n -th termbegins with q − n ( n − for negative n . In the analysis of bulk indices, such behaviour is usuallyindicative of a “bad” setup where some operators, such as these boundary monopoles, hit theunitarity bound along the RG flow and the U (1) R in the IR contains emergent symmetries.Our tools are thus insufficient to study the problem.There is one choice of D b.c. for which the half-index does make sense. If we give D c b.c.to the chiral (possibly with a vortex of charge n ) and break U (1) ∂ flavor symmetry, we find II D ,D c,n ( x ; q ) = 1( q ) ∞ (cid:88) m ∈ Z x − m ( q n − m ; q ) ∞ = 1( q ) ∞ (cid:88) m ≥ x m − n ( q m +1 ; q ) ∞ (4.33)= x − n (cid:88) m ≥ x m ( q ) m = x − n II N ( x ; q )Thus, we seem to recover N b.c. for a free chiral, with a flavor Wilson line of charge − n .Summarizing: B (theory T ) B (cid:48)(cid:48) (theory T (cid:48)(cid:48) )N ( D , D c )N + Wilson − n ( D , D c ) + Vortex n D ( N , N) + fermiD + Vortex − n ( N , N) + fermi, Wilson n D c ( N , D)D c + Vortex n ( N , D) + Wilson n (4.34) It may appear that there is a curious asymmetry between the available boundary conditionsfor theories T (cid:48) and T (cid:48)(cid:48) , and their respective duals for the free chiral T . One may well wonderhow the sign of a Chern-Simons level, + for T (cid:48) and − for T (cid:48)(cid:48) can make such a difference.The distinction is indeed irrelevant in the 3d N = 2 bulk. However, when constructingboundary conditions, we implicitly chose • a “right” (vs. left) boundary condition, for a 3d theory on x ⊥ ≤ x ⊥ ≥ a 2d N = (0 ,
2) (vs. N = (2 , N = (0 ,
2) vs. N = (2 , Both choices break the symmetry betweenpositive and negative Chern-Simons levels in the bulk. If we were to reverse both choicessimultaneously, then the behavior of boundary conditions for T (cid:48) and T (cid:48)(cid:48) would be exchanged. We would next like to construct a duality interface between the free chiral T and the abeliangauge theories T (cid:48) or T (cid:48)(cid:48) from section 4. In preparation, we discuss some of the generalstructure of a duality interface. A systematic way to construct a duality interface for a pair of dual 3d theories T and T ∨ (these could be any theories) involves starting with a pair of UV boundary conditions B and B ! for T alone that can be coupled together in such a way that they flow to the trivial/identityinterface in the IR. This “factorization of the identity” is illustrated on the LHS of Figure 3,with B as a right b.c. and B ! as a left b.c. It is not always guaranteed that a suitable pair B , B ! exists. If it does, however, the duality interface can be obtained by dualizing ( T , B )to a dual boundary condition ( T ∨ , B ∨ ) for the dual theory — i.e. dualizing T on the lefthalf-space — and then coupling ( T ∨ , B ∨ ) back to the boundary condition B ! on the righthalf-space, as on the RHS of Figure 3.In 3d N = 2 UV gauge theories, we can always find suitable ( B , B ! ), often in many dif-ferent ways. The pair is built up from certain complementary choices of boundary conditionsfor the matter multiplets and for the gauge multiplets.Consider a free 3d chiral multiplet. It is not hard to see that taking B = N and B ! = D(or vice versa) factorizes the identity as desired, if the two are coupled together by a quadraticsuperpotential. To be explicit, let us denote the 2d N = (0 ,
2) multiplets on the left of theinterface ( x ⊥ ≤
0) as (Φ , Ψ), and on the right ( x ⊥ ≥
0) as (Φ (cid:48) , Ψ (cid:48) ). We may deform theproduct of N b.c. for (Φ , Ψ) and D b.c. for (Φ (cid:48) , Ψ (cid:48) ) by the superpotential (cid:90) d x dθ + Φ (cid:12)(cid:12) ∂ Ψ (cid:48) (cid:12)(cid:12) ∂ . (5.1)In the IR, this has the effect of simultaneously setting Ψ (cid:12)(cid:12) ∂ = Ψ (cid:48) (cid:12)(cid:12) ∂ and Φ (cid:12)(cid:12) ∂ = Φ (cid:48) (cid:12)(cid:12) ∂ , therebygluing the theory back together.We may also check anomalies. Suppose there is a background Chern-Simons level k forthe chiral multiplet’s U (1) flavor symmetry. Then the flavor symmetry has boundary ’t Hooft– 49 – id TT IR T IR dualize = flow flow T B ! B TT B ! couple T B ! B _ T _ couple TT _ I id T IR T IR := coupling of , B _ B ! Figure 3 . Defining a duality interface I between dual 3d theories T ∨ and T by starting with afactorization of the identity interface in T (on the LHS) and dualizing half the space. The interface hasthe necessary property (that it flows to the identity in the IR) provided that the “diagram commutes”;in particular the coupling on the RHS must involve the appropriate duals of the operators on the LHS. anomalies that depend both on the boundary condition and whether it is on the left or right:b.c. \ location x ⊥ ≤ x ⊥ ≥ k − − k − D k + − k + (5.2)Thus B = N b.c. on x ⊥ ≤ B ! = D b.c. on x ⊥ ≥ B = D can be paired with B ! = N.)In a similar way, the N and D boundary conditions for a 3d gauge multiplet constitutea suitable ( B , B ! ) pair. Indeed, there is a canonical coupling between N and D , given byusing the dynamical G gauge symmetry on N to gauge the G ∂ boundary flavor symmetryon D . This obviously glues the gauge theory back together, and extends supersymmetricallyto identify the entire 3d gauge multiplet across the interface. Moreover, anomalies cancel inessentially the same way as above. For example, for simple G at Chern-Simons level k , the– 50 –oundary gauge or ’t Hooft anomalies areb.c. \ location x ⊥ ≤ x ⊥ ≥ N ( k + h )Tr( f ) ( − k + h )Tr( f ) D ( k − h )Tr( f ) ( − k − h )Tr( f ) (5.3)and cancel between N and D .In a full 3d gauge theory, we may now construct a suitable pair ( B , B ! ) by simply combin-ing a choice of (N,D) or (D,N) for each matter multiplet and a choice of ( N , D ) or ( D , N ) forthe gauge group. The procedure should work even in the presence of a bulk superpotential W ,as long as sufficiently many D b.c. can be chosen on each side to ensure the vanishing of W at the interface. Let’s illustrate the above construction for theory T (a free chiral) and its duals T (cid:48) ( U (1) +achiral) and T (cid:48)(cid:48) ( U (1) − +a chiral) from Section 4.The simplest way to begin is by “factorizing the identity” in theory T (cid:48) , using the com-plementary boundary conditions B (cid:48) = ( D , D) (on x ⊥ ≤
0) and B (cid:48) ! = ( N , N) (on x ⊥ ≥ U (1) ∂ flavor symmetry on the left, and by a superpotential of the form (5.1). Then we dualize the x ⊥ ≤ B (cid:48) = ( D , D) is B ∨ = (N + 2d fermi).Explicitly, if we denote the bulk 3d chiral multiplet of T (cid:48) as (Φ (cid:48) , Ψ (cid:48) ), the bulk 3d chiral of T as (Φ , Ψ), and the 2d Fermi as Γ, the map of boundary operators relatesTheory T (cid:48) : Ψ (cid:48) ↔ Theory T : ΦΓ . (5.4)After re-coupling the B ∨ = (N + 2d fermi) b.c. for T to B (cid:48) ! = ( N , N) b.c. for T (cid:48) , we findthat the interface has a fairly symmetric description. It is built by: • Placing T on a half-space x ⊥ ≤ • Placing T (cid:48) on a half-space x ⊥ ≥ B (cid:48) ! = ( N ,N) b.c. (leaving the chiral Φ (cid:48) uncon-strained at the boundary) • Adding a 2d Fermi multiplet Γ at the interface x ⊥ = 0 • Coupling the two halves via a cubic superpotential (cid:82) dθ + ΦΓΦ (cid:48) , and identifying (gaug-ing) the U (1) ∂ flavor symmetry on the left with the U (1) gauge symmetry on the right.We schematically represent this interface as T N (cid:12)(cid:12) Γ (cid:12)(cid:12) T (cid:48) ( N , N) ( J Γ = ΦΦ (cid:48) ) (5.5)– 51 –otice that the charges of the various multiplets Φ Γ Φ (cid:48) U (1) ∂ = U (1) gauge − U (1) x − U (1) R (cid:48) preserves gauge, flavor, andR-symmetry. More so, by construction, anomalies for all symmetries cancel perfectly: from Nb.c. on the left we have − ( f x − r ) ; from ( N ,N) b.c. on the right we have − (cid:2) ( f − r ) + 2 f f x − r (cid:3) + r − ( f − r ) (note the opposite sign from bulk CS levels, due to the orientation),and from Γ we have ( f + f x − r ) , which add up to zero. Interestingly, anomaly cancellationforces the interface Fermi multiplet Γ to be charged under the topological U (1) x symmetryof T (cid:48) . The cubic superpotential then identifies U (1) x with the usual flavor symmetry of T .Notice how the Fermi multiplet Γ mediates the map of operators under the duality. Thesuperpotential (cid:82) dθ + ΦΓΦ (cid:48) modifies the N b.c. for the chirals on either side, from the usualΨ (cid:12)(cid:12) ∂ = 0, Ψ (cid:48) (cid:12)(cid:12) ∂ = 0 to Ψ = ΓΦ (cid:48) , ΦΓ = Ψ (cid:48) . (5.7)In addition, if we move onto the (bulk) moduli space of T by giving Φ a vev, the Fermimultiplet Γ will become massive. Integrating it out at one loop generates a “field-dependentFI term” on the interface, encoded in the superpotential (cid:90) dθ + Υ (cid:48) log Φ , (5.8)where Υ (cid:48) is the field strength multiplet in theory T (cid:48) , restricted to the boundary. The super-potential (5.8) now carries the contribution to the mixed gauge-topological anomaly from Γ;in particular, by itself, (5.8) breaks the U (1) x symmetry under which Φ is charged. We nowrecall from Section 2.5.3 that N b.c. on the gauge multiplet of Theory T (cid:48) is a D b.c. on itsdual-photon multiplet S (cid:48)∨ . In turn, the effective superpotential (5.8) modifies the D b.c. onthe dual photon to S (cid:48)∨ (cid:12)(cid:12) ∂ = J Υ (cid:48) = log Φ. Since the monopole operator of T (cid:48) may be definedin the UV as V (cid:48) ∼ exp( S (cid:48)∨ ), this induces a relation V (cid:48) ∼ Φ (5.9)at the interface, which identifies the vev of the monopole operator of T (cid:48) with the vev of thefree chiral of T .We may similarly define a duality interface between T and T (cid:48)(cid:48) ( U (1) − +a chiral). Nowthe simplest procedure is to “factorize the identity” for T by coupling D b.c. on x ⊥ ≤ x ⊥ ≥
0, and then dualize the left half to (cid:2) ( N , N)+fermi (cid:3) for T (cid:48)(cid:48) . We obtain: Superpotentials like this were described and generalized in many ways in, e.g. , [52, 137–141]. At its heart,the superpotential is an N = (0 ,
2) version of the “Σ log Σ” effective twisted superpotential that appears in N = (2 ,
2) theories and played a fundamental role in 2d mirror symmetry [83, 142]. – 52 – T (cid:48)(cid:48) on x ⊥ ≤ N , N) b.c. • T on x ⊥ ≥ • a 2d Fermi multiplet Γ of charge ( − , − ,
1) for U (1) ∂ = U (1) gauge × U (1) x × U (1) R ,coupled to the bulk via a cubic superpotential (cid:82) dθ + Φ (cid:48)(cid:48) ΓΦ.It is again straightforward to check that all anomalies cancel. The map of operators alsoproceeds the same way, mediated by Γ.The collision between the T (cid:48)(cid:48) (cid:12)(cid:12) T interface and the T (cid:12)(cid:12) T (cid:48) interface now suggests a UVdefinition for a duality interface between T (cid:48)(cid:48) and T (cid:48) . Since sandwiching T between two Nb.c. leaves behind an ordinary 2d chiral multiplet Φ, we expect the T (cid:48)(cid:48) (cid:12)(cid:12) T (cid:48) duality interfaceto contain two 2d Fermi multiplets Γ , Γ (cid:48) and the 2d chiral Φ, coupled to the bulk chirals Φ (cid:48)(cid:48) ,Φ (cid:48) on either side by a superpotential (cid:90) dθ + (cid:0) Φ (cid:48)(cid:48) ΓΦ + Φ˜ΓΦ (cid:48) ) . (5.10)We emphasize that the duality interfaces constructed here are all uni-directional . Forexample, the interface above between T and T (cid:48) requires T to sit on the half-space x ⊥ ≤ T (cid:48) to sit on x ⊥ ≥
0, and not the other way around. The asymmetry comes from ourchoice in preserving 2d N = (0 ,
2) SUSY (rather than (2 , We hope to pursue this furtherelsewhere.
We may extend the techniques used to understand particle-vortex duality to produce dualboundary conditions and a duality interface for the pair of dual 3d N = 2 theoriesSQED U (1) w/ 2 chirals of charge ± ↔ XYZ model3 chirals, W = X d Y d Z d . (6.1)This basic duality was introduced in [3], and may be derived from abelian 3d N = 4 mirrorsymmetry. One new ingredient is the bulk superpotential W in the XYZ model. As discussedin Section 2.3, boundary conditions for the XYZ model may require extra boundary degreesof freedom to factorize W . We shall mainly consider boundary conditions are “sufficientlyDirichlet,” as in Section 2.3.1, so that W is automatically set to zero on the boundary.However, the duality interface will indeed involve a nontrivial factorization. This is reminiscent of the 3d N = 4 abelian duality interfaces defined in [60]. We indeed expect that the3d N = 4 interfaces can be reduced to 3d N = 2 interfaces, which may occasionally involve nonperturbativecouplings. – 53 –e use the following conventions. In SQED, the chirals, decomposed into N = (0 , , Ψ), ( ˜Φ , ˜Ψ). In XYZ the chirals are ( X, Ψ X ), ( Y, Ψ Y ), ( Z, Ψ Z ),and duality maps the meson Φ ˜Φ of SQED to X , and the monopole operators of SQED to Y and Z . The global symmetry group of SQED comprises a U (1) a axial symmetry, U (1) y topological symmetry, and the usual U (1) R symmetry, under which the various chiral fieldshave charges Φ ˜Φ X Y ZU (1) gauge − U (1) a − − U (1) y − U (1) R U (1) y symmetry, and may be encoded in the anomaly polynomial 2 f y (where y is the U (1) y fieldstrength). This matches identically zero bulk CS terms for XYZ. This identification followsfrom a careful comparison of sphere partition functions, which are sensitive to backgroundChern-Simons terms (contact terms), see e.g. [18]. The most interesting dual pair of boundary conditions — and the one that we use to build theduality interface — involves Dirichlet ( D ) for the gauge multiplet in SQED and D for bothchirals. We’ll call this ( D ,D,D). We can derive its mirror in the XYZ model from a simplephysical analysis!The ( D ,D,D) b.c. in SQED should leave the two monopole operators unconstrained atthe boundary, while setting the meson Φ ˜Φ to zero. Therefore, we expect that the dual b.c. inthe XYZ model is Neumann for Y and Z , and Dirichlet for X . We call this putative dual(D,N,N). Note that, a priori, it requires no further boundary d.o.f., since it sets W | ∂ = 0.However, just as in particle-vortex duality:- ( D ,D,D) b.c. for SQED has a boundary U (1) ∂ flavor symmetry that’s missing in XYZ;- there is a mismatch in boundary ’t Hooft anomalies between ( D ,D,D) and (D,N,N).There is an easy way to fix both of these problems by adding an extra boundary Fermimultiplet on the XYZ side. Indeed, the difference in boundary ’t Hooft anomalies (with f , a , y denoting the field strenghts for U (1) ∂ , U (1) a , U (1) y ) I D ,D,D = 2 f y − r + ( f + a − r ) + ( − f + a − r ) I D,N,N = (2 a − r ) − ( y − a ) − ( y + a ) (6.3)is I D ,D,D − I D,N,N = ( f + y ) , and is precisely made up by a 2d Fermi multiplet Γ withcharges ( − , , − ,
0) for U (1) ∂ × U (1) a × U (1) y × U (1) R . This multiplet “carries” the U (1) ∂ symmetry on the XYZ side. We surmise that( D , D,D) ↔ (D,N,N) + Fermi Γ . (6.4) The same conclusion was reached in [53]. – 54 –his putative duality of boundary conditions immediately leads to a construction of theduality interface, following the logic of Section 5.1. We may start with a factorization of theidentity interface in SQED that couples the ( D ,D,D) b.c. on x ⊥ ≤ N ,N,N) on x ⊥ ≥ x ⊥ ≤ N ,N,N) on x ⊥ ≥
0. We might schematically, and more symmetrically, denote this as
XYZ (D,N,N) (cid:12)(cid:12) Γ (cid:12)(cid:12) SQED ( N , N,N) . (6.5)Here the U (1) gauge symmetry on the right is used to gauge the U (1) ∂ flavor symmetry of Γ.In addition, there must be some boundary superpotential couplings! There is essentially asingle possibility compatible with the global symmetries and with factorization of W ; wepropose (cid:90) dθ + (cid:2) Ψ X Φ ˜Φ + Y ΓΦ (cid:3) , E Γ = Z ˜Φ , (6.6)where Γ is given both J and E terms J Γ = Y Φ, E Γ = Z ˜Φ. This could actually be deduced fromthe original coupling in the factorization of the identity interface and the map of operatorsacross the duality. The coupling (6.6) actually treats Φ , ˜Φ and Y, Z symmetrically, though itdoes not look so, since we may always replace Γ with its “T-dual” Γ † whose E and J termsare swapped ( cf. Appendix A). To check that (6.6) provides a matrix factorization of thebulk superpotential on the XYZ side, we first observe that the Ψ X Φ ˜Φ coupling sets X = Φ ˜Φ (6.7)at the interface; then J Γ E Γ = ( Y Φ)( Z ˜Φ) (6.7) = XY Z (6.8)as required.The proposed interface gives the expected map of operators. The relation between X and the meson Φ ˜Φ is explicit in (6.7). In addition, the E and J terms for Γ modify the N b.c.for Y, Z, Φ, ˜Φ as in Section 2.2:Ψ Y = ΓΦ , Ψ Z = Γ ˜Φ ; Y Γ = Ψ , Z
Γ = ˜Ψ . (6.9)Finally, the relations involving monopole operators may be deduced from quantum correctionsto the interface superpotential (just like in (5.8) for particle-vortex duality). We may give avev to either Y or Z (but not both, due to the bulk X d Y d Z d superpotential) and make Γmassive at the boundary. Integrating Γ out at one-loop generates interface superpotentials Y (cid:54) = 0 ⇒ (cid:90) dθ + Υ log
Y , Z (cid:54) = 0 ⇒ − (cid:90) dθ + Υ log
