PPrepared for submission to JHEP
Vertex Algebras at the Corner
Davide Gaiotto, Miroslav Rapˇc´ak Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada N2L 2Y5
Abstract:
We introduce a class of Vertex Operator Algebras which arise at junc-tions of supersymmetric interfaces in N = 4 Super Yang Mills gauge theory. Thesevertex algebras satisfy non-trivial duality relations inherited from S-duality of the four-dimensional gauge theory. The gauge theory construction equips the vertex algebraswith collections of modules labelled by supersymmetric interface line defects. We dis-cuss in detail the simplest class of algebras Y L,M,N , which generalizes W N algebras.We uncover tantalizing relations between Y L,M,N , the topological vertex and the W ∞ algebra. a r X i v : . [ h e p - t h ] S e p ontents W ∞ ( p, q ) -fivebrane interfaces and their junctions. 6 L = 0 junctions 82.1.1 The N = M case 82.1.2 The N (cid:54) = M cases 92.2 The L > N = M case 102.2.2 The N (cid:54) = M cases 11 L = 0 and N > M cases 143.2.2 The L = 0 and N = M case 153.2.3 The L > Y L,M,N [Ψ]
216 Modules 23 U (1) Ψ U (1) corners 307.2.1 The Y , , [Ψ] description 337.2.2 “Degenerate” modules 357.2.3 Y , , [Ψ − ] and relatives 357.2.4 Degenerate Modules 367.2.5 General Modules 377.3 Three U (1) corners 37 U (2) gauge groups. 39 × U (1) 398.1.1 Y , , [Ψ] 398.1.2 Modules 408.1.3 Y , , [Ψ] 418.1.4 Modules 428.2 Parafermions × U (1) 438.2.1 Y , , and parafermions. 438.2.2 Degenerate modules 458.2.3 Y , , and parafermions. 468.2.4 Degenerate modules 488.2.5 Y , , and parafermions. 508.2.6 Degenerate modules 51 Y ,M,N . 529.2.2 The vacuum character of Y L, ,N . 549.2.3 A general conjecture 559.3 Crystal melting and vacuum characters 569.3.1 Melting crystals for Y L, , Y , ,N Y ,N,N Y ,M,N , M < N Y L, ,N Y L,M,N Y -algebras 61 Y -algebras 6310.3 Super-Virasoro 6610.4 Central charge 67
11 From Junctions to Webs 68 N = 2 super-Virasoro 7011.2 A trivalent junction with multiple D5 branes 71 A Conventions for current algebras 72
A.1 Free fermions 72A.1.1 A single real fermion 72A.1.2 SO ( n ) fermions 72A.1.3 A single complex fermion and the bc system 73A.1.4 U ( n ) fermions 74A.2 Symplectic fermions VOAs Sf and Sf βγ systems. 76A.3.2 Sp (2 n ) and U ( n ) symplectic bosons 77A.3.3 OSp ( n | m ) fermions 78A.3.4 U ( n | m ) symplectic bosons 79A.4 U ( N ) κ currents 80A.5 U ( M | N ) κ currents 81A.6 The bosonization of U (1 | κ B General central charge 83
B.1 Unitary case 83B.2 Ortho-symplectic case 86
C Series, products and contour integrals 89D Characters for W N − M, , ··· , U ( N )
89E Boundary conditions for hypermultiplets 91
E.1 Neumann boundary VOA 93E.2 Dirichlet boundary VOA 94– 1 –
Boundary conditions for gauge theory 95
The objective of this paper is to introduce a new class of Vertex Operator Algebras Y L,M,N [Ψ] labelled by three integers and a continuous coupling Ψ, which generalize thestandard W-algebras W N of type sl ( N ). These algebras are important building blocksof a general class of VOA’s which can be defined in terms of junctions of boundaryconditions and interfaces in the GL-twisted N = 4 Super Yang Mills gauge theory [1].The concrete definition of Y L,M,N [Ψ] is somewhat laborious: it involves a BRSTreduction of the combination of WZW algebras for super-groups U ( N | L ) Ψ × U ( M | L ) − Ψ and a certain collection of bc and βγ systems. We will introduce it and motivate it inthe main text of the paper. Our definition will be manifestly symmetric under the reflection Ψ ↔ − Ψ accom-panied by the exchange N ↔ M . Our main conjecture is that our definition is alsosymmetric under an “S-duality” transformation Ψ ↔ Ψ − accompanied by the ex-change L ↔ M . The two transformations combine into an S triality symmetry whichacts by permuting the three integral labels L , M , N while acting on the coupling Ψ byappropriate P SL (2 , Z ) duality transformations. In particular, we have cyclic rotations: Y L,M,N [Ψ] = Y N,L,M [ 11 − Ψ ] = Y M,N,L [1 −