Z , (6.10)where Υ is the field strength in SQED, restricted to the interface. These modify the D b.c.for the monopoles V ± of SQED to V + ∼ Y , V − ∼ Z . (6.11)– 55 –nlike the case of particle-vortex dualities, the duality interface between SQED and XYZis actually symmetric. That is, we can sensibly define an interface in the opposite direction
SQED ( N ,N,N) (cid:12)(cid:12) Γ (cid:12)(cid:12) XYZ (D,N,N) (6.12)in essentially the same way as above, but now with a 2d Fermi multiplet Γ of charges( − , , ,
0) under U (1) gauge × U (1) a × U (1) y × U (1) R . The reader may verify that all anoma-lies cancel. The important difference between the SQED-XYZ duality and the particle-vortexduality of Section 4 is that the former has vanishing bulk Chern-Simons levels for dynamicalgauge symmetry. This makes the SQED-XYZ duality more symmetric with respect to leftvs. right boundary conditions.Finally, we may check the duality (6.4) with a half-index computation. For SQED,we compute the “effective” boundary CS levels from the anomaly (6.3), which simplifies to f + 2 f y + ( a − r ) − r . Following the rules of Section 3.4, this leads to a half-index II SQED( D , D,D) ( s, a, y ; q ) = 1( q ) ∞ (cid:88) n ∈ Z q n s n y n II D ( q n sa ; q ) (cid:124) (cid:123)(cid:122) (cid:125) ( q − n s − a − ; q ) ∞ II D ( q − n s − a ; q ) (cid:124) (cid:123)(cid:122) (cid:125) ( q n sa − ; q ) ∞ , (6.13)where s, a, y are the U (1) ∂ , U (1) a , U (1) y fugacities. This turns out to equal II XYZ(D,N,N)+Γ ( s, a, y ; q ) = ( − q sy ) ∞ ( − q s − y − ) ∞ ( qa − ; q ) ∞ ( − q a − y ; q ) ∞ ( − q a − y − ; q ) ∞ (6.14)= F (cid:16) − q sy (cid:17) II D ( a ; q ) II N (cid:16) − q ya ; q (cid:17) II N (cid:16) − q ay ; q (cid:17) as desired.Just as in (4.10), the equivalence of (6.13) and (6.14) may be established by showingthat both expressions obey the same first-order difference equations in s , a , and y , and thatthey are equal at a particular point. The difference equations, which may all be interpretedas identities for line operators (Section 3.7), are II ( q − s, a, y ; q ) = q − syII ( q − s, a, y ; q ) , II ( q − s, q a, q y ; q ) = − a q − a − y − II ( s, a, y ; q ) ,II ( q − s, a, qy ; q ) = q a − y q − a − y − II ( s, a, y ; q ) (6.15)and follow either directly from (6.14) or by simple manipulations inside the sum of (6.13).A convenient evaluation point is s = a = y = 1, where both (6.13) and (6.14) reduce fairlytrivially to ( q ) ∞ . Alternatively, by sending s/a → sa and sy fixed, we mayreduce to the identity in (4.10)–(4.11). This corresponds physically to turning on a largereal mass to integrate out ˜Φ in SQED and X, Z in XYZ, recovering the basic particle-vortexduality. This “reverse” interface may be derived systematically by starting with a factorization of the identityinterface in XYZ, of the form (N,D,D) | (D,N,N), then using a duality from Section 6.2 to dualize the left sideto SQED with ( N , N,N) b.c. coupled to a boundary Fermi Γ. – 56 – .2 Other dual boundary conditions
There are many other dual pairs of boundary conditions for SQED and XYZ. As we havealready mentioned, this pair of 3d theories is even better behaved than the particle-vortexduality of Section 4, due to vanishing bulk Chern-Simons levels. Thus both Dirichlet and Neu-mann b.c. are available for the gauge multiplet of SQED, with interesting duals on the XYZside. We summarize the proposed dualities in Table 1. For each of these pairs of boundaryconditions, the boundary anomalies match up perfectly, and there is an associated half-indexidentity. We discuss a few of the more interesting cases below. Various vortex/Wilson linescan also be incorporated in a fairly straightforward way, following the models of Section 4.SQED XYZ( D , D,D) (D,N,N)+Γ / ( sy ) ( D , D,D c ) (D,N,D)( D , D c ,D) (D,D,N)( D , D c ,D c ) (D c ,N,D) (cid:39) (D c ,D,N)( D , N,D) (D,N,N) + Γ / ( sy ) + C as ; J Γ = CY ( D , D,N) (D,N,N) + Γ / ( sy ) + ˜ C a/s ; E Γ = ˜ CZ ( D , N,N) (D,N,N) + Γ / ( sy ) + C as + ˜ C a/s ; J Ψ X = C ˜ C, J Γ = CY, E Γ = ˜ CZ ( N , N,N) + Γ s/y (N,D,D)( N , N,N) + Γ s/y + Γ (cid:48)− q / /a ; J Γ (cid:48) = Φ ˜Φ (D,D,D)( N , N,D) (N,D c ,D) (cid:39) (D,D c ,N)( N , D,N) (N,D,D c ) (cid:39) (D,N,D c ) Table 1 . Elementary pairs of dual boundary conditions for SQED and XYZ. Additional boundaryFermi and chiral multiplets are denoted Γ and C, ˜ C , respectively, with their charge under U (1) gauge (or U (1) ∂ ), U (1) a and U (1) y encoded by fugacities s , a , and y in the subscripts. ( D , D c ,D c ) ↔ (D c ,N,D) or (D c ,D,N) The new feature here is that there seem to be two distinct UV boundary conditions for theXYZ model that are IR equivalent. This becomes less surprising upon closer inspection. The(D c ,N,D) b.c. sets X | ∂ = c (cid:54) = 0 and Z | ∂ = 0 while seemingly leaving Z unconstrained.However, the bulk F-term ∂W/∂Z = XY (identified as the N = (0 ,
2) E-term for Ψ Z ) mustalso vanish in a supersymmetric vacuum. Since at the boundary the F-term is XY | ∂ = cY | ∂ ,we see that Y | ∂ is also set to zero in the IR. Thus, the (D c ,N,D) b.c. seems to look identicalto (D c ,D,N). As mentioned in the introduction, some of the dualities involving Neumann b.c. for the gauge multipletof SQED have appeared previously in [50]. – 57 –e can verify that boundary ’t Hooft anomalies match. We compute:(D c ,N,D) : (2 a − r ) − ( y − a ) + ( y + a ) (cid:12)(cid:12) a =0 = r , (D c ,D,N) : (2 a − r ) + ( y − a ) − ( y + a ) (cid:12)(cid:12) a =0 = r , (6.16)which agree. Moreover, these match the boundary anomaly of ( D ,D c ,D c ) b.c. for SQED:( D ,D c ,D c ) : 2 f y − r + ( f + a − r ) + ( − f + a − r ) (cid:12)(cid:12) f = a =0 = r . (6.17)Curiously, the anomaly for (D c ,D,D), which might also have been expected to be IR equivalentto these b.c., equals (2 a − r ) + ( y − a ) + ( y + a ) (cid:12)(cid:12) a =0 = y + r , and does not agree.The half-index identity corresponding to the putative duality of boundary conditions isobtained by taking the identity (6.13)-(6.14) of Section 6.1 and setting to one the fugacitiesfor both of the chirals in SQED, which are given D c b.c., namely as = 1 and a/s = 1, orsimply a = s = 1. This results fairly trivially in a half-index on either side equal to ( q ) ∞ ,independent of y .A stronger argument for the duality can be made by deforming the ( D , D,D) ↔ (D,N,N)+Γduality of Section 6.1 by boundary superpotentials. To reach ( D , D c ,D c ) in SQED, we add asuperpotential (cid:90) dθ + ( c Ψ + c (cid:48) ˜Ψ) . (6.18)If we turn these terms on one at a time, we may view the c Ψ coupling as modifying the b.c.to Φ | ∂ = c , and then the c (cid:48) ˜Ψ coupling as subsequently modifying the boundary vev of the meson operator to Φ ˜Φ | ∂ = cc (cid:48) . Correspondingly, in SQED we add superpotentials for thedual operators: first we add c Γ Y (since Γ Y is dual to Ψ | ∂ ), and then we add cc (cid:48) Ψ X (since X is dual to the meson). The superpotential (cid:90) dθ + ( c Γ Y + cc (cid:48) Ψ X ) (6.19)has the effect of “flipping” the b.c. on Y from N to D, and deforming the b.c. on X to X | ∂ = cc (cid:48) , which is precisely (D c ,D,N). Alternatively, had we turned on the superpotentialterms in the opposite order, we would have obtained (D c ,N,D). ( D ,N,N) ↔ (D,N,N) with a Fermi and two chirals The boundary condition here for XYZ is a true matrix factorization, very similar to the onethat entered the duality interface. Indeed, this duality can neatly be obtained by collidingthe interface (6.5) with ( D ,N,N) b.c. for SQED. The “sandwich” between ( N ,N,N) (on theright side of the interface) and ( D ,N,N) kills the U (1) gauge symmetry, and leaves behindtwo 2d chirals C, ˜ C , coming from the reductions of Φ , ˜Φ on the segment. The interface itselfcontributes the 2d Fermi Γ, and we are left with (D,N,N) b.c. for XYZ coupled to Γ , C, ˜ C viathe superpotential (cid:90) dθ + (cid:2) Ψ X C ˜ C + Γ CY (cid:3) , E Γ = ˜ CZ . (6.20)– 58 –s in the interface, this sets X | ∂ = C ˜ C , so that E Γ J Γ = C ˜ CY Z = XY Z = W .The relevant index identity is1( q ) ∞ (cid:88) n ∈ Z q − n s − n y n II N ( q n sa ; q ) II N ( q − n s − a ; q ) (6.21)= F (cid:16) − q sy (cid:17) C( as )C( a/s ) II D ( a ; q ) II N (cid:16) − q ya ; q (cid:17) II N (cid:16) − q ay ; q (cid:17) , and follows easily from (6.13)-(6.14) by rewriting II N ( q n sa ; q ) II N ( q − n s − a ; q )as C( as )C( a/s ) q n s n II D ( q n sa ; q ) II D ( q − n s − a ; q ) on the LHS. ( N ,N,N)+Fermi ↔ (N,D,D) The ( N ,N,N) boundary condition in SQED kills the monopole operators while leaving themeson unconstrained, so we would naively expect it to be dual to (N,D,D) in XYZ. In fact,cancellation of a gauge anomaly on the SQED side requires the introduction of an additionalboundary Fermi multiplet Γ. Together with this modification, the duality seems to hold asexpected.Let us consider the anomalies explicitly. On the SQED side, we have r + 2 f y − ( f + a − r ) − ( − f + a − r ) = − f + 2 f y − ( a − r ) + r . (6.22)Adding a 2d Fermi multiplet Γ of charge ( − , , ,
0) for U (1) gauge × U (1) a × U (1) y × U (1) R contributes ( f − y ) , modifying the anomaly to y − ( a − r ) + r , and in particular cancelingthe terms involving the dynamical field strength f . We may compare this with (N,D,D) b.c.for SQED, which has anomaly − (2 a − r ) + ( y − a ) + ( y + a ) = y − ( a − r ) + r , (6.23)matching perfectly.The relevant half-index identity is( q ) ∞ (cid:73) ds πis II N ( as ; q ) II N ( a/s ; q )F( − q y/s ; q ) = ( q ) ∞ (cid:73) ds πis F( − q y/s ; q )( as ; q ) ∞ ( a/s ; q ) ∞ residues = (cid:88) n ≥ ( − n q n ( n +1) F( − q − n y/a ; q )( q ) n ( q n a ; q ) ∞ = F( − q y/a ; q )( a ; q ) ∞ (cid:88) n ≥ ( − q y/a ) n ( a ; q ) n ( q ) n q-binomial = F( − q y/a ; q )( a ; q ) ∞ ( − q ay ; q ) ∞ ( − q y/a ; q ) ∞ = ( − q a/y ; q ) ∞ ( − q ay ; q ) ∞ ( a ; q ) ∞ = II N ( a ; q ) II D ( − q y/a ; q ) II D ( − q / ( ay ); q ) . (6.24)The same identity appeared when studying holomorphic block dualities [67], later interpretedvia half-indices in [50]. – 59 – N ,N,N)+2 fermis ↔ (D,D,D) This is a simple flip of the preceding boundary condition: an additional 2d Fermi multipletΓ (cid:48) is used to flip the meson in SQED, and to flip N to D b.c. for X in XYZ. ( N ,D,N) ↔ (N,D,D c ) or (D,N,D c ) Finally, we take a look at another duality where the XYZ boundary condition has multipleUV descriptions. In SQED, the ( N ,D,N) b.c. does not have a gauge anomaly, but it doeshave a mixed anomaly: we compute a boundary polynomial r + 2 f y + ( f + a − r ) − ( − f + a − r ) = r + 2 f ( y + a − r ) . (6.25)This breaks the symmetry with field strength y + a − r , which is precisely the symmetryunder which the chiral Z in the XYZ model is charged. Thus, in XYZ, we expect a D c b.c.for Z . Moreover, ( N ,D,N) sets the meson to zero so we expect its dual to give D b.c. to X .Comparing ’t Hooft anomalies then suggests that the dual b.c. should be (D,N,D c ).As before, the BPS equations on the (D,N,D c ) b.c. actually set Y | ∂ = 0 as well in asupersymmetric vacuum. This fits the proposed duality, since N b.c. in SQED certainly killsboth monopole operators, including the one dual to Y . We suspect that the (D,N,D c ) b.c.can equivalently be described as (N,D,D c ).The half-index identity is fairly simple:( q ) ∞ (cid:73) ds πis II D ( as ; q ) II N ( a/s ; q ) = ( q ) ∞ (cid:73) ds πis ( q/ ( as ); q ) ∞ ( a/s ; q ) ∞ residues = (cid:88) n ≥ q n ( − q ) n ( q − n a − ; q ) ∞ ( q ) n = ( qa − ; q ) ∞ (cid:88) n ≥ ( qa − ) n ( a ; q ) n ( q ) n q-binomial = ( q ) ∞ = II N ( a ; q ) II D ( − q y/a ; q ) II D ( − q / ( ay ); q ) (cid:12)(cid:12) y = − q /a . (6.26) D and 2d dualities It is again instructive to take D -type boundary conditions and gauge the 2d U (1) ∂ symmetry,comparing the result with proposed mirrors of N -type boundary conditions.The richest example is to gauge ( D , N,N) + Γ s/y in order to obtain ( N , N,N) + Γ s/y . Onthe mirror side, we get ( D, N,N) coupled to an intricate boundary theory: a 2d U (1) s gaugetheory with two Fermi multiplets Γ s/y and Γ / ( sy ) and two chiral multiplets C as and ˜ C a/s .From Table 1, we expected this mirror to be equivalent to (N,D,D). This is only possible ifthe 2d boundary gauge theory flows in the IR to a combination of a chiral multiplet and twoFermi multiplets.The picture is supported by a 2d index calculation, which involves picking a single polefrom the positively charged C as . Indeed, if we add an extra Fermi multiplet to analyze– 60 – N , N,N) + Γ s/y + Γ (cid:48)− q / /a , the RG flow of the 2d theory to the two Fermi multiplets thatflip (D,N,N) to (D,D,D) is the very simplest example of (0 ,
2) triality [52, Fig. 2], with N = N = 1 and N = 0. U ( N ) SQCD with N f = N The basic SQED ↔ XYZ duality has a natural generalization, in which SQED is replaced by U ( N ) SQCD with N f = N fundamental and antifundamental chirals (quarks and antiquarks),and the dual theory is another “Landau-Ginzburg model” with a cubic superpotential cap-turing the low-energy dynamics of SQCD [3, 4]. Specifically, the dual theory contains chirals Y d , Z d that match the monopole operators of SQCD, an N × N matrix M d matching themesons of SQCD, and a superpotential W = det( M d ) Y d Z d . (6.27)We’ll call this the “detYZ” model. Just as in SQED ↔ XYZ, the bulk UV Chern-Simonslevels are identically zero on both sides. The dual boundary conditions for SQCD and detYZend up following an identical pattern as those of SQED ↔ XYZ, and we describe a few ofthem here.Let us denote the quarks of SQED, decomposed into N = (0 ,
2) multiplets, as ( Q ia , Ψ ai ),( ˜ Q ¯ ai , ˜Ψ i ¯ a ), where i, a, ¯ a are indices for U ( N ) gauge and SU ( N ) , (cid:94) SU ( N ) flavor symmetries, respec-tively. Similarly, we decompose the chirals of detYZ into ( M a ¯ a , µ a ¯ a ), ( Y, Ψ Y ), and ( Z, Ψ Z ).The charges of the (0 ,
2) chiral halves are U ( N ) gauge SU ( N ) (cid:94) SU ( N ) U (1) a U (1) y U (1) R Q N N 1 Q N 1 N M N N Y − N Z − N − D ,D,D) ↔ (D,N,N) + Γ( N ,N,N) + Γ (cid:48) ↔ (N,D,D) , (6.29)where Γ and Γ (cid:48) are 2d Fermi multiplets in representations U ( N ) or U ( N ) ∂ SU ( N ) (cid:94) SU ( N ) U (1) a U (1) y U (1) R Γ det − − (cid:48) det − D ,D,D), i.e. Dirichlet for the U ( N ) gauge multiplet as well as the quarks and anti-quarks of SQED.Following (2.51) for the computation of U ( N ) anomalies, we find I ( D ,D,D) = − N Tr( s ) + (Tr s ) − N r + 2(Tr s ) y (gauginos, FI) (6.31)+ ( N Tr( x ) + N Tr( s ) + 2 N (Tr s )( a − r ) + N ( a − r ) ) ( Q )+ ( N Tr(˜ x ) + N Tr( s ) − N (Tr s )( a − r ) + N ( a − r ) ) ( ˜ Q )= (Tr s ) + 2(Tr s ) y + N (cid:2) Tr( x ) + Tr(˜ x ) (cid:3) + N ( a − r ) − N r where, in order to match fugacities in the index, we have used s to denote the field strength of U ( N ) ∂ , and x , ˜ x to denote the field strengths of the SU ( N ) , (cid:94) SU ( N ) flavor symmetry. On thedual side, the (D,N,N) boundary condition (D b.c. for M , N b.c. for Y, Z ) has an anomaly I (D,N,N) = ( N Tr x + N Tr ˜ x + N (2 a − r ) ) ( M ) (6.32) − ( − N a + y ) − ( − N a − y ) ( Y, Z )The difference in these two quantities, I ( D ,D,D) = I (D,N,N) + (Tr s + y ) , (6.33)is just right to match the additional contribution of the Fermi multiplet Γ on the detYZside. Similarly, the anomalies for the other pair of boundary conditions are I ( N ,N,N) = −I ( D ,D,D) + 4(Tr s ) y and I (N,D,D) = −I (D,N,N) , so I ( N ,N,N) + (Tr s − y ) = I (N,D,D) , (6.34)with the difference made up for by the Γ (cid:48) Fermi multiplet. In particular, the anomaly of Γ (cid:48) is just right to cancel the gauge and mixed gauge-topological anomaly for N b.c. in SQCD.We may also check these proposed dual boundary conditions by a half-index computation.Following the rules of Section 3, we compute II ( D ,D,D) = 1( q ) N ∞ (cid:88) m ∈ Z N q m · m s m y (cid:80) i m i (cid:81) i (cid:54) = j ( q m i − m j s i /s j ; q ) ∞ N (cid:89) i,α =1 II D (cid:16) q m i ax α s i ; q (cid:17) II D (cid:16) aq m i ˜ x α s i ; q (cid:17) , (6.35) II (D,N,N)+Γ = F (cid:16) − q y (cid:81) i s i ; q (cid:17) II N ( − q a − N y ; q ) II N ( − q a − N y − ; q ) N (cid:89) α,β =1 II D (cid:16) a x α ˜ x β ; q (cid:17) , (6.36)where s = ( s , ..., s N ) are the U ( N ) ∂ fugacities, m = ( m , ..., m N ) ∈ Z N (cid:39) cochar( U ( N ))are the monopole charges, a, y are the axial and topological fugacities, and x = ( x , ..., x N ),˜ x = (˜ x , ..., ˜ x N ) are the SU ( N ) , (cid:94) SU ( N ) fugacities, constrained to satisfy (cid:81) α x α = (cid:81) β ˜ x β = 1.We have checked these formulas for a variety of N in Mathematica up to order q . It seems feasible that the equality II ( D ,D,D) = II (D,N,N)+Γ could be established in general by comparingdifference equations, much like (6.15). This approach is promising because the index (6.36) is simply a productof q -Pochhammer symbols, which obey simple first-order equations. – 62 –y following the general prescription of Section 5.1, we may now use (6.29) to constructduality interfaces. They have the same form as in SQED ↔ XYZ. In one direction, detYZ (D,N,N) (cid:12)(cid:12) Γ (cid:12)(cid:12) SQCD ( N , N,N) , (6.37)with an interface superpotential (cid:82) dθ + (cid:2) Y Γdet( Q ) + Qµ ˜ Q (cid:3) and E Γ = Z det( ˜ Q ). Note that atthe interface this sets M = Q ˜ Q and provides a matrix factorization of the bulk superpotential E Γ J Γ = Y Z det( Q ) det( ˜ Q ) = det( M ) Y Z . In the other direction,
SQCD ( N , N,N) (cid:12)(cid:12) Γ (cid:48) (cid:12)(cid:12) detYZ (D,N,N) , (6.38)with superpotential (cid:82) dθ + (cid:2) Y Γ (cid:48) det( ˜ Q ) + Qµ ˜ Q (cid:3) and E Γ (cid:48) = Z det( Q ). The Fermi muliplets Γand Γ (cid:48) mediate the map of bulk operators across the duality interface, essentially the sameway as in SQED ↔ XYZ. In particular, giving a vev to Y or Z generates a superpoten-tial (cid:82) dθ + (TrΥ) log Y or − (cid:82) dθ + (TrΥ) log Z , generalizing (6.11), which sets the monopoleoperators of SQCD equal to Y and Z .Simple modifications of (6.29) and/or collisions with the interfaces may be used to con-struct many, many other dual pairs of boundary conditions, analogous to all of Table 1.It is also instructive to take D -type boundary conditions and gauge the 2d U (1) ∂ , com-paring the result with proposed mirrors of N -type boundary conditions. Again, we can startfrom ( D , D,D), add fundamental and anti-fundamental chiral fields to flip to ( D , N,N) andFermi multiplet Γ (cid:48) .On the mirror side we get (
D, N, N ) coupled to a 2d U ( N ) ∂ gauge theory with funda-mental and anti-fundamental chiral fields and two Fermi multiplets in the determinant rep-resentation. Consistency of our proposal requires the 2d theory to flow to a bi-fundamentalchiral and two Fermi multiplets to flip ( D, N, N ) to (
N, D, D ).This is reasonable. Indeed, if we add an extra bi-fundamental Fermi multiplet, the RGflow of the 2d theory to the two Fermi multiplets which flip (
D, N, N ) to (
D, D, D ) is a simpleexample of (0 ,
2) triality [52], with N = N = N and N = 0. So far we have discussed pairs of 3d N = 2 dual theories with nontrivial gauge degrees offreedom on one side only, but potentially interesting matter content on both sides. We nowturn to the opposite scenario: gauge groups on both sides, and trivial matter. We investigateboundary conditions in 3d N = 2 level-rank duality.Level-rank dualities comprise a familiar set of tractable examples that have their origins inequivalences of two-dimensional chiral algebras. Viewing the associated conformal field theo-ries as the edge modes of three dimensional topological theories, the level-rank duality ascendsto a duality of Chern-Simons theories. The classic non-supersymmetric dualities [61–65] (and– 63 –ore modern extensions, e.g. [4, 37]) readily admit N = 2 supersymmetric completions[11, 14]. For instance, one basic equivalence in 3d N = 2 Chern-Simons theory is U ( N ) k + N ↔ U ( k ) − k − N . (7.1)Another is SU ( N ) k + N ↔ U ( k ) − k − N, − N , where the notation on the left indicates a differentlevel for the U (1) factor, discussed at the end of Section 7.1.These level-rank dualities follow from several dualities we will meet later with additionalmatter, such as Aharony, Giveon-Kutasov, or other Seiberg-like dualities, either by setting N f = 0 or by adding masses to the matter multiplets and integrating them out until oneobtains a pure N = 2 gauge theory.The supersymmetric dualities are related to non-supersymmetric ones in a straightfor-ward way: by integrating out (or in!) massive vectormultiplet scalars and gauginos, andshifting the Chern-Simons level due to the gauginos. The shift is always controlled by the ad-joint anomaly. If G is a simple group, an N = 2 G κ Chern-Simons theory flows to G κ − h sign( κ ) non-supersymmetric Chern-Simons in the IR. If G is not simple, the level shift may be morecomplicated, as in Section 2.4.In the deep infrared, both the supersymmetric and non-supersymmetric Chern-Simonstheories (which we think of as defined by a path integral over gauge connections, with theusual Chern-Simons action) flow to a massive, topological quantum field theory. We will referto the topological theory corresponding to G k non-SUSY Chern-Simons as T F T [ G k ]. It hasno local degrees of freedom, but does have a category of line operators.The level-rank duals of boundary conditions may be understood to follow from two morefundamental UV-IR relations. Let G be a simple group and k >
0. We propose that1) 3d N = 2 G k + h Yang Mills-Chern-Simons theory with a Dirichlet ( D ) boundary condi-tion flows to a left-moving (chiral) G k WZW model coupled to
T F T [ G k ].2) 3d N = 2 G − k − h Yang-Mills-Chern-Simons theory with a Neumann ( N ) boundarycondition must be coupled to a left-moving chiral algebra T d (which may be part ofan N = (0 ,
2) boundary theory) in order to cancel the gauge anomaly. In the IR, thisboundary condition flows to the coset T d /G − k coupled to T F T [ G − k ].Both relations are fully compliant with ’t Hooft anomaly matching. Analogous statementsshould hold for non-simple G with modified level shifts.The idea that Neumann b.c. for Yang-Mills-Chern-Simons theory leads to coset modelsis familiar in the literature, cf. [51, 68, 69], though we believe the statement about Dirichletboundary conditions is new. We will give evidence of both statements for G = U ( N ) and SU ( N ) (and, in the case of D b.c., general simple G ) by computing half-indices. With D b.c.,the half-index constitutes an “abelianized” formula for WZW characters. We will also checkthe identification of bulk (UV) Wilson lines with modules for the boundary chiral algebras.We note that the sign of the Chern-Simons level is important in the above statements. Asdiscussed in Section 4.4, the choice of left vs. right boundary condition breaks the symmetry– 64 –etween positive and negative bulk levels. Recall that 3d N = 2 G κ YM-CS theory breaksSUSY in the bulk if | κ | < h [128, 143–145].With our conventions for boundary conditions, the half-indices of G k + h theory with D b.c. are badly behaved. Presumably, boundary monopole operators hit some unitarity boundalong the RG flow. Notice that there are no known N = 2 supersymmetric anti-chiral WZWmodels, which would have been an obvious candidate for the IR boundary physics.On the other hand, N b.c. for k < ,
2) chiral multiplets, making index calculations somewhat trickier. The IRphysics should involve some (0 , G gauge theory coupled both to a G − k WZW model andthe UV boundary matter. It goes beyond the scope of this paper.The supersymmetric statements above have obvious non-supersymmetric versions:1’) Non-SUSY G k YM-CS theory with a Dirichlet b.c. flows to a chiral G k WZW modelcoupled to
T F T [ G k ].2’) Non-SUSY G − k YM-CS theory with Neumann b.c. coupled to chiral algebra T d tocancel the gauge anomaly flows to a chiral T d /G − k coset, coupled to T F T [ G − k ].(Now there are no subtleties about simple vs. non-simple G .) We became aware of theserelations in discussions with Kevin Costello and in other projects (such as [146]).The supersymmetric and non-supersymmetric statements about RG flows lead to level-rank-dual boundary conditions when appropriate choices for G and T d are made — so that aparticular coset T d /G k (coming from N b.c.) happens to be equivalent to a ˜ G ˜ k WZW model(coming from D b.c.). We will discuss this in Section 7.2. We present a few examples of statement (1) about RG flows with Dirichlet boundary condi-tions. In the IR, the bulk theory is purely topological and has no local operators. We shouldthus find that half-indices compute the characters of appropriate WZW models that appearin the IR on the boundary. We indeed find a match with the Weyl-Kac character formula[147–149] for general simple G . U(1)
Let us first consider 3d N = 2 G = U (1) k Yang-Mills-Chern-Simons theory. The half-indexof D b.c. is very simply II D [ U (1) k ] = 1( q ) ∞ (cid:88) m ∈ Z q km x km y k , (7.2)where x is the fugacity for the U (1) ∂ boundary flavor symmetry, and y is the topological U (1) y fugacity. For k = 1, this is an ordinary theta-function II D [ U (1) ] = ( − q xy ; q ) ∞ ( − q / ( xy ); q ) ∞ =F( − q xy ; q ), which is the vacuum character of U (1) WZW, or equivalently a left-handed 2d– 65 –ermion, a.k.a an N = (0 ,
2) Fermi multiplet. For general k >
0, (7.2) is a higher-level theta-function II D [ U (1) k ] = ( q k ; q k ) ∞ ( q ) ∞ ( − q k x k y ; q k ) ∞ ( − q k x k y − ; q k ) ∞ , which matches the vacuumcharacter for U (1) k WZW (intended as a lattice VOA), II D [ U (1) k ] = χ [ U (1) k ] . (7.3)Notice that the sum over boundary monopole sectors is crucial in reproducing the WZWcharacter, rather than the Kac-Moody current algebra character. The current algebra char-acter just counts modes of the chiral U (1) current J ∼ i∂φ , reproducing the 1 / ( q ) ∞ prefactor.The WZW character further includes the vertex operators e imφ , which correspond to bound-ary U (1) monopoles.When k <
0, the monopole sum diverges badly, similar to what we saw in particle-vortexduality (4.31) when the boundary anomaly was negative. We will restrict ourselves to k > n ∈ Z in the UV G = U (1) k Yang-Mills-Chern-Simons theorymodifies the half-index as II D [ U (1) k , n ] = 1( q ) ∞ (cid:88) m ∈ Z q km x km y km × ( q m x ) n = 1( q ) ∞ (cid:88) m ∈ Z q km + mn x km + n y km . (7.4)This agrees with the WZW character up to a mild prefactor, χ n [ U (1) k ] = q − n k y − nk II D [ U (1) k , n ] . (7.5) SU(2)
Next, consider SU (2) k , at k >
0. A general spin j character may be written as χ j [ SU (2) k ] = 1( q ) ∞ ( qx ; q ) ∞ ( qx − ; q ) ∞ ∞ (cid:88) m = −∞ q ( k +2) m +(2 j +1) m χ ( k +2) m + j ( x ) (7.6)where χ j ( x ) = x j +1 − x − j − x − x − . Notice that there are negative terms in the sum, as χ − j ( x ) = − χ j − ( x ). In particular, for the vacuum character we can write χ [ SU (2) k ] = 1( q ) ∞ ( qx ; q ) ∞ ( qx − ; q ) ∞ (cid:34) ∞ (cid:88) m =0 q ( k +2) m + m χ ( k +2) m ( x ) − ∞ (cid:88) m =1 q ( k +2) m − m χ ( k +2) m − ( x ) (cid:35) (7.7)This shows the sequence of nested Verma modules of spin 0, k + 1, k + 2, 2 k + 3 , . . . whichgive a resolution of the vacuum module. Henceforth, we will write all characters without the conventional prefactor, sometimes called the modularanomaly, so that the q -series begins at order q . For a character associated to an integrable representation ofsome level k affine Kac-Moody algebra with highest weight Λ the modular characteristic may be written as q s (Λ) where s (Λ) = | Λ + ρ | / (2( k + h )) − | ρ | / h and ρ denotes the Weyl vector. – 66 –nother simple manipulation of (7.6) expresses the vacuum character as χ [ SU (2) k ] = 1( q ) ∞ ∞ (cid:88) m = −∞ q ( k +2) m x k +2) m ( q − m − x q m )(1 − x )( qx ; q ) ∞ ( qx − ; q ) ∞ = 1( q ) ∞ ∞ (cid:88) m = −∞ q km x km q m x ; q ) ∞ ( q − m x − ; q ) ∞ = II D [ SU (2) k +2 ] , (7.8)where in the middle step we employed the usual identity F( x ; q ) = q n ( − q − x ) n F( q n x ; q ) forF( x ; q ) = ( x ; q ) ∞ ( qx − ; q ) ∞ . This final expression agrees precisely with the half-index of 3d N = 2 SU (2) k +2 CS-YM theory with D b.c., as we had hoped.In a similar way, the spin- j character may be rewritten as χ j [ SU (2) k ] = 1( q ) ∞ ∞ (cid:88) m = −∞ q km x km q m x ; q ) ∞ ( q − m x − ; q ) ∞ χ j ( q m x ) (7.9)which agrees with the half-index II D [ SU (2) k +2 , j ] in the presence of a spin- j Wilson line.Notice also the difference equation χ j [ SU (2) k ]( q ∓ x ) = x ± k q k χ j [ SU (2) k ]( x ) , (7.10)which follows from the discussion in Sections 3.6, 3.7.As a concrete example, the half-index of the theory G = SU (2) with Dirichlet boundaryconditions with a Wilson line of spin 1 / II D [ SU (2) , ] = χ ( x )+( q +2 q ) (cid:2) χ ( x )+ χ ( x ) (cid:3) + q (cid:2) χ ( x )+3 χ ( x )+ χ ( x ) (cid:3) + . . . (7.11)One can check explicitly that this is exactly the character in (cid:98) su (2) with highest weight vectorΛ = (1 ,
1) (written via its Dynkin labels), as expected. (The vacuum, or basic, representationof (cid:98) su (2) has highest weight vector Λ = ( k,
0) = (2 ,
0) in this notation). One can performan identical check for the j = 1 Wilson line against the representation with highest weightvector Λ = (0 , k, j . SO(3)
Similarly, we can begin with the WZW vacuum character for SO (3) k . This can be obtainedby combining the vacuum character of SU (2) k with the spin k character of SU (2) k , whichhas dimension k ( k +1)2 k +2 = k . It can be compactly written as a sum over a lattice refined by afactor of two: χ [ SO (3) k ] = 1( q ) ∞ ( qx ; q ) ∞ ( qx − ; q ) ∞ ∞ (cid:88) m = −∞ ( − m q k +12 m + m χ ( k +1) m ( x ) , (7.12)and rearranged the same way as before to χ [ SO (3) k ] = 1( q ) ∞ ∞ (cid:88) m = −∞ q k m x k m q m x ; q ) ∞ ( q − m x − ; q ) ∞ = II D [ SO (3) k +2 ] , (7.13)which matches the Dirichlet half-index. – 67 – ny simple G Analogously, we expect the vacuum WZW character for general simple group G at level k > II D [ G k + h ] = 1( q ) r ∞ (cid:88) m ∈ Λ ∨ q k ( m,m ) x km (cid:81) α ∈ Φ ( q m · α x α ; q ) ∞ , (7.14)where r = rank( G ), Λ ∨ denotes the cocharacter lattice of G , and Φ is the set of roots of G .Just as in the SU (2) example above, this can be rewritten as II D [ G k + h ] = 1( q ) r ∞ (cid:88) m ∈ Λ ∨ q k ( m,m ) x km (cid:81) α ∈ Φ + (1 − q m · α x α ) (cid:81) α ∈ Φ + F( q m · α x α ; q ) . (7.15)Then using F( q m · α x α ; q ) = q − ( m · α ) ( − q x − α ) m · α F( x α ; q ) as well as (cid:80) α ∈ Φ + ( m · α ) = h ( m, m ) and (cid:81) α ∈ Φ + x α ( m · α ) = x hm this becomes II D [ G k + h ] = 1( q ) r ∞ (cid:81) α ∈ Φ + ( qx α ; q ) ∞ ( qx − α ; q ) ∞ (cid:88) m ∈ Λ ∨ q ( k + h )( m,m ) x ( k + h ) m (cid:89) α ∈ Φ + ( − q ) − m · α − q m · α x α − x α . (7.16)In Appendix B, we show that this expression is equivalent to the usual expression for theWeyl-Kac character formula (up to the overall factor of the modular anomaly) when G issimply connected.We can also write down the half-index with a Wilson loop insertion in representation R : II D [ G k + h , R ] = 1( q ) r ∞ (cid:88) m ∈ Λ ∨ q k ( m,m ) x km (cid:81) α ∈ Φ + ( q m · α x α ; q ) ∞ Tr R ( q m x ) , (7.17)which we expect to coincide with the character of other highest-weight modules. Of course,there should be identifications between Wilson lines. These are captured by the identity II D [ G k + h , R ]( q ( δ, · ) x ; q ) = x kδ q − k ( δ,δ ) II D [ G k + h , R ]( x ; q ) , (7.18)which follows easily from a shift of summation in (7.17) . U(N)
For level-rank duality, it is useful to consider U ( N ) groups as well. There are two well-behavedpossibilities.Suppose that we wish to reproduce the U ( N ) k WZW characters from a half-index, at k >
0. Then we need to look for a 3d N = 2 theory whose boundary ’t Hooft anomaly is k Tr( f ). A naive guess is N = 2 U ( N ) k + N theory; but the computation in (2.51) shows thatthis won’t quite work, due to an extra (Tr f ) contribution from the gauginos. What worksinstead is an N = 2 YM-CS theory with a different level for the U (1) part of U ( N ).– 68 –e use the following notation (coincident with [21, 37]): We write U ( N ) κ = U ( N ) κ,κ fora Chern-Simons theory whose (UV) action corresponds to the anomaly polynomial U ( N ) κ : κ Tr( f ) , (7.19)where f is the U ( N ) field strength. We write U ( N ) κ,κ + p for the more general possibility U ( N ) κ,κ + p : κ Tr( f ) + pN (Tr f ) . (7.20)Notice that if we restrict f to be diagonal with all entries equal to f / √ N (corresponding tothe U (1) field strength inside U ( N )), the polynomial (7.20) reduces to ( κ + p ) f . Thus theeffective U (1) level is κ + p .In order to reproduce the U ( N ) k WZW vacuum character, we should use 3d N = 2 U ( N ) k + N,k theory. The polynomial encoding its bulk CS levels is I bulk = ( k + N )Tr( f ) − (Tr f ) , so that the boundary anomaly on a D b.c. is I D = I bulk − (cid:2) N Tr( f ) − (Tr f ) + N r (cid:3) = k Tr( f ) − N r , (7.21)which (modulo the r term) matches the anomaly of U ( N ) k WZW. If we include a topological U (1) y symmetry as well, which gives an extra term 2(Tr f ) y in the anomaly polynomial, thenthe half-index is II D [ U ( N ) k + N,k ] = 1( q ) N ∞ (cid:88) m ∈ Z q k m · m x km y (cid:80) i m i (cid:81) i (cid:54) = j ( q m i − m j x i /x j ; q ) ∞ ? = χ [ U ( N ) k ] . (7.22)Alternatively, we may consider 3d N = 2 U ( N ) k + N YM-CS theory. Its bulk CS levelsare encoded by ( k + N )Tr( f ), so the boundary anomaly on D b.c. is now I D = k Tr( f ) + (Tr f ) + 2(Tr f ) y − N r . (7.23)The half-index II D [ U ( N ) k + N ] = 1( q ) N ∞ (cid:88) m ∈ Z q k m · m + ( (cid:80) i m i ) x km ( xy ) (cid:80) i m i (cid:81) i (cid:54) = j ( q m i − m j x i /x j ; q ) ∞ ? = χ [ U ( N ) k,k + N ] (7.24)is now expected to compute the vacuum character of U ( N ) k,k + N WZW, which matches theanomaly (7.23).