1Ψ ] (1.1)An alternative, instructive way to describe the S symmetry is to introduce threeparameters (cid:15) i which satisfy (cid:15) + (cid:15) + (cid:15) = 0 Ψ = − (cid:15) (cid:15) (1.2) As pointed out by Mikhail Bershtein, algebras with similar structure were defined in [2, 3] in termsof a kernel of screening charges. It would be interesting to explore the relation further. Here and throughout the main text we will follow a somewhat unusual notation for the levelof unitary (super) WZW algebras, such that the current algebra U ( N | L ) Ψ contains a standard SU ( N | L ) Ψ+ L − N WZW current sub-algebra at level Ψ + L − N . In particular, the critical levelcorresponds to Ψ = 0. We will review our definitions in Appendix A and scatter frequent remindersabout this notational choice throughout the text. – 2 –hen the S symmetry acts on Y (cid:15) ,(cid:15) ,(cid:15) N ,N ,N ≡ Y N ,N ,N [ − (cid:15) (cid:15) ] (1.3)by a simultaneous permutation of the (cid:15) i and N i labels.We can illustrate this type of relations for W N ≡ Y , ,N . The Y , ,N [Ψ] VOA isdefined as the regular quantum Drinfeld-Sokolov reduction of U ( N ) Ψ and thus coincideswith the standard W-algebra W N with parameter b = − Ψ combined with a free U (1) current. The W N algebra has a symmetry b → b − known as Feigin-Frenkelduality, demonstrating immediately the expected S-duality relation between Y , ,N [Ψ]and Y , ,N [Ψ − ].On the other hand, our definition of Y N, , [1 − Ψ − ] involves a BRST reduction ofa product of elementary VOAs U ( N ) − − Ψ × U ( N ) Ψ1 − Ψ × Ff U ( N ) × bc u ( N ) , (1.4)where Ff U ( N ) denotes the VOA of N complex free fermions transforming in a funda-mental representation of U ( N ) and bc u ( N ) a bc ghost system valued in the u ( N ) Liealgebra. The BRST complex is essentially a symmetric description of a coset construction,leading to a third realization of W N as W N = SU ( N ) Ψ1 − Ψ − N × SU ( N ) SU ( N ) − Ψ − N (1.5)which is the analytic continuation of the well-known coset definition of W N minimalmodels. See e.g. [4] for a review and further references on this “triality” enjoyed by W N algebras.One of the most important features of the W N W-algebra is the existence of twodistinct collections of degenerate modules labelled by weights of SU ( N ) and permutedby the Feigin-Frenkel duality. These degenerate modules have very special fusion andbraiding properties.An extension of our main conjecture is the claim that Y L,M,N [Ψ] will have threecollections D ν , H λ , W µ of degenerate modules which are permuted by the S trial-ity symmetry and are labelled respectively by weights of U ( L | M ), U ( M | N ), U ( N | L ).These modules will also have special fusion and braiding properties. Recall our choice of notation in Appendix A which defines U ( N ) Ψ in terms of SU ( N ) Ψ − N and U (1) WZW currents. In terms of the (cid:15) i , the WZW levels become U ( N ) (cid:15) (cid:15) × U ( N ) (cid:15) (cid:15) . – 3 – .2 Gauge theory construction Our conjecture