We now consider statement (2): that 3d N = 2 YM-CS theory with N b.c., and appropriatedegrees of freedom to cancel the gauge anomaly, flows to a coset model. Since this has alreadybeen studied in the literature, we only give a few examples, and focus on explaining how thestatement may be used productively to generate dual boundary conditions.– 69 – (1) − k ↔ SU(k) k +1 Consider 3d N = 2 U (1) − k theory, now with N b.c. The boundary anomaly is simply I N = − k f . (7.25)If k >
0, a simple way to cancel the anomaly is by introducing k boundary Fermi multipletsΓ i of gauge charge +1 (and R-charge zero). The corresponding half-index is II N +Γ [ U (1) − k ] = ( q ) ∞ (cid:73) ds πis k (cid:89) i =1 F( − q x i s ; q ) , (7.26)where we have introduced additional fugacities x = ( x , ..., x k ) with (cid:81) i x i = 1 for the newboundary SU ( k ) flavor symmetry that rotates the Fermi multiplets. We now observe thatthe boundary anomaly has been shifted to a ’t Hooft anomaly I N + I Γ = Tr( x ) (7.27)for the SU ( k ) flavor symmetry. Such an anomaly could be matched by an SU ( k ) WZWmodel in the IR. Indeed the SU ( k ) WZW is exactly what we would expect from the cosetconstruction k free fermions U (1) − k = U ( k ) U (1) − k = SU ( k ) . (7.28)The index (7.26) is actually easy to evaluate directly, since F( − q z ) = q ) ∞ (cid:80) n ∈ Z q n z n and taking the coefficient of s in the integrand gives II N +Γ [ U (1) − k ] = 1( q ) k − ∞ (cid:88) n ∈ Z k (cid:80) i ni =0 q n · n x n = χ [ SU ( k ) ] . (7.29)This is the lattice sum for the vacuum character of SU ( k ) WZW. For example, at k = 2 wemay use SU (2) fugacities ( x , x ) = ( x, x − ) and obtain χ [ SU (2) ] = 1+ qχ ( x )+ q (cid:2) χ ( x )+ χ ( x ) (cid:3) + q (cid:2) χ ( x )+ χ ( x )]+ q (cid:2) χ ( x )+2 χ ( x )+2 χ ( x ) (cid:3) + ... (7.30)where χ j ( x ) is the spin- j character of SU (2), just as in (7.6).In Section 7.1 we encountered another boundary condition whose half-index was an SU ( k ) character, namely D b.c. for 3d N = 2 SU ( k ) k +1 theory. The Dirichlet half-index ofthis latter theory matches (7.29). We deduce a duality of SUSY boundary conditions U (1) − k N + ( k fund. fermis) ↔ SU ( k ) k +1 D . (7.31)We could in principle have considered k < N b.c. gauge anomalyfor U (1) − k would have to be cancelled by chiral rather than Fermi multiplets. However, thisleads to the same sort of difficulties in the index computation that we encountered in particle-vortex duality, cf. (4.23). To avoid potential problems with free chirals on the boundary, werestrict ourselves to k >
0. – 70 – (N) − k − N, − k ↔ SU(k) k + N To generalize the U (1) − k example above, let us consider 3d N = 2 U ( N ) − k − N, − k with k, N > I bulk = − ( k + N )Tr( f ) + (Tr f ) . (7.32)The boundary anomaly on N b.c. due to gauginos then becomes I N = I bulk + N Tr( f ) − (Tr f ) + N r = − k Tr( f ) + N r . (7.33)We may cancel this by introducing k fundamental Fermi multiplets Γ i (of R-charge zero),transforming under an SU ( k ) boundary flavor symmetry. They shift the anomaly by I Γ = k Tr( f ) + N Tr( x ), where x is the SU ( k ) field strength, to I N + I Γ = N Tr( x ) + N r . (7.34)This matches perfectly with the ’t Hooft anomaly of a Dirichlet b.c. on 3d N = 2 SU ( k ) k + N YM-CS theory (up to a shift of the R-R anomaly, which can be put in by hand). Thus wemay expect a duality of boundary conditions U ( N ) − k − N, − k N + k fund. fermis ↔ SU ( k ) k + N D . (7.35)We are claiming slightly more than this, namely that both sides flow to an SU ( k ) N WZWmodel on the boundary. On the right side this follows from Section 7.1. On the left side, weexpect a coset model, which agrees by standard level-rank duality of chiral algebras:
N k free fermions U ( N ) − k = U ( N k ) U ( N ) − k (cid:39) SU ( k ) N (7.36)In addition, the Neumann b.c. for U ( N ) − k − N, − k theory has a half-index II N +Γ [ U ( N ) − k − N, − k ] = ( q ) N ∞ (cid:73) N (cid:89) i =1 ds i πis i (cid:89) i (cid:54) = j (cid:16) q s i s j ; q (cid:17) ∞ N (cid:89) i =1 k (cid:89) a =1 F( − q x a s i ; q ) (7.37)whose agreement with the vacuum character χ [ SU ( k ) N ] we have checked computationallyfor various values of N and k . SU(N) − k − N ↔ U(k) k + N,N
Conversely, we might start with Neumann b.c. for 3d N = 2 SU ( N ) − k − N . The boundarygauge anomaly is again cancelled by k fundamental Fermi multiplets ( i.e. coupling to N k freefermions), which now introduce a U ( k ) boundary flavor symmetry. We expect the Neumannb.c. to flow to the coset model N k free fermions SU ( N ) − k (cid:39) U ( k ) N . (7.38)The same result is obtained on a Dirichlet b.c. for 3d N = 2 U ( k ) k + N,N , whence SU ( N ) − k − N N + k fund. fermis ↔ U ( k ) k + N,N D . (7.39)– 71 – (N) − k − N ↔ U(k) k + N Finally, let us apply the same analysis starting with a Neumann b.c. for 3d N = 2 U ( N ) − k − N theory, with k, N >
0. It is useful to keep track of the U (1) y topological symmetry, for whichwe introduce a field strength y as usual. Then the bulk CS levels are I bulk = − ( k + N )Tr( f ) + 2(Tr f ) y , (7.40)leading to a boundary anomaly I N = I bulk + N Tr( f ) − (Tr f ) + N r = − k Tr( f ) − (Tr f ) + 2(Tr f ) y + N r . (7.41)Adding k fundamental Fermi multiplets Γ on the boundary is not sufficient to cancel the − (Tr f ) part of the gauge anomaly. However, we can deal with this by introducing one more“baryonic” Fermi multiplet η , which transforms under the U (1) part of U ( N ). We can evencancel the mixed gauge-topological anomaly by giving η nontrivial U (1) y charge.The multiplets Γ and η introduce an extra U ( k ) flavor symmetry on the boundary. Thisis the commutant U ( N ) × U (1) y in U ( N k + 1). Under both gauge and flavor symmetry, theseFermi multiplets may be taken to have charges U ( N ) U ( k ) U (1) y U (1) R Γ N k η det det − − I Γ + I η = (cid:2) k Tr( f ) + N Tr( x ) + 2(Tr f )(Tr x ) (cid:3) + (Tr f − Tr x − y ) (7.43)= k Tr( f ) + N Tr( x ) + (Tr f ) + (Tr x ) + 2 y (Tr x − Tr f ) − y , which cancels the gauge part of the N b.c. anomaly and leaves behind I N + I Γ + I η = N Tr( x ) + (Tr x ) + 2 y (Tr x ) − y + N r . (7.44)The ’t Hooft anomaly (7.44) matches that of Dirichlet b.c. in U ( N ) k + N theory (7.23),up to a shift of the r and y terms. Thus, we may surmise that U ( k ) − k − N N + Γ , η ↔ U ( k ) k + N D . (7.45)In the IR, we expect the right side to flow to a U ( k ) k + N,N
WZW model, and the left side toflow to a coset, essentially
N k + 1 free fermions U ( N ) − k − N, − k (cid:39) U ( k ) k + N,N . (7.46)This is the duality of chiral algebras responsible for level-rank duality of non-superymmetric U ( N ) − k − N, − k and U ( k ) k + N,N
Chern-Simons theories, cf. [21].– 72 –e may also compare half-indices. The Neumann b.c. has II N +Γ+ η [ U ( N ) − k − N ] = ( q ) N ∞ (cid:73) N (cid:89) i =1 ds i πis i (cid:89) i (cid:54) = j (cid:16) q s i s j ; q (cid:17) ∞ F (cid:16) − q (cid:81) i s i y (cid:81) a x a ; q (cid:17) N (cid:89) i =1 k (cid:89) a =1 F( − q x a s i ; q ) , (7.47)and we have checked computationally that this indeed matches (7.24) in several cases. We may use the proposed dual boundary conditions above to construct duality interfacesfor level-rank dual theories. The construction works the same way as in Sections 5 and 6.Just as in the case of particle-vortex duality, the interfaces turn out to be directional: in ourconventions, they must necessarily have negative CS levels on the left to positive CS levelson the right.For example, the duality interfaces for U ( N ) − k − N, − k ↔ SU ( k ) k + N and SU ( N ) − k − N ↔ U ( k ) k + N,N have N b.c. for the gauge fields on either side, coupled to N k free Fermi multipletsΓ in a bi-fundamental representation of the gauge groups. Schematically, U ( N ) − k − N, − k (cid:12)(cid:12) Γ | SU ( k ) k + N , SU ( N ) − k − N (cid:12)(cid:12) Γ (cid:12)(cid:12) U ( k ) k + N,N . (7.48)Again we emphasize that anomalies on the interfaces only cancel for k, N > U ( N ) − k − N ↔ U ( k ) k + N involves both bi-fundamentaland bi-baryonic Fermi multiplets Γ , η as in (7.42): U ( N ) − k − N (cid:12)(cid:12) Γ , η (cid:12)(cid:12) U ( k ) k + N . (7.49)It is worth pointing out that this is a “supersymmetrization” of a non-supersymmetriclevel-rank duality interface, defined in terms of bi-fundamental fermions rather than Fermimultiplets. In bulk 3d dualities, one has a great deal of freedom to “move” U (1) gauge groups around.This typically proceeds by gauging topological symmetries, which can have the effect of un-gauging dynamical U (1)’s [15, 119, 120]. The bulk U ( N ) and SU ( N ) level-rank dualitiesabove are known to be related by such operations. The gauging and un-gauging operationsmay be applied in the presence of dual boundary conditions as well, being careful to accountfor the boundary degrees of freedom.We illustrate this by reducing the duality of boundary conditions in U ( N ) − k − N ↔ U ( k ) k + N theories (7.45) to U ( N ) − k − N, − k ↔ SU ( k ) k + N as in (7.35). For a perfect match ofboundary ’t Hooft anomalies, the bulk CS levels in U ( N ) − k − N ↔ U ( k ) k + N duality may bedefined as U ( N ) − k − N : I bulk = − ( k + N )Tr( f ) + 2(Tr f ) y + y − N r ,U ( k ) k + N : I bulk = ( k + N )Tr( x ) + 2(Tr x ) y + k r . (7.50)– 73 –ow we simultaneously gauge the U (1) y topological symmetry in both theories. In the U ( k ) k + N theory, this has the effect of un-gauging the U (1) part of U ( N ), to which thetopological symmetry has a mixed CS coupling. Thus we are left with an SU ( k ) k + N bulktheory. The boundary condition is still D .On the other hand, in the U ( N ) − k − N theory, due to the extra y CS term, gauging the U (1) y symmetry has virtually no effect in the bulk. Indeed, once U (1) y is gauged, we canredefine abelian gauge charges in a way that corresponds to sending y → y − Tr f . The bulkCS levels become I (cid:48) bulk = − ( k + N )Tr( f ) + (Tr f ) + y − N r , (7.51)and we see that the bulk theory has become a decoupled U ( N ) − k − N, − k × U (1) . On theboundary, the extra 2d Fermi multiplets now have charges U ( N ) SU ( k ) U (1) y U (1) R Γ N k η − U ( N ) and η only under the new U (1), giving decoupled bound-ary conditions (cid:0) U ( N ) − k − N, − k w/ N + Γ (cid:1) ⊗ (cid:0) U (1) w/ N + η (cid:1) . (7.53)However, U (1) with a boundary Fermi multiplet η is a trivial theory in the IR, both on thebulk and on the boundary: the coset chiral algebra is (1 free fermion) /U (1) = U (1) /U (1) ,and we can easily verify that the half-index is II N + η [ U (1) ] = ( q ) ∞ (cid:73) dy πiy F( − q /y ) = (cid:73) dy πiy (cid:88) n ∈ Z q n y n = 1 . (7.54)Altogether, the effect of gauging the U (1) y symmetry is to reduce the U ( N ) − k − N side of theduality to U ( N ) − k − N, − k with N b.c. coupled to Γ. This is precisely what we want for (7.35). In this section we identify several dual pairs of boundary conditions for Aharony-dual theorieswith U ( N ) gauge groups [4]. Aharony dualities generalize the SQED ↔ XYZ and SQCD ↔ detYZ dualities from Section 6, and flow from 4d Seiberg duality [150] ( cf. [15]). Just asin SQED and SQCD in Section 6, all the bulk UV Chern-Simons levels in Aharony dualityare identically zero, and the matter fields are “non-chiral” — in that equally numbers offundamental and anti-fundamental 3d chiral multiplets occur. This leads to a structure ofdual boundary conditions that is very similar to that found in Section 6, and duality interfacesthat are effectively bi-directional. – 74 – .1 Fundamental duality and interface Aharony duality relates the 3d N = 2 theoriesTheory A : (cid:40) U ( N c ) gauge theorywith N f fundamental ( Q ) and N f anti-fundamental ( ˜ Q ) chiralsTheory B : U ( N (cid:48) c ) gauge theory, N (cid:48) c := N f − N c with N f fundamental ( Q (cid:48) ) and N f anti-fundamental ( ˜ Q (cid:48) ) chirals,a N f × N f matrix of singlets ( M ), and two more singlets ( V ± ),and W = M Q (cid:48) ˜ Q (cid:48) + U − V +3d + U +3d V − (8.1)We assume that N f > N c (so N (cid:48) c > N f = N c one finds instead theSQED ↔ XYZ and SQCD ↔ detYZ dualities of Section 6.Recall that the bulk duality maps the mesons Q ˜ Q of Theory A to the singlets M ofTheory B, and it maps the two scalar monopole operators of Theory A to the singlets V ± ofTheory B. Theory B has its own mesons Q (cid:48) ˜ Q (cid:48) and scalar monopole operators U ± ; however,these operators are killed by the bulk superpotential.The global symmetry is SU ( N f ) × (cid:94) SU ( N f ) × U (1) a × U (1) y × U (1) R , where as usualthe latter three U (1)’s are axial, topological, and R-symmetry. We split the bulk chirals into N = (0 ,
2) chiral and Fermi multiplets as usual, namely( Q, Ψ) , ( ˜ Q, ˜Ψ) ; ( Q (cid:48) , Ψ (cid:48) ) , ( ˜ Q (cid:48) , ˜Ψ (cid:48) ) , ( M, µ ) , ( V ± , υ ± ) . (8.2)The (0 ,
2) chirals can be taken to have charges U ( N c ) U ( N (cid:48) c ) SU ( N f ) (cid:94) SU ( N f ) U (1) a U (1) y U (1) R Q N c f Q N c f Q (cid:48) c N f − Q (cid:48) c f − M f N f V ± − N f ± N (cid:48) c + 1 (8.3)and as usual the (0,2) fermis have the conjugate flavor charges and (1 − ρ chiral ) R-charge. Weagain emphasize that the bulk UV Chern-Simons levels (both for gauge and global symme-tries) are identically zero.We propose the following two pairs of dual boundary conditions:( D , Q D , ˜ Q D) ↔ ( N , Q (cid:48) N , ˜ Q (cid:48) N , M D , V ± N ) + Γ , η ( N , Q N , ˜ Q N) + Γ (cid:48) , η (cid:48) ↔ ( D , Q (cid:48) D , ˜ Q (cid:48) D , M N , V ± D ) , (8.4)– 75 –here Γ , η, Γ (cid:48) , η (cid:48) are 2d boundary Fermi multiplets with charges U ( N c ) U ( N (cid:48) c ) SU ( N f ) (cid:94) SU ( N f ) U (1) a U (1) y U (1) R Γ N c N (cid:48) c η det det − (cid:48) N c N (cid:48) c η (cid:48) det det − − , η are charged under the dynamical U ( N (cid:48) c ) of Theory B, but also carry chargeunder a U ( N c ) ∂ boundary flavor symmetry, consistent with D b.c. in Theory A. The oppositeis true of Γ (cid:48) , η (cid:48) . The presence of these 2d fermis and their charges can be inferred from match-ing boundary anomalies, as we show momentarily. There are also two general consistencychecks showing that the pairs of boundary conditions in (8.4) are reasonable :1) Neither requires a matrix factorization of the superpotential in Theory B. In particular,both ( Q (cid:48) N , ˜ Q (cid:48) N , M D) and ( Q (cid:48) N , ˜ Q (cid:48) N , M D) set the cubic term in the superpotential to zero, satisfyingthe requirements of Section 2.3. In addition, ( D , V ± D ) explicitly sets the monopole term inthe superpotential to zero, while ( N , V ± N ) does so by killing the U ± monopole operatorswith N b.c. on the gauge multiplet.2) The choice of D vs. N b.c. for the various matter fields either fully aligned or fullyanti-aligned with the signs of the fields’ axial charges. This turns out to be important inrecovering pure level-rank duality by turning on a large axial mass and integrating outall the bulk matter — in particular, it ensures that after the bulk matter is integratedout, no edge modes are left behind. We will say more Section 9.Two natural duality interfaces can be produced from (8.4) by applying the procedure ofSection 5.1: Theory B ( N , Q (cid:48) N , ˜ Q (cid:48) N , M D , V ± N ) (cid:12)(cid:12) Γ , η (cid:12)(cid:12) Theory A ( N , Q N , ˜ Q N) or
Theory A ( N , Q N , ˜ Q N) (cid:12)(cid:12) Γ (cid:48) , η (cid:48) (cid:12)(cid:12) Theory B ( N , Q (cid:48) N , ˜ Q (cid:48) N , M D , V ± N ) . (8.6)Each interface carries a superpotential, (cid:90) dθ + (cid:2) µQ ˜ Q + Q (cid:48) Γ Q ] (cid:0) E Γ = ˜ Q (cid:48) ˜ Q ) or (cid:90) dθ + (cid:2) Q ˜ Qµ + Q Γ (cid:48) Q (cid:48) ] (cid:0) E Γ (cid:48) = ˜ Q ˜ Q (cid:48) ) (8.7)that factorizes the cubic part of the bulk superpotential from Theory B, by a now familiarmechanism. Namely, the µQ ˜ Q coupling sets M = Q ˜ Q at the interface, and then E Γ · J Γ = Q ˜ QQ (cid:48) ˜ Q (cid:48) (with all gauge and flavor indices contracted), which equals the bulk W = M Q (cid:48) ˜ Q (cid:48) .We would expect the Fermi multiplet η to play a role in 1) factorizing the U − V + + U + V − part of the bulk superpotential at the interface; and 2) mediating the relation betweenmonopole operators of Theory A and the V ± singlets of Theory B at the interface. The– 76 –atter function was accomplished by Γ in SQED ↔ XYZ and SQCD ↔ deyYZ in Section 6, cf. (6.11). The precise couplings of η to the bulk and the mechanism by which it accomplishes(1) and (2) are not yet clear, and would be interesting to understand.Other dual pairs of boundary conditions can be generated from (8.4), either by deformingboth sides, or by colliding with the interface (8.6). The pattern is a generalization of Table 1for SQED ↔ XYZ.
Now, let us check more closely that boundary anomalies match for the proposed dual pairs(8.4). We denote the field strengths for various gauge and global symmetries as U ( N c ) U ( N (cid:48) c ) SU ( N f ) (cid:94) SU ( N f ) U (1) a U (1) y U (1) R s s (cid:48) x ˜ x a y r (8.8)As usual, ‘ s ’ may refer to the dynamical field strength on N b.c., or the U ( N c ) ∂ flavor fieldstrength on D b.c., and similarly for s (cid:48) . Then for ( D , Q D , ˜ Q D), the boundary anomaly is I ( D ,D,D) = − N c Tr( s ) + (Tr s ) − N c r + 2(Tr s ) y (gauginos, FI) (8.9)+ (cid:2) N c Tr( x ) + N f Tr( s ) + 2 N f (Tr s )( a − r ) + N c N f ( a − r ) (cid:3) ( Q )+ (cid:2) N c Tr(˜ x ) + N f Tr( s ) − N f (Tr s )( a − r ) + N c N f ( a − r ) (cid:3) ( ˜ Q )= N (cid:48) c Tr( s ) + (Tr s ) + 2(Tr s ) y + N c (cid:2) Tr( x ) + Tr(˜ x ) (cid:3) + N c N f ( a − r ) − N c r , and I ( N ,N,N) = −I ( D ,D,D) + 4(Tr s ) y . Similarly, the ( N , Q (cid:48) N , ˜ Q (cid:48) N , M D , V ± N ) anomaly is I ( N ,N,N,D,N) = + N (cid:48) c Tr( s (cid:48) ) − (Tr s (cid:48) ) + N (cid:48) c r + 2(Tr s (cid:48) ) y (gauginos, FI) (8.10) − (cid:2) N (cid:48) c Tr( x ) + N f Tr( s (cid:48) ) + 2 N f (Tr s )( − a ) + N (cid:48) c N f ( − a ) (cid:3) ( Q (cid:48) ) − (cid:2) N (cid:48) c Tr(˜ x ) + N f Tr( s (cid:48) ) − N f (Tr s )( − a ) + N (cid:48) c N f ( − a ) (cid:3) ( ˜ Q (cid:48) )+ (cid:2) N f (Tr( x ) + Tr(˜ x ) + N f (2 a − r ) (cid:3) ( M ) − (cid:2) ( − N f a + y + N (cid:48) c r ) + ( − N f a − y + N (cid:48) c r ) (cid:3) ( V ± ) , and I ( D ,D,D,N,D) = −I ( N ,N,N,D,N) + 4(Tr s (cid:48) ) y . Quite beautifully, we find I ( D ,D,D) = I ( N ,N,N,D,N) + I Γ + I η , I ( N ,N,N) + I Γ (cid:48) + I η (cid:48) = I ( D ,D,D,N,D) , (8.11)where I Γ = I Γ (cid:48) = N (cid:48) c Tr( s ) + 2(Tr s )(Tr s (cid:48) ) + N c Tr( s (cid:48) ) and I η = (Tr s − Tr s (cid:48) + y ) , I η (cid:48) =(Tr s − Tr s (cid:48) − y ) are precisely the anomalies of the additional 2d Fermi multiplets we proposed.In particular, the ( N ,N,N,D,N) boundary condition in Theory B has gauge and mixedgauge-topological anomalies that are precisely cancelled by coupling to the boundary fermisΓ , η ; and the ( N ,N,N) b.c. in theory A has similar anomalies that are cancelled by couplingto Γ (cid:48) , η (cid:48) . – 77 – .3 Half-indices To compute half-indices of the boundary conditions in (8.4), we introduce fugacities U ( N c ) U ( N (cid:48) c ) SU ( N f ) (cid:94) SU ( N f ) U (1) a U (1) y U (1) R s, m z, n x ˜ x a y − q (8.12)constrained so that (cid:81) N f α =1 x α = (cid:81) N f α =1 ˜ x α = 1. Then we find II ( D ,D,D) = 1( q ) N c ∞ (cid:88) m ∈ Z N q (cid:2) N (cid:48) c m · m +( (cid:80) i m i ) (cid:3) s N (cid:48) c m (cid:0) y (cid:81) i s i (cid:1) (cid:80) i m i (cid:81) i (cid:54) = j ( q m i − m j s i /s j ; q ) ∞ (cid:89) ≤ i ≤ N ≤ α ≤ Nf II D ( q m i as i x α ; q ) II D (cid:16) aq m i s i ˜ x α ; q (cid:17) , (8.13) II ( N ,N,N,D,N)+Γ ,η = II N (cid:16) ( − q ) N (cid:48) c +1 ya N f (cid:17) II N (cid:16) ( − q ) N (cid:48) c +1 a N f y (cid:17) N f (cid:89) α,β =1 II D (cid:16) a x α ˜ x β (cid:17) (8.14) × ( q ) N (cid:48) c ∞ N (cid:48) c ! (cid:73) N (cid:48) c (cid:89) i =1 dz i πiz i (cid:89) i (cid:54) = j ( qz i /z j ; q ) ∞ (cid:89) ≤ i ≤ N (cid:48) c ≤ α ≤ Nf II N (cid:16) − q z i ax α ; q (cid:17) II N (cid:16) − q ˜ x α az i ; q (cid:17) × F (cid:16) − q y (cid:81) i s i (cid:81) j z j ; q (cid:17) (cid:89) ≤ i ≤ Nc ≤ j ≤ N (cid:48) c F( − q s i z j ; q ) , as well as II ( N ,N,N)+Γ (cid:48) ,η (cid:48) = ( q ) N c ∞ N c ! (cid:73) N c (cid:89) i =1 ds i πis i (cid:89) i (cid:54) = j ( qs i /s j ; q ) ∞ (cid:89) ≤ i ≤ N ≤ α ≤ Nf II N ( as i x α ; q ) II N (cid:16) as i ˜ x α ; q (cid:17) (8.15) × F (cid:16) − q (cid:81) i s i y (cid:81) j z j ; q (cid:17) (cid:89) ≤ i ≤ Nc ≤ j ≤ N (cid:48) c F( − q s i z j ; q ) ,II ( D ,D,D,N,D) = II D (cid:16) ( − q ) N (cid:48) c +1 ya N f (cid:17) II D (cid:16) ( − q ) N (cid:48) c +1 a N f y (cid:17) N f (cid:89) α,β =1 II N (cid:16) a x α ˜ x β (cid:17) (8.16) × q ) N (cid:48) c ∞ (cid:88) n ∈ Z N (cid:48) c q (cid:2) N c n · n +( (cid:80) i n i ) (cid:3) z N c n (cid:0) y (cid:81) i z i (cid:1) (cid:80) i n i (cid:81) i (cid:54) = j ( q n i − n j z i /z j ; q ) ∞ (cid:89) ≤ i ≤ N (cid:48) c ≤ α ≤ Nf II D (cid:16) − q + n i z i ax α ; q (cid:17) II D (cid:16) − q − n i ˜ x α az i ; q (cid:17) . We expect that II ( D ,D,D) ( s, x, ˜ x, q, y ; q ) = II ( N ,N,N,D,N)+Γ ,η ( s, x, ˜ x, q, y ; q ) II ( N ,N,N)+Γ (cid:48) ,η (cid:48) ( z, x, ˜ x, q, y ; q ) = II ( D ,D,D,N,D) ( z, x, ˜ x, q, y ; q ) , (8.17)– 78 –nd have checked these and have checked these in a variety of examples by comparing (atleast) the first ten terms in the q -series expansion. For illustrative purposes, let us examine one nontrivial example, the duality II ( D ,D,D) ( s, x, ˜ x, q, y ; q ) = II ( N ,N,N,D,N)+Γ ,η ( s, x, ˜ x, q, y ; q ) with N c = N (cid:48) c = 2 , N f = 4. For those who wish to reproducesome of our computations in Mathematica , we note that in most examples it is convenient tomake the substitution a → a/q , which has the nice feature of improving convergence in themonopole sum. The expressions quickly become unwieldy, so we only display a few terms: II ( s, x, ˜ x, q, y ; q ) =1 + (cid:18) s s + s s + 2 (cid:19) q + q / (cid:0) s s y + s s y + s s y + s + s s + s (cid:1) s s y + q (6 − s ˜ x a − s ˜ x a ˜ x − s ˜ x a ˜ x − s a ˜ x − s ˜ x a − s ˜ x a ˜ x − s ˜ x a ˜ x − s a ˜ x − x as − as x − x as x − x as x − x as − as x − x as x − x as x + 2 (cid:18) s s + s s (cid:19) + s + s s s + s + s s s ) + . . . It is well known ( cf. [14, 15]) that Aharony dualities can be deformed by real masses insuch a way that they flow to interesting “chiral” dualities with non-zero Chern-Simons levels,including Giveon-Kutasov duality [11] and level-rank dualities in pure gauge theory fromSection 7. With some care, the flows can be performed in the presence of boundary conditions.Then the basic dual boundary conditions (8.4) in Aharony duality can be used to derive (orre-derive) dual boundary conditions for other pairs of bulk theories. We discuss the basictechnique, and illustrate it in a few examples. U ( N ) k + N ↔ U ( k ) − k − N level-rank duality Consider the Aharony duality of Section 8 with N c = N and N f = k + N , where k, N > U (1) a symmetry allows all matter fields to beintegrated out in the bulk of both Theory A and Theory B. Integrating out fermions shiftsthe Chern-Simons level in a manner that depends on the sign of the mass: • With a large positive axial mass, the bulk Theory A flows to pure N = 2 U ( N ) k + N Yang-Mills-Chern-Simons theory, and Theory B flows to a pure N = 2 U ( k ) − k − N YM-CS theory. The global symmetry is reduced to U (1) y × U (1) R (the remainder of theflavor group acts trivially). There are some non-trivial background Chern-Simons terms;after an equal shift of the R-R term on both sides (by − ( N + kN ) r ) we find thatthe bulk CS terms correspond to the anomaly polynomialsTheory A: U ( N ) k + N I A bulk = ( k + N )Tr( s ) + 2 y Tr s + N r , Theory B: U ( k ) − k − N I B bulk = − ( k + N )Tr( s (cid:48) ) + 2 y Tr s (cid:48) − y + N r . (9.1) For comparison, the full superconformal index for many Aharony dual pairs was computed in [151]. – 79 –
With a large negative axial mass, we instead findTheory A: U ( N ) − k − N I A bulk = − ( k + N )Tr( s ) + 2 y Tr s − N r , Theory B: U ( k ) k + N I B bulk = ( k + N )Tr( s (cid:48) ) + 2 y Tr s (cid:48) + y − N r . (9.2)These coincide with the level-rank duality of Section 7.We can derive the dual boundary conditions in level-rank duality from dual boundaryconditions in Aharony duality, but in doing so we must be a little careful. Turning on a realmass m Φ for a 3d N = 2 chiral multiplet allows it to be integrated out in the bulk. However,in the presence of a boundary condition, it may acquire a massless edge mode, in the form ofa purely 2d N = (0 ,
2) chiral or Fermi multiplet. We recall from Section 2.4 that • a 3d N = 2 chiral with N b.c. has an N = (0 ,
2) chiral edge mode if m Φ > • a 3d N = 2 chiral with D b.c. has an N = (0 ,
2) chiral edge mode if m Φ < D , Q D , ˜ Q D) ↔ ( N , Q (cid:48) N , ˜ Q (cid:48) N , M D , V ± N ) + Γ , η ( N , Q N , ˜ Q N) + Γ (cid:48) , η (cid:48) ↔ ( D , Q (cid:48) D , ˜ Q (cid:48) D , M N , V ± D ) , (9.3)We use the top pair of boundary conditions when turning on a large, positive axial mass m a .Since the effective real mass of each 3d chiral is proportional to its axial charge, this meansthat Q, ˜ Q, M will have positive mass while Q (cid:48) , ˜ Q (cid:48) , V ± have negative mass; therefore, no edgemodes will arise during the flow. We deduce that U ( N ) k + N YM-CS with D b.c. ↔ U ( k ) − k − N YM-CS with N b.c. + Γ , η , (9.4)where Γ , η are bifundamental and bi-det Fermi multiplets, just as in level-rank duality, Section7.2. Alternatively, we can use the bottom pair of b.c. when turning on a large, negative axialmass, resulting in U ( N ) − k − N YM-CS with N b.c. + Γ (cid:48) , η (cid:48) ↔ U ( k ) k + N YM-CS with D b.c. . (9.5)This is of course the same as (9.4).In a similar way, we can flow from the interfaces (8.6) in Aharony duality to the interface(7.45) in level-rank duality. No matter whether we turn on a large positive mass in the firstinterface of (8.6) or a large negative mass in the second interface, we end up with the samelevel-rank interface ( U ( N ) − k − N , N ) (cid:12)(cid:12) Γ , η (cid:12)(cid:12) ( U ( k ) k + N , D ) , (9.6)– 80 –hich necessarily has a negative level on the left and a positive level on the right.In principal, one could consider turning on an axial mass of the “wrong” sign. This doesgenerate edge modes, which need to be kept track of carefully. In particular, it may introduceadditional 2d N = (0 ,
2) chiral multiplets on the boundary. Charged 2d N = (0 ,
2) chiralscan be problematic, as we first found in particle-vortex duality, cf. (4.23).For example, suppose that we introduce a large negative axial mass in the top pair ofboundary conditions in (9.3). This will lead to chiral edge modes of Q (cid:48) d , ˜ Q (cid:48) d , which arepotentially problematic. A possibly way to remedy the problem is to first flip the D b.c. onthe Q d , ˜ Q d chirals in Theory A, by introducing boundary (0,2) chiral multiplets q, ˜ q of thesame charges. Then, in the presence of negative axial mass, Theory A flows to U ( N ) − k − N CS-YM theory with D b.c., while Theory B flows to U ( N ) k + N YM-CS theory with N b.c.coupled to a large collection of 2d multiplets:- The fermis Γ , η - A Fermi µ that’s a surviving edge mode of M d - Chirals Q (cid:48) , ˜ Q (cid:48) , V ± that are surviving edge modes of the corresponding bulk fields- Chirals q, ˜ q from flipping Q, ˜ Q in Theory AThese various 2d multiplets are coupled together by a boundary superpotential (cid:82) dθ + (cid:2) µq ˜ q + q Γ Q (cid:48) (cid:3) , as well as E µ = − Q (cid:48) ˜ Q (cid:48) and E Γ = ˜ q ˜ Q (cid:48) . The V ± are free 2d chirals, but not necessarilyproblematic because they are uncharged under the gauge symmetry. U ( N ) k + N − Nf ↔ U ( k ) − k − N + Nf with N f fundamentals It is not necessary to integrate out all the matter when flowing from Aharony duality. Hereand in Section 9.4 we discuss two particularly well-behaved scenarios in which some 3d chiralmatter remains.Consider the basic bulk Aharony duality of Section 8, and write N c = N , N f = k + N , N (cid:48) c = k (9.7)as before. Suppose we want integrate out all the anti-fundamental chirals ˜ Q d in TheoryA, leaving behind fundamentals Q d . This can be done by first redefining axial charge inTheory A by adding − U (1) part of the gauge group, and by N f times the topological charge. Then introducing a large positive (say) axial mass gives˜ Q d a positive mass, keeps Q d massless, and (thanks to the mixing with topological charge)maintains a vanishing effective FI parameter. The bulk Theory A flows toTheory A: U ( N ) Nf = U ( N ) k + N − Nf with N f fundamental chirals Q d , (9.8)and the flow generates bulk Chern-Simons terms for gauge and global symmetries encoded inthe polynomial I A bulk = N f Tr( s ) + (Tr s ) (cid:2) y + N f r (cid:3) + NN f r . (9.9)By inspecting effective Chern-Simons levels on the Coulomb branch of the theory, one alsofinds that there is a single scalar, gauge-invariant monopole remaining.– 81 –e may track the effect of the same deformation in Theory B. We similarly start byredefining axial charge by adding N f times topological charge and (now) +1 times U (1) (cid:48) gauge charge. Then introducing a large positive axial mass gives the ˜ Q (cid:48) d anti-fundamentalchirals and the V − d singlet a negative real mass, gives all the singlets M d a positive mass,and keeps Q (cid:48) d and V +3 d massless. The bulk theory flows toTheory B: U ( k ) − Nf = U ( k ) − k − N + Nf with N f fund’s Q (cid:48) d and a singlet V +3 d , (9.10)with bulk Chern-Simons terms I B bulk = − N f Tr( s (cid:48) ) + 2(Tr s (cid:48) ) y + k r y − N r − y . (9.11)We summarize the charges of the remaining matter fields in Theories A and B: U ( N ) U ( k ) SU ( N f ) U (1) y U (1) R Q N 1 N f Q (cid:48) f V + ± k + 1 (9.12)These bulk flows are nicely compatible with the first pair of dual boundary conditionsin Aharony duality (8.4), namely ( D ,D,D) in Theory A and ( N ,N,N,D,N)+Γ , η in TheoryB. These boundary conditions ensure that no edge modes arise as some of the bulk fieldsbecome massive, with positive axial mass as above. The boundary Fermi multiplets Γ , η areuntouched by the flow, and continue to have charges U ( N ) U ( k ) SU ( N f ) U (1) y U (1) R Γ N k 1 η det det − D , Q D) ↔ ( N , Q (cid:48) N , V + N ) + Γ , η . (9.14)Similarly, we obtain a duality interface
Theory B ( N , N , N) (cid:12)(cid:12) Γ , η (cid:12)(cid:12) Theory A ( N , N) (9.15)with superpotential (cid:82) dθ + Γ QQ (cid:48) .Note that the boundary anomalies in the dual pair (9.14) are guaranteed to match byconsistency of the flow. Indeed, the usual computation of anomalies gives I ( D , D) = I A bulk − N c Tr( s ) + (Tr s ) − N c r + (cid:2) N c Tr( x ) + N f Tr( s ) + 2 N f (Tr s )( − r ) + N c N f ( − r ) (cid:3) = I Aharony( D , D , D) , (9.16) I ( N , N , N) = I A bulk + k Tr( s (cid:48) ) − (Tr s (cid:48) ) + k r − (cid:2) k Tr( x ) + N f Tr( s (cid:48) ) (cid:3) − ( y + k (cid:48) r ) = I Aharony( N ,N,N,D,N) , (9.17)– 82 –s required for anomaly matching in the RG flows. Therefore, (8.11) implies the desired I ( D , D) = I ( N , N , N) + I Γ + I η . (9.18)The equality of half-indices corresponding to the pair of b.c. (9.14) also follows fromequality of the half-indices in Aharony duality (8.13), (8.14), by applying a limit to fugacitiesthat mimics the limit of real masses in the flow. Specifically, we should redefine s i → s i /a , z i → az i , y → a N f y (9.19)and then send a → ∞ . Conveniently, the prefactor in the II ( D , D,D) index (that depends onthe boundary anomaly) remains independent of a , thanks to our careful mixing of axial andtopological symmetries. Moreover, a only appears in the arguments of quantum dilogarithmswith negative exponents, thanks to our careful alignment of N vs. D b.c. and axial charges.Thus the a → ∞ is well defined, and simply sends every quantum dilogarithm containing anegative power of a to ‘1’. This just means that the contributions of all massive bulk fieldsdisappear. Assuming equality of the half-indices (8.13), (8.14), we derive an equality II ( D ,D) = 1( q ) N ∞ (cid:88) m ∈ Z N q (cid:2) km · m +( (cid:80) i m i ) (cid:3) s km (cid:0) y (cid:81) i s i (cid:1) (cid:80) i m i (cid:81) i (cid:54) = j ( q m i − m j s i /s j ; q ) ∞ (cid:89) ≤ i ≤ N ≤ α ≤ Nf II D ( q m i s i x α ; q ) (9.20)= ( q ) k ∞ k ! (cid:73) k (cid:89) i =1 dz i πiz i (cid:89) i (cid:54) = j ( qz i /z j ; q ) ∞ (cid:89) ≤ i ≤ k ≤ α ≤ Nf II N (cid:16) − q z i x α ; q (cid:17) F (cid:16) − q y (cid:81) i s i (cid:81) j z j ; q (cid:17) (cid:89) ≤ i ≤ N ≤ j ≤ k F( − q s i z j ; q )= II ( N ,N,N)+Γ ,η . Finally, note that we could equally well have triggered the above flows with a largenegative axial mass. This leads toTheory A : U ( N ) − k − N + N f with N f fundamentals Q d Theory B : U ( N ) k + N − N f with N f fundamentals Q (cid:48) d + a singlet V − d (9.21)Now the Aharony-dual boundary conditions that avoid edge modes are ( N ,N,N)+Γ (cid:48) , η (cid:48) and( D ,D,D,N,D), which flow to ( N , Q N) + Γ (cid:48) , η (cid:48) ↔ ( D , Q (cid:48) D , V + D ) , (9.22)and the opposite interface Theory A ( N , N) (cid:12)(cid:12) Γ (cid:48) , η (cid:48) (cid:12)(cid:12) Theory B ( N , N , N). SU ( N ) k + N − Nf ↔ U ( k ) − k − N + Nf , − k + Nf with N f fundamentals Just as in level-rank duality, one may obtain a U-SU duality with matter from U-U bygauging the topological U (1) y symmetry. If we start from the duality with fundamental At the level of the full superconformal index, such gauging procedures for Seiberg-like dual pairs werediscussed in [152]. – 83 –atter from Section 9.2, gauging the topological symmetry results in a SU ( N ) k + N − Nf ↔ U ( k ) − k − N + Nf , − k + Nf duality in the bulk, with N f fundamental chirals on each side.On the boundary, the baryonic Fermi multiplet η is lost by the same mechanism as inSection 7.4. The N fundamental Fermi multiplets Γ coupled to Neumann b.c. are sufficientto cancel gauge anomalies. We expect, for example, that SU ( N ) k + N − Nf w/ ( D ,D) ↔ U ( k ) − k − N + Nf , − k + Nf w/ ( N , N) + Γ (9.23)A full match of ’t Hooft anomalies proceeds exactly as in U-SU level-rank duality with theadditional fundamental contributions.The half-indexes are II ( D ,D) = 1( q ) N − ∞ (cid:88) m ∈ Z N , (cid:80) i m i =0 q (cid:2) km · m (cid:3) s km (cid:81) α ( q m.α s α ; q ) ∞ (cid:89) ≤ i ≤ N ≤ α ≤ Nf II D ( q m i s i x α ; q ) (9.24)= ( q ) k ∞ k ! (cid:73) k (cid:89) i =1 dz i πiz i (cid:89) i (cid:54) = j ( qz i /z j ; q ) ∞ (cid:89) ≤ i ≤ k ≤ α ≤ Nf II N (cid:16) − q z i x α ; q (cid:17) (cid:89) ≤ i ≤ N ≤ j ≤ k F( − q s i z j ; q )= II ( N ,N)+Γ . where now the s fugacities parameterize the torus of SU ( N ) and m are vectors in the cochar-acter lattice of SU ( N ).Let us present a simple non-abelian example. On the Neumann side we consider U (2) − / , − / with one fundamental and three boundary Fermi multiplets, and on the Dirichlet side we con-sider its dual SU (3) / with one fundamental.The Neumann side of the half-index is II ( N ,N ) = ( q ) ∞ (cid:73) (cid:89) i =1 dz i πiz i (cid:89) i (cid:54) = ji,j =1 ( z i z − j ; q ) ∞ (cid:81) i =1 F ( − q / z i s ) F ( − q / z i s /s ) F ( − q / z i /s ) (cid:81) i =1 ( − q / a − z i ; q ) ∞ while the Dirichlet side is II ( D ,D ) = 1( q ) ∞ (cid:88) m ,m ∈ Z q m − m m +2 m s m − m s m − m ( q m − m s /s ) ∞ ( q − m − m /s s ) ∞ ( q m − m s /s ) ∞ × II D (( a/q ) q − m s ) II D (( a/q ) q − − m /s ) II D (( a/q ) q − − m + m s /s )( q − m + m s /s ) ∞ ( q m + m s s ) ∞ ( q m +2 m +2 s /s ) ∞ where, as in other examples, we have shifted the fugacity for the axial symmetry a → a/q forconvenience. One can check that these expressions match order by order in q . The expansionmay be written compactly in terms of SU (3) characters II T , ( N ,N ) = II T , ( D ,D ) = − χ (0 , ( c , c ) − qχ (1 , ( c , c )+ q (cid:2) − χ (0 , ( c , c ) + 1 a χ (0 , ( c , c ) − χ (2 , ( c , c ) − χ (1 , ( c , c ) (cid:3) + . . . – 84 –here χ ( n ,n ) ( t , t ) = 1 (cid:0) t − t (cid:1) ( t t − (cid:0) t − t (cid:1) (cid:104) − t t n +11 (cid:18) t t (cid:19) n + n + t t n + n +31 (cid:18) t t (cid:19) n − (cid:18) t (cid:19) n − t n + n +41 + t n +31 (cid:18) t (cid:19) n + n + (cid:18) t (cid:19) n − (cid:18) t t (cid:19) n + n − t (cid:18) t (cid:19) n + n − (cid:18) t t (cid:19) n (cid:105) . U ( N ) k + N − nf + na ↔ U ( k ) − k − N + nf + na with n f fundamentals, n a anti-fundamentals Generalizing the previous example, we may partially integrate out some fundamentals andantifundamentals from Aharony duality. If we wish to avoid edge modes, we may use (say)the ( D ,D,D) ↔ ( N ,N,N,D,N)+Γ , η duality of boundary conditions, and make sure that bothfundamentals and antifundamentals are integrated out with positive mass in Theory A, andnegative mass in Theory B.Suppose that we wish to integrate out N f − n f fundamentals (leaving n f behind) and N f − n a anti-fundamentals (leaving n a behind), all with positive mass in Theory A. Following[14], this can be accomplished by breaking the flavor symmetry SU ( N f ) → SU ( n f ) × SU ( N f − n f ) × U (1) f and (cid:94) SU ( N f ) → (cid:94) SU ( n a ) × (cid:94) SU ( N f − n a ) × (cid:94) U (1) f , and, roughly speaking, using acombination of axial, U (1) f , and (cid:94) U (1) f masses to give the last N f − n f fundamentals ( N f − n a anti-fundamentals) a positive mass while keeping the rest massless. As long as n a , n f < N f ,no scalar, gauge-invariant monopole operators remain.The corresponding transformation in Theory B gives the last N f − n f fundamentals( N f − n a anti-fundamentals) a negative mass. It also gives some of the mesons a positivemass, and (due to mixing with the topological symmetry) it turns out to give the V ± singletsa negative mass so long as n a , n f < N f . This aligns perfectly with the (N,N,D,N) b.c. inTheory B, so no edge modes are generated.After integrating out all massive fields in the bulk, we arrive atTheory A: U ( N ) k + N − nf + na with n f fund’s Q d , n a anti-fund’s ˜ Q d Theory B: U ( k ) − k − N + nf + na with n f fund’s Q (cid:48) d , n a anti-fund’s ˜ Q (cid:48) d , n f × n a singlets M d (9.25)with a superpotential W = M d ˜ Q d ˜ Q (cid:48) d in Theory B, and gauge and flavor charges U ( N ) U ( k ) SU ( n f ) (cid:94) SU ( n a ) U (1) a U (1) y U (1) R Q N 1 n f Q N 1 1 n a Q (cid:48) f − Q (cid:48) a − M f n a I A bulk = (cid:0) k + N − n f + n a (cid:1) Tr( s ) + (Tr s ) (cid:2) y + ( n a − n f )( a − r ) (cid:3) (9.27) I B bulk = − (cid:0) k + N − n f + n a (cid:1) Tr( s (cid:48) ) + (Tr s (cid:48) ) (cid:2) y + ( n a − n f ) a (cid:3) (9.28)+ ( k + N − n a )Tr x + ( k + N − n f )Tr ˜ x + (cid:2) ( n a + n f )( k + N ) − n a n f (cid:3) a + (cid:2) − ( n a + n f ) N + 2 n a n f (cid:3) a r − (cid:2) k + ( N − n a )( N − n f ) (cid:3) r − y The Aharony-dual boundary conditions ( D ,D,D) ↔ ( N ,N,N,D,N)+Γ , η now flow to( D , Q D , ˜ Q D) ↔ ( N , Q (cid:48) N , ˜ Q (cid:48) N , M D) + Γ , η , (9.29)with the Fermi multiplets Γ , η remaining untouched — they have the same bifundamental andbi-det charges as in (8.5). Anomaly matching along the RG flow on either side ensures thatboundary anomalies will still match perfectly; and an identity of half-indices may be derivedfrom the Aharony half-indices (8.13)–(8.14) by sending appropriate fugacities to infinity, justas in (9.20). Finally, from (9.29) (or by flowing with the Aharony interface (8.6)), we obtainthe duality interface
Theory B ( N , N,N,D) (cid:12)(cid:12) Γ , η (cid:12)(cid:12) Theory A ( N , N,N) (9.30)with a superpotential (cid:82) dθ + (cid:2) Γ QQ (cid:48) + µQ ˜ Q (cid:3) and E Γ = ˜ Q ˜ Q (cid:48) , so as to factorize the bulk super-potential M ˜ Q ˜ Q (cid:48) .Alternatively, we could have used real masses of the opposite signs, and the Aharony-dual boundary conditions ( N ,N,N)+Γ (cid:48) , η (cid:48) ↔ ( D ,D,D,N,D), to obtain dual boundary con-ditions for the bulk theories with opposite Chern-Simons levels, and an opposite interface( N , N,N) (cid:12)(cid:12) Γ (cid:48) , η (cid:48) (cid:12)(cid:12) ( N , N,N,D).
There are many more Seiberg-like dualities in the bulk that flow from Aharony duality byintegrating out chirals with a combination of positive and negative masses. This generatesbulk Chern-Simons levels that are smaller than those in (9.25). In the presence of our usualboundary conditions, such a flow will necessarily produce edge modes, which must be incor-porated into a duality of boundary conditions in the way that was outlined at the end ofSection 9.1. This leads to boundary conditions that appear rather complicated, though inprincipal their analysis is systematic.
10 Adjoint matter
We conclude with a simple example of a gauge theory coupled to adjoint matter: SU (2) k plus a single adjoint chiral multiplet Φ . If the Chern-Simons level is equal to ±
1, then To obtain this form, we shifted the global CS levels for both theories A and B by a constant amount. – 86 –ccording to the “duality appetizer” of Jafferis and Yin [66] this theory is dual to a single free N = 2 chiral multiplet. We consider the following duality amuse-bouche by finding simpledual boundary conditions.We assume that the chiral Φ has R-charge zero, and charge +1 for a U (1) x flavorsymmetry. We also set all the background CS levels in the bulk to zero.First suppose that the dynamical Chern-Simons level is non-negative, k ≥
0, so thatDirichlet b.c. are well behaved. Taking D b.c. for the gauge multiplet and D b.c. for thechiral, the anomaly polynomial is I D , D = k Tr( f ) − (cid:2) f ) + r (cid:3) + (cid:2) f ) + ( x − r ) (cid:3) = k Tr( f ) + x − xr . (10.1)The corresponding half-index is II D , D ( x, y ; q ) = 1( q ) ∞ II D ( x ; q ) (cid:88) m ∈ Z q km y km II D ( q m xy ; q ) II D ( q − m xy − ; q )( q m y ; q ) ∞ ( q − m y − ; q ) ∞ , (10.2)and converges nicely as long as k ≥
0. (A substitution x → x/q aids computations). Here x is the fugacity for the bulk U (1) x flavor symmetry and ( y, y − ) are the fugacities for the SU (2) ∂ boundary flavor symmetry.For k = 1 we find II k =1 D , D ( x, y ; q )= II D ( x ; q ) χ [ SU (2) ]( y ) , (10.3) i.e. the index of D b.c. for a free chiral Tr(Φ ) (as predicted by the duality appetizer),together with a decoupled SU (2) WZW model on the boundary. The WZW model carriesthe Tr( f ) boundary ’t Hooft anomaly. In order to get a perfect match of boundary anomalies,the RHS also requires background CS terms − ( x + r ) in the bulk.Amusingly, at k = 0 we also find II k =0 D , D ( x, y ; q ) = II D ( x ; q ) II N ( x − ; q ) , (10.4)suggesting that the bulk theory is dual to two free chirals of U (1) x charge +2 , −
1, with N andD b.c., respectively. Indeed, we can identify the +2 chiral with Tr(Φ ), and the − I D + I N = (2 x − r ) − ( x + r ) = x − xr .We may similarly consider Neumann b.c., assuming that the Chern-Simons level is non-positive, k ≤
0. At k = 0, the anomaly polynomial of N b.c. on the gauge multiplet and Nb.c. on the adjoint chiral is I N , N k =0 = − x + 3 xr . (10.5)Happily, there is no gauge anomaly to cancel. We suspect by comparison to (10.4) that inthe IR we should find two chirals of U (1) x charges +1 , −
1, with N and D b.c., respectively.Indeed, the half-index is II k =0 N , N ( x ; q ) = ( q ) ∞ (cid:73) ds πis ( s ; q ) ∞ ( s − ; q ) ∞ ( x ; q ) ∞ ( xs ; q ) ∞ ( xs − ; q ) ∞ = II N ( x ; q ) II D ( x − ; q ) . (10.6)– 87 –t k = −
1, the anomaly polynomial is I N , N k = − = − Tr( f ) − x + 3 xr , (10.7)and a gauge anomaly must be cancelled. This can be done by adding two boundary Fermimultiplets Γ , Γ of R-charge zero, transforming as a doublet of SU (2). These Fermi multipletsalso transform with charge +1 (say) under a boundary U (1) z flavor symmetry. The totalboundary anomaly becomes I N , N + I Γ = − Tr( f ) − x + 3 xr + (cid:2) Tr( f ) + 2 z (cid:3) = 2 z − x + 3 xr . (10.8)We find that the corresponding half-index at k = − II k = − N , N+Γ ( x, z ; q ) = ( q ) ∞ (cid:73) ds πis ( s ; q ) ∞ ( s − ; q ) ∞ ( x ; q ) ∞ ( xs ; q ) ∞ ( xs − ; q ) ∞ F( − q sz ; q )F( − q s − z ; q )= II N ( x ; q ) χ [ SU (2) ]( z ) . (10.9)This shows the expected Tr(Φ ) chiral with N b.c., in addition to — surprisingly — a decou-pled SU (2) WZW model. The WZW model indicates an enhancement of the U (1) z boundaryflavor symmetry to SU (2) z , and carries the boundary ’t Hooft anomaly 2 z = Tr (cid:0) z − z (cid:1) . Acknowledgements
We would like to thank Kevin Costello, Sergei Gukov, Pavel Putrov, and Nathan Seiberg forvaluable discussions and inspiration. T.D. is partially supported by ERC Starting Grant no.335739 “Quantum fields and knot homologies,” funded by the European Research Councilunder the European Union’s Seventh Framework Programme. T.D. and N.P. would liketo thank the Perimeter Institute for Theoretical Physics for hospitality and support duringsome phases of this project. The work of D.G. was supported by the Perimeter Institute forTheoretical Physics. Research at the Perimeter Institute is supported by the Government ofCanada through Industry Canada and by the Province of Ontario through the Ministry ofEconomic Development & Innovation. N.P. is supported by a Sherman Fairchild PostdoctoralFellowship. This material is based upon work supported by the U.S. Department of Energy,Office of Science, Office of High Energy Physics, under Award Number DE-SC0011632.
A Fermionic T-duality
There is a well known symmetry in 2d N = (0 ,
2) GLSM’s that exchanges a Fermi multipletand its conjugate Γ ↔ Γ † , while at the same time swapping its E and J terms E ↔ J . Moreprecisely, this is a duality of off-shell Fermi superfields that replaces a Fermi superfield Γconstrained to obey D + Γ = E and entering in a superpotential coupling (cid:82) dθ + Γ J (for somespecified functions E, J of the chirals) with another Fermi superfield Λ constrained to obey D + Λ = J and a coupling (cid:82) dθ + Λ E . Going on shell, these superfields are related byΛ = Γ † . (A.1)– 88 –e show here that this can be viewed as a fermionic analogue of T-duality. We thank N.Seiberg for suggesting this perspective to us.We begin by considering a 2d N = (2 ,
2) theory with a free, periodic chiral multipletΦ, satisfying D ± Φ = 0. With respect to a N = (0 ,
2) subalgebra, the multiplet splits into achiral and a fermi C := Φ (cid:12)(cid:12) θ − =¯ θ − =0 , Γ := D − Φ (cid:12)(cid:12) θ − =¯ θ − =0 . (A.2)The T-dual of Φ is a twisted-chiral multiplet ˜Φ, which also splits into a N = (0 ,
2) chiral anda fermi ˜ C := ˜Φ (cid:12)(cid:12) θ − =¯ θ − =0 , Λ := D − ˜Φ (cid:12)(cid:12) θ − =¯ θ − =0 . (A.3)Note that Γ and Λ are ordinary (0,2) Fermi multiplets, both with E = J = 0, which mustcapture the same degrees of freedom. Since the (2,2) chiral and its T-dual are related on shellby ( cf. [142]) Φ + Φ † = ˜Φ + ˜Φ † , (A.4)we immediately see that, on shell, D − (Φ + Φ † ) (cid:12)(cid:12) θ − =¯ θ − =0 = D − ( ˜Φ + ˜Φ † ) (cid:12)(cid:12) θ − =¯ θ − =0 ⇒ Γ = Λ † , (A.5)which is the same as (A.1).To incorporate general E and J terms, we work directly in an interacting 2d N = (0 , N = (0 ,
2) theory with a distinguished Fermi multiplet Γ,which is constrained to satisfy D + Γ = E Γ ( C ) (a function of the chiral multiplets) and hasa superpotential interaction (cid:82) dθ + Γ J Γ ( C ). We assume that the kinetic term for Γ has beendiagonalized, so that the part of the superspace Lagrangian involving Γ takes the form L (Γ) = (cid:90) dθ + d ¯ θ + h ( C, C † , ... )ΓΓ † + (cid:90) dθ + Γ J Γ + c.c. (A.6)Note that supersymmetry requires the total E · J to vanish (or more generally to be constant,see Footnote 5), but as long as there are other Fermi multiplets in the theory we may assumethat E Γ and J Γ are totally generic.We can rewrite the action in terms of a general, unconstrained superfield Γ by introducinga second Fermi multiplet Λ to act as a Lagrange multiplier: L (Γ , Λ) = (cid:90) dθ + d ¯ θ + h ( C, C † , ... )ΓΓ † + (cid:90) dθ + (cid:2) Γ J Γ − Λ( D + Γ − E Γ ) (cid:3) + c.c. (A.7)A priori, D + Λ could be arbitrary (the chiral constraint D + Γ = E Γ is implemented regardless);but new superpotential term preserves SUSY only if the total ‘ E · J ’ is unchanged (up to aconstant). The new effective contribution to E · J , replacing E Γ J Γ , is D + (cid:2) Γ J Γ − Λ( D + Γ − E Γ ) (cid:3) = D + Γ J Γ − D + Λ D + Γ + D + Λ E Γ . (A.8) We assume this only for clarity of presentation. The duality transformation below can be generalized instraightforward way to theories with a general Fermi kinetic term. – 89 –ather nicely, this precisely equals E Γ J Γ if we choose D + Λ = J Γ . With this choice, we mayalso conveniently rewrite the superpotential asΓ J Γ − Λ( D + Γ − E Γ ) = − D + (ΓΛ) + Λ E Γ , (A.9)so that the Lagrangian takes the form L (Γ , Λ) = (cid:90) dθ + d ¯ θ + (cid:2) h ΓΓ † − ΓΛ − Λ † Γ † (cid:3) + (cid:90) dθ + Λ E Γ + c.c. (A.10)Finally, we integrate out the unconstrained superfield Γ to obtain a dual Lagrangian in termsof Λ, L (Λ) = (cid:90) dθ + d ¯ θ + h ΛΛ † + (cid:90) dθ + Λ E Γ + c.c. (A.11)As desired, the E and J terms have been swapped. More interestingly, the metric has alsobeen inverted. It is also easy to see from (A.10) that, on shell, h Γ † = Λ . (A.12)Note that in a gauge theory, if Γ has charge q under an abelian (say) gauge symmetry,the effective metric in the Lagrangian looks like h ∼ e qA + . Its inverse in the Λ kinetic termis 1 /h ∼ e − qA + , consistent with the fact that Γ and Λ must have opposite charges. B Character of the Vacuum Module
In this section, we show that equation (7.16) is equivalent to the standard Weyl-Kac characterformula for the vacuum module.The standard Weyl-Kac character formula for an integrable representation of an affineKac-Moody algebra of highest weight ˆ λ is ch ˆ λ [ G k ] = (cid:80) w ∈ ˆ W (cid:15) ( w ) e w (ˆ λ +ˆ ρ ) e ˆ ρ (cid:81) ˆ α> (1 − e − ˆ α ) mult (ˆ α ) (B.1)where we can write the affine weight and Weyl vectors in terms of their finite counterpartsas ˆ λ = ( λ, k, , ˆ ρ = ( ρ, h, . In particular, the highest weight vector corresponding tothe vacuum module is λ = (0 , k, r with all components equal to zero. In general, a hatted quantity denotes its affinecounterpart. It is convenient to rewrite the ‘normalized character’ following [148] as χ ˆ λ [ G k ] = q s (ˆ λ ) ch ˆ λ = (cid:80) w ∈ W (cid:15) ( w )Θ w (ˆ λ +ˆ ρ ) (cid:80) w ∈ W (cid:15) ( w )Θ w ˆ ρ = (cid:80) w ∈ W (cid:15) ( w )Θ w (ˆ λ +ˆ ρ ) q | ρ | h e ˆ ρ (cid:81) ˆ α> (1 − e − ˆ α ) mult (ˆ α ) (B.2) The third entry of the vector, called the grade of the representation, is zero for an integrable highestweight. – 90 –here the theta functions, defined below, in the numerator and denominator together includethe overall factor of the modular anomaly, see footnote 28. Notice that the sum in thenumerator is now over the finite Weyl group. The theta functions areΘ ˆ λ ( q, x ) = (cid:88) α ∨ ∈ Q ∨ e ( λ + kα ∨ ,k, | λ | / k −| λ + kα ∨ | / k ) =: (cid:88) m ∈ Q ∨ x km x λ q k | m + λ/k | = (cid:88) m ∈ Q ∨ x km x λ q k ( m,m ) q ( m,λ ) q | λ | / k (B.3)with the sum taken over the coroot lattice Q ∨ .We can use this to rewrite the numerator of the Weyl-Kac formula for the vacuum moduleas q −| ρ | h e − ˆ ρ (cid:88) w ∈ W (cid:15) ( w )Θ w (ˆ λ vac +ˆ ρ ) = q −| ρ | h x − ρ (cid:88) w ∈ W (cid:15) ( w )Θ ( wρ,k + h, = q −| ρ | h x − ρ (cid:88) w ∈ W (cid:88) m ∈ Q ∨ (cid:15) ( w ) x wρ q ( m,wρ ) x ( k + h ) m q ( k + h )( m,m ) q | ρ | / ( k + h ) = q −| ρ | h q | ρ | / ( k + h ) (cid:88) m ∈ Q ∨ q ( k + h ) | m | x ( k + h ) m q ( m,ρ ) (cid:88) w ∈ W (cid:15) ( w ) x wρ − ρ q ( m,wρ − ρ ) = q s (ˆ λ vac ) (cid:88) m ∈ Q ∨ q ( k + h ) | m | x ( k + h ) m q ( m,ρ ) (cid:89) α ∈ Φ + (1 − q m.α x α )= q s (ˆ λ vac ) (cid:88) m ∈ Q ∨ q ( k + h ) | m | x ( k + h ) m (cid:89) α ∈ Φ + ( − q / ) − m.α (1 − q m.α x α )In the second line we used the fact that Weyl translates of ρ have the same length and inthe second-to-last line we have used the Weyl denominator formula: (cid:80) w ∈ W (cid:15) ( w ) e w ( ρ ) − ρ = (cid:81) α ∈ Φ + (1 − e − α ). Finally, in the last line, we used the identity (cid:81) α ∈ Φ + ( q / ) − m.α = q ( m,ρ ) (after accounting for our particular sign conventions). The extra factor of ( − m.α is trivialin the case of G simply connected, since m.ρ ∈ Z , m ∈ Λ ∨ = Q ∨ so that ( − m.α = 1.This is precisely the numerator of equation (7.16)—except for the modular characteristic q | ρ | / ( k + h ) − | ρ | /h which we drop—if the group G is simply connected and hence the corootlattice can be identified with the cocharacter lattice.Manipulating the denominator formula of the affine Weyl-Kac character formula is eveneasier: (cid:89) ˆ α> (1 − e − ˆ α ) mult (ˆ α ) = ( q ) r ∞ (cid:89) α ∈ Φ + ( qx α ; q ) ∞ ( qx − α ; q ) ∞ (1 − x α )where we recall that the set of positive affine roots ˆ α = ( α, n,
0) includes roots with positive,negative, and zero α with positive n , as well as roots of the form ˆ α = ( α, ,
0) with α ∈ Φ + .For non-simply connected G our half-index formulae suggest that one should just replacethe coroot lattice with the cocharacter lattice in the definition of our theta function, as wellas honestly include the ( − m factor, which may be nontrivial in general. The former replace-ment indeed occurs in the work of [153], who proposes a non-simply connected generalizationof the Weyl-Kac formula in Theorem 5.5. It would be interesting to precisely match our– 91 –ormula to the literature in the case of simple but not simply connected G (or else refine ourconjecture). References [1] J. de Boer, K. Hori, Y. Oz, and Z. Yin,
Branes and mirror symmetry in N=2 supersymmetricgauge theories in three-dimensions , Nucl. Phys.
B502 (1997) 107–124, [ hep-th/9702154 ].[2] J. de Boer, K. Hori, and Y. Oz,
Dynamics of N=2 supersymmetric gauge theories inthree-dimensions , Nucl. Phys.
B500 (1997) 163–191, [ hep-th/9703100 ].[3] O. Aharony, A. Hanany, K. A. Intriligator, N. Seiberg, and M. J. Strassler,
Aspects of N=2supersymmetric gauge theories in three-dimensions , Nucl. Phys.
B499 (1997) 67–99,[ hep-th/9703110 ].[4] O. Aharony,
IR duality in d = 3 N=2 supersymmetric
U Sp (2 N c ) and U ( N c ) gauge theories , Phys. Lett.
B404 (1997) 71–76, [ hep-th/9703215 ].[5] A. Karch,
Seiberg duality in three-dimensions , Phys. Lett.
B405 (1997) 79–84,[ hep-th/9703172 ].[6] S. Elitzur, A. Giveon, D. Kutasov, E. Rabinovici, and A. Schwimmer,
Brane dynamics andN=1 supersymmetric gauge theory , Nucl. Phys.
B505 (1997) 202–250, [ hep-th/9704104 ].[7] N. Dorey and D. Tong,
Mirror symmetry and toric geometry in three-dimensional gaugetheories , JHEP (2000) 018, [ hep-th/9911094 ].[8] K. A. Intriligator and N. Seiberg, Mirror symmetry in three-dimensional gauge theories , Phys.Lett.
B387 (1996) 513–519, [ hep-th/9607207 ].[9] J. de Boer, K. Hori, H. Ooguri, and Y. Oz,
Mirror symmetry in three-dimensional gaugetheories, quivers and D-branes , Nucl. Phys.
B493 (1997) 101–147, [ hep-th/9611063 ].[10] J. de Boer, K. Hori, H. Ooguri, Y. Oz, and Z. Yin,
Mirror symmetry in three-dimensionaltheories, SL(2,Z) and D-brane moduli spaces , Nucl. Phys.
B493 (1997) 148–176,[ hep-th/9612131 ].[11] A. Giveon and D. Kutasov,
Seiberg Duality in Chern-Simons Theory , Nucl. Phys.
B812 (2009) 1–11, [ arXiv:0808.0360 ].[12] A. Kapustin,
Seiberg-like duality in three dimensions for orthogonal gauge groups , arXiv:1104.0466 .[13] B. Willett and I. Yaakov, N=2 Dualities and Z Extremization in Three Dimensions , arXiv:1104.0487 .[14] F. Benini, C. Closset, and S. Cremonesi, Comments on 3d Seiberg-like dualities , JHEP (2011) 075, [ arXiv:1108.5373 ].[15] O. Aharony, S. S. Razamat, N. Seiberg, and B. Willett,
3d dualities from 4d dualities , JHEP (2013) 149, [ arXiv:1305.3924 ].[16] O. Aharony and D. Fleischer, IR Dualities in General 3d Supersymmetric SU(N) QCDTheories , JHEP (2015) 162, [ arXiv:1411.5475 ]. – 92 –
17] Y. Terashima and M. Yamazaki,
SL(2,R) Chern-Simons, Liouville, and Gauge Theory onDuality Walls , JHEP (2011) 135, [ arXiv:1103.5748 ].[18] T. Dimofte, D. Gaiotto, and S. Gukov, Gauge Theories Labelled by Three-Manifolds , Commun.Math. Phys. (2014) 367–419, [ arXiv:1108.4389 ].[19] S. Cecotti, C. Cordova, and C. Vafa,
Braids, Walls, and Mirrors , arXiv:1110.2115 .[20] T. Dimofte, D. Gaiotto, and R. van der Veen, RG Domain Walls and Hybrid Triangulations , Adv. Theor. Math. Phys. (2015) 137–276, [ arXiv:1304.6721 ].[21] O. Aharony, Baryons, monopoles and dualities in Chern-Simons-matter theories , JHEP (2016) 093, [ arXiv:1512.0016 ].[22] S. Jain, S. Minwalla, and S. Yokoyama, Chern Simons duality with a fundamental boson andfermion , JHEP (2013) 037, [ arXiv:1305.7235 ].[23] G. Gur-Ari and R. Yacoby, Three Dimensional Bosonization From Supersymmetry , JHEP (2015) 013, [ arXiv:1507.0437 ].[24] S. Kachru, M. Mulligan, G. Torroba, and H. Wang, Bosonization and Mirror Symmetry , Phys.Rev.
D94 (2016), no. 8 085009, [ arXiv:1608.0507 ].[25] S. Kachru, M. Mulligan, G. Torroba, and H. Wang,
Nonsupersymmetric dualities from mirrorsymmetry , Phys. Rev. Lett. (2017), no. 1 011602, [ arXiv:1609.0214 ].[26] A. M. Polyakov,
Fermi-Bose Transmutations Induced by Gauge Fields , Mod. Phys. Lett. A3 (1988) 325.[27] M. Barkeshli and J. McGreevy, Continuous transition between fractional quantum Hall andsuperfluid states , Phys. Rev.
B89 (2014), no. 23 235116, [ arXiv:1201.4393 ].[28] S. Giombi, S. Minwalla, S. Prakash, S. P. Trivedi, S. R. Wadia, and X. Yin,
Chern-SimonsTheory with Vector Fermion Matter , Eur. Phys. J.
C72 (2012) 2112, [ arXiv:1110.4386 ].[29] O. Aharony, G. Gur-Ari, and R. Yacoby, d=3 Bosonic Vector Models Coupled toChern-Simons Gauge Theories , JHEP (2012) 037, [ arXiv:1110.4382 ].[30] O. Aharony, G. Gur-Ari, and R. Yacoby, Correlation Functions of Large NChern-Simons-Matter Theories and Bosonization in Three Dimensions , JHEP (2012) 028,[ arXiv:1207.4593 ].[31] J. Maldacena and A. Zhiboedov, Constraining Conformal Field Theories with A Higher SpinSymmetry , J. Phys.
A46 (2013) 214011, [ arXiv:1112.1016 ].[32] J. Maldacena and A. Zhiboedov,
Constraining conformal field theories with a slightly brokenhigher spin symmetry , Class. Quant. Grav. (2013) 104003, [ arXiv:1204.3882 ].[33] N. Seiberg, T. Senthil, C. Wang, and E. Witten, A Duality Web in 2+1 Dimensions andCondensed Matter Physics , Annals Phys. (2016) 395–433, [ arXiv:1606.0198 ].[34] A. Karch and D. Tong,
Particle-Vortex Duality from 3d Bosonization , Phys. Rev. X6 (2016),no. 3 031043, [ arXiv:1606.0189 ].[35] A. Karch, B. Robinson, and D. Tong, More Abelian Dualities in 2+1 Dimensions , JHEP (2017) 017, [ arXiv:1609.0401 ].[36] J. Murugan and H. Nastase, Particle-vortex duality in topological insulators and – 93 – uperconductors , JHEP (2017) 159, [ arXiv:1606.0191 ].[37] P.-S. Hsin and N. Seiberg, Level/rank Duality and Chern-Simons-Matter Theories , JHEP (2016) 095, [ arXiv:1607.0745 ].[38] O. Aharony, F. Benini, P.-S. Hsin, and N. Seiberg, Chern-Simons-matter dualities with SO and U Sp gauge groups , JHEP (2017) 072, [ arXiv:1611.0787 ].[39] F. Benini, P.-S. Hsin, and N. Seiberg, Comments on global symmetries, anomalies, and dualityin (2 + 1)d , JHEP (2017) 135, [ arXiv:1702.0703 ].[40] D. Radicevic, D. Tong, and C. Turner, Non-Abelian 3d Bosonization and Quantum HallStates , JHEP (2016) 067, [ arXiv:1608.0473 ].[41] C. Xu and Y.-Z. You, Self-dual Quantum Electrodynamics as Boundary State of the threedimensional Bosonic Topological Insulator , Phys. Rev.
B92 (2015), no. 22 220416,[ arXiv:1510.0603 ].[42] M. Cheng and C. Xu,
Series of (2+1)-dimensional stable self-dual interacting conformal fieldtheories , Phys. Rev.
B94 (2016), no. 21 214415, [ arXiv:1609.0256 ].[43] D. T. Son,
Is the Composite Fermion a Dirac Particle? , Phys. Rev. X5 (2015), no. 3 031027,[ arXiv:1502.0344 ].[44] C. Wang and T. Senthil, Dual Dirac Liquid on the Surface of the Electron TopologicalInsulator , Phys. Rev. X5 (2015), no. 4 041031, [ arXiv:1505.0514 ].[45] M. A. Metlitski and A. Vishwanath, Particle-vortex duality of two-dimensional Dirac fermionfrom electric-magnetic duality of three-dimensional topological insulators , Phys. Rev.
B93 (2016), no. 24 245151, [ arXiv:1505.0514 ].[46] D. F. Mross, J. Alicea, and O. I. Motrunich,
Explicit derivation of duality between a free Diraccone and quantum electrodynamics in (2+1) dimensions , Phys. Rev. Lett. (2016), no. 1016802, [ arXiv:1510.0845 ].[47] M. E. Peskin,
Mandelstam ’t Hooft Duality in Abelian Lattice Models , Annals Phys. (1978) 122.[48] C. Dasgupta and B. I. Halperin,
Phase Transition in a Lattice Model of Superconductivity , Phys. Rev. Lett. (1981) 1556–1560.[49] M. P. A. Fisher and D. H. Lee, Correspondence between two-dimensional bosons and a bulksuperconductor in a magnetic field , Phys. Rev.
B39 (1989) 2756.[50] A. Gadde, S. Gukov, and P. Putrov,
Walls, Lines, and Spectral Dualities in 3d GaugeTheories , JHEP (2014) 047, [ arXiv:1302.0015 ].[51] A. Gadde, S. Gukov, and P. Putrov, Fivebranes and 4-manifolds , arXiv:1306.4320 .[52] A. Gadde, S. Gukov, and P. Putrov, (0, 2) trialities , JHEP (2014) 076, [ arXiv:1310.0818 ].[53] T. Okazaki and S. Yamaguchi, Supersymmetric boundary conditions in three-dimensional N=2theories , Phys. Rev.
D87 (2013), no. 12 125005, [ arXiv:1302.6593 ].[54] S. Sugishita and S. Terashima,
Exact Results in Supersymmetric Field Theories on Manifoldswith Boundaries , JHEP (2013) 021, [ arXiv:1308.1973 ].[55] Y. Yoshida and K. Sugiyama, Localization of 3d N = 2 Supersymmetric Theories on S × D , – 94 – rXiv:1409.6713 .[56] F. Aprile and V. Niarchos, N =2 supersymmetric field theories on 3-manifolds with A-typeboundaries , JHEP (2016) 126, [ arXiv:1604.0156 ].[57] D. Gaiotto and E. Witten, Janus Configurations, Chern-Simons Couplings, And Thetheta-Angle in N=4 Super Yang-Mills Theory , JHEP (2010) 097, [ arXiv:0804.2907 ].[58] D. Gaiotto and E. Witten, Supersymmetric Boundary Conditions in N=4 Super Yang-MillsTheory , J. Statist. Phys. (2009) 789–855, [ arXiv:0804.2902 ].[59] D. Gaiotto and E. Witten,
S-Duality of Boundary Conditions In N=4 Super Yang-MillsTheory , Adv. Theor. Math. Phys. (2009), no. 3 721–896, [ arXiv:0807.3720 ].[60] M. Bullimore, T. Dimofte, D. Gaiotto, and J. Hilburn, Boundaries, Mirror Symmetry, andSymplectic Duality in 3d N = 4 Gauge Theory , JHEP (2016) 108, [ arXiv:1603.0838 ].[61] S. G. Naculich, H. A. Riggs, and H. J. Schnitzer, Group Level Duality in WZW Models andChern-Simons Theory , Phys. Lett.
B246 (1990) 417–422.[62] E. J. Mlawer, S. G. Naculich, H. A. Riggs, and H. J. Schnitzer,
Group level duality of WZWfusion coefficients and Chern-Simons link observables , Nucl. Phys.
B352 (1991) 863–896.[63] T. Nakanishi and A. Tsuchiya,
Level rank duality of WZW models in conformal field theory , Commun. Math. Phys. (1992) 351–372.[64] M. Camperi, F. Levstein, and G. Zemba,
The Large N Limit of Chern-Simons Gauge Theory , Phys. Lett.
B247 (1990) 549–554.[65] S. G. Naculich and H. J. Schnitzer,
Level-rank duality of the U(N) WZW model,Chern-Simons theory, and 2-D qYM theory , JHEP (2007) 023, [ hep-th/0703089 ].[66] D. Jafferis and X. Yin, A Duality Appetizer , arXiv:1103.5700 .[67] C. Beem, T. Dimofte, and S. Pasquetti, Holomorphic Blocks in Three Dimensions , JHEP (2014) 177, [ arXiv:1211.1986 ].[68] R. Dijkgraaf, L. Hollands, P. Sulkowski, and C. Vafa, Supersymmetric gauge theories,intersecting branes and free fermions , JHEP (2008) 106, [ arXiv:0709.4446 ].[69] A. Armoni and V. Niarchos, Defects in Chern-Simons theory, gauged WZW models on thebrane, and level-rank duality , JHEP (2015) 062, [ arXiv:1505.0291 ].[70] V. Pestun et. al. , Localization techniques in quantum field theories , J. Phys.
A50 (2017),no. 44 440301, [ arXiv:1608.0295 ].[71] N. Drukker, D. Gaiotto, and J. Gomis,
The Virtue of Defects in 4D Gauge Theories and 2DCFTs , JHEP (2011) 025, [ arXiv:1003.1112 ].[72] K. Hosomichi, S. Lee, and J. Park, AGT on the S-duality Wall , JHEP (2010) 079,[ arXiv:1009.0340 ].[73] T. Dimofte, D. Gaiotto, and S. Gukov, , Adv. Theor. Math. Phys. (2013), no. 5 975–1076, [ arXiv:1112.5179 ].[74] D. Gang, E. Koh, and K. Lee, Superconformal Index with Duality Domain Wall , JHEP (2012) 187, [ arXiv:1205.0069 ]. – 95 –
75] A. Kapustin, B. Willett, and I. Yaakov,
Exact Results for Wilson Loops in SuperconformalChern-Simons Theories with Matter , JHEP (2010) 089, [ arXiv:0909.4559 ].[76] A. Kapustin, B. Willett, and I. Yaakov, Nonperturbative Tests of Three-Dimensional Dualities , JHEP (2010) 013, [ arXiv:1003.5694 ].[77] A. Kapustin, B. Willett, and I. Yaakov, Tests of Seiberg-like Duality in Three Dimensions , arXiv:1012.4021 .[78] N. Hama, K. Hosomichi, and S. Lee, Notes on SUSY Gauge Theories on Three-Sphere , JHEP (2011) 127, [ arXiv:1012.3512 ].[79] Y. Imamura and S. Yokoyama, Index for three dimensional superconformal field theories withgeneral R-charge assignments , JHEP (2011) 007, [ arXiv:1101.0557 ].[80] A. Kapustin and B. Willett, Generalized Superconformal Index for Three Dimensional FieldTheories , arXiv:1106.2484 .[81] M. Aganagic, K. Costello, J. McNamara, and C. Vafa, Topological Chern-Simons/MatterTheories , arXiv:1706.0997 .[82] E. Witten, Mirror manifolds and topological field theory , hep-th/9112056 . [AMS/IP Stud.Adv. Math.9,121(1998)].[83] E. Witten, Phases of N=2 theories in two-dimensions , Nucl. Phys.
B403 (1993) 159–222,[ hep-th/9301042 ]. [AMS/IP Stud. Adv. Math.1,143(1996)].[84] A. N. Schellekens and N. P. Warner,
Anomalies and Modular Invariance in String Theory , Phys. Lett.
B177 (1986) 317–323.[85] A. N. Schellekens and N. P. Warner,
Anomalies, Characters and Strings , Nucl. Phys.
B287 (1987) 317.[86] E. Witten,
Elliptic Genera and Quantum Field Theory , Commun. Math. Phys. (1987) 525.[87] A. Gadde and S. Gukov,
2d Index and Surface operators , JHEP (2014) 080,[ arXiv:1305.0266 ].[88] F. Benini, R. Eager, K. Hori, and Y. Tachikawa, Elliptic genera of two-dimensional N=2 gaugetheories with rank-one gauge groups , Lett. Math. Phys. (2014) 465–493,[ arXiv:1305.0533 ].[89] F. Benini, R. Eager, K. Hori, and Y. Tachikawa,
Elliptic Genera of 2d N = 2 Gauge Theories , Commun. Math. Phys. (2015), no. 3 1241–1286, [ arXiv:1308.4896 ].[90] J. Kinney, J. M. Maldacena, S. Minwalla, and S. Raju,
An Index for 4 dimensional superconformal theories , Commun. Math. Phys. (2007) 209–254, [ hep-th/0510251 ].[91] S. Kim,
The Complete superconformal index for N=6 Chern-Simons theory , Nucl. Phys.
B821 (2009) 241–284, [ arXiv:0903.4172 ]. [Erratum: Nucl. Phys.B864,884(2012)].[92] N. Hama, K. Hosomichi, and S. Lee,
SUSY Gauge Theories on Squashed Three-Spheres , JHEP (2011) 014, [ arXiv:1102.4716 ].[93] S. Cecotti, D. Gaiotto, and C. Vafa, tt ∗ geometry in 3 and 4 dimensions , JHEP (2014) 055,[ arXiv:1312.1008 ].[94] L. F. Alday, D. Martelli, P. Richmond, and J. Sparks, Localization on Three-Manifolds , JHEP – 96 – (2013) 095, [ arXiv:1307.6848 ].[95] F. Benini and W. Peelaers, Higgs branch localization in three dimensions , JHEP (2014)030, [ arXiv:1312.6078 ].[96] M. Fujitsuka, M. Honda, and Y. Yoshida, Higgs branch localization of 3d N=2 theories , PTEP (2014), no. 12 123B02, [ arXiv:1312.3627 ].[97] V. Pestun,
Localization of gauge theory on a four-sphere and supersymmetric Wilson loops , Commun. Math. Phys. (2012) 71–129, [ arXiv:0712.2824 ].[98] S. Pasquetti,
Factorisation of N = 2 Theories on the Squashed 3-Sphere , JHEP (2012) 120,[ arXiv:1111.6905 ].[99] C. Hwang, H.-C. Kim, and J. Park, Factorization of the 3d superconformal index , JHEP (2014) 018, [ arXiv:1211.6023 ].[100] M. Taki, Holomorphic Blocks for 3d Non-abelian Partition Functions , arXiv:1303.5915 .[101] C. Hwang, P. Yi, and Y. Yoshida, Fundamental Vortices, Wall-Crossing, and Particle-VortexDuality , JHEP (2017) 099, [ arXiv:1703.0021 ].[102] M. Aganagic and A. Okounkov, Elliptic stable envelope , arXiv:1604.0042 .[103] S. Gukov, D. Pei, P. Putrov, and C. Vafa, BPS spectra and 3-manifold invariants , arXiv:1701.0656 .[104] D. Gaiotto, Boundaries, interfaces and dualities , talk at Natifest, Princeton (September 2016).[105] K. Aitken, A. Baumgartner, A. Karch, and B. Robinson,
3d Abelian Dualities withBoundaries , arXiv:1712.0280 .[106] H.-J. Chung and T. Okazaki, (2,2) and (0,4) Supersymmetric Boundary Conditions in 3d N =4 Theories and Type IIB Branes , arXiv:1608.0536 .[107] D. Gaiotto, Twisted compactifications of 3d N = 4 theories and conformal blocks , arXiv:1611.0152 .[108] K. Costello and D. Gaiotto, To appear , .[109] T. Dimofte, M. Gabella, and A. B. Goncharov,
K-Decompositions and 3d Gauge Theories , JHEP (2016) 151, [ arXiv:1301.0192 ].[110] A. Gadde, S. Gukov, and P. Putrov, Duality Defects , arXiv:1404.2929 .[111] D. Gaiotto, G. W. Moore, and A. Neitzke, Wall-Crossing in Coupled 2d-4d Systems , JHEP (2012) 082, [ arXiv:1103.2598 ].[112] D. Gaiotto, S. Gukov, and N. Seiberg, Surface Defects and Resolvents , JHEP (2013) 070,[ arXiv:1307.2578 ].[113] D. Gaiotto and H.-C. Kim, Surface defects and instanton partition functions , JHEP (2016)012, [ arXiv:1412.2781 ].[114] M. Aganagic, K. Hori, A. Karch, and D. Tong, Mirror symmetry in (2+1)-dimensions and(1+1)-dimensions , JHEP (2001) 022, [ hep-th/0105075 ].[115] O. Aharony, S. S. Razamat, and B. Willett, From 3d duality to 2d duality , arXiv:1710.0092 .[116] D. Honda and T. Okuda, Exact results for boundaries and domain walls in 2d supersymmetric – 97 – heories , JHEP (2015) 140, [ arXiv:1308.2217 ].[117] K. Hori and M. Romo, Exact Results In Two-Dimensional (2,2) Supersymmetric GaugeTheories With Boundary , arXiv:1308.2438 .[118] J. Wess and J. Bagger, Supersymmetry and Supergravity . Princeton, USA: Univ. Pr. (1992)259 p, 1992.[119] E. Witten,
SL(2,Z) action on three-dimensional conformal field theories with Abeliansymmetry , hep-th/0307041 .[120] A. Kapustin and M. J. Strassler, On mirror symmetry in three-dimensional Abelian gaugetheories , JHEP (1999) 021, [ hep-th/9902033 ].[121] N. P. Warner, Supersymmetry in boundary integrable models , Nucl. Phys.
B450 (1995)663–694, [ hep-th/9506064 ].[122] A. J. Niemi and G. W. Semenoff,
Axial Anomaly Induced Fermion Fractionization andEffective Gauge Theory Actions in Odd Dimensional Space-Times , Phys. Rev. Lett. (1983)2077.[123] A. N. Redlich, Gauge Noninvariance and Parity Violation of Three-Dimensional Fermions , Phys. Rev. Lett. (1984) 18.[124] A. N. Redlich, Parity Violation and Gauge Noninvariance of the Effective Gauge Field Actionin Three-Dimensions , Phys. Rev.
D29 (1984) 2366–2374.[125] D.-E. Diaconescu,
D-branes, monopoles and Nahm equations , Nucl. Phys.
B503 (1997)220–238, [ hep-th/9608163 ].[126] N. R. Constable, R. C. Myers, and O. Tafjord,
The Noncommutative bion core , Phys. Rev.
D61 (2000) 106009, [ hep-th/9911136 ].[127] J.L.Cardy,
Scaling and Renormalization in Statistical Physics , Cambridge Lecture Notes inPhysics (1996).[128] K. Intriligator and N. Seiberg,
Aspects of 3d N=2 Chern-Simons-Matter Theories , JHEP (2013) 079, [ arXiv:1305.1633 ].[129] D. Gaiotto, L. Rastelli, and S. S. Razamat, Bootstrapping the superconformal index withsurface defects , JHEP (2013) 022, [ arXiv:1207.3577 ].[130] C. Cordova, D. Gaiotto, and S.-H. Shao, Surface Defects and Chiral Algebras , JHEP (2017)140, [ arXiv:1704.0195 ].[131] T. Dimofte and S. Gukov, Chern-Simons Theory and S-duality , JHEP (2013) 109,[ arXiv:1106.4550 ].[132] S. Gukov and E. Witten, Gauge Theory, Ramification, And The Geometric LanglandsProgram , hep-th/0612073 .[133] D. Gaiotto, G. W. Moore, and A. Neitzke, Framed BPS States , Adv. Theor. Math. Phys. (2013), no. 2 241–397, [ arXiv:1006.0146 ].[134] S. Gukov, Three-dimensional quantum gravity, Chern-Simons theory, and the A polynomial , Commun. Math. Phys. (2005) 577–627, [ hep-th/0306165 ].[135] S. Garoufalidis,
On the characteristic and deformation varieties of a knot , Geom. Topol. – 98 – onogr. (2004) 291–309, [ hep-th/0306230 ].[136] T. Dimofte, Quantum Riemann Surfaces in Chern-Simons Theory , Adv. Theor. Math. Phys. (2013), no. 3 479–599, [ arXiv:1102.4847 ].[137] A. Adams, A. Basu, and S. Sethi, (0,2) duality , Adv. Theor. Math. Phys. (2003), no. 5865–950, [ hep-th/0309226 ].[138] J. McOrist and I. V. Melnikov, Half-Twisted Correlators from the Coulomb Branch , JHEP (2008) 071, [ arXiv:0712.3272 ].[139] D. Tong, The Quantum Dynamics of Heterotic Vortex Strings , JHEP (2007) 022,[ hep-th/0703235 ].[140] M. Blaszczyk, S. Groot Nibbelink, and F. Ruehle, Green-Schwarz Mechanism in Heterotic(2,0) Gauged Linear Sigma Models: Torsion and NS5 Branes , JHEP (2011) 083,[ arXiv:1107.0320 ].[141] C. Quigley and S. Sethi, Linear Sigma Models with Torsion , JHEP (2011) 034,[ arXiv:1107.0714 ].[142] K. Hori and C. Vafa, Mirror symmetry , hep-th/0002222 .[143] E. Witten, Supersymmetric index of three-dimensional gauge theory , hep-th/9903005 .[144] O. Bergman, A. Hanany, A. Karch, and B. Kol, Branes and supersymmetry breaking inthree-dimensional gauge theories , JHEP (1999) 036, [ hep-th/9908075 ].[145] K. Ohta, Supersymmetric index and s rule for type IIB branes , JHEP (1999) 006,[ hep-th/9908120 ].[146] D. Gaiotto and M. Rapˇc´ak, Vertex Algebras at the Corner , arXiv:1703.0098 .[147] V. G. Kac, Infinite-dimensional Lie algebras , vol. 44. Cambridge university press, 1994.[148] P. Francesco, P. Mathieu, and D. S´en´echal,
Conformal field theory . Springer Science &Business Media, 2012.[149] S. Kass, R. V. Moody, J. Patera, and R. Slansky,
Affine Lie algebras, weight multiplicities,and branching rules . Univ of California Press, 1990.[150] N. Seiberg,
Electric - magnetic duality in supersymmetric nonAbelian gauge theories , Nucl.Phys.
B435 (1995) 129–146, [ hep-th/9411149 ].[151] D. Bashkirov,
Aharony duality and monopole operators in three dimensions , arXiv:1106.4110 .[152] J. Park and K.-J. Park, Seiberg-like dualities for 3d N = 2 theories with SU(N) gauge group , Journal of High Energy Physics (2013), no. 10 198, [ hep-th/1305.6280 ].[153] R. Wendt,
A character formula for representations of loop groups based on non-simplyconnected Lie groups , Mathematische Zeitschrift (2004), no. 3 549–580.(2004), no. 3 549–580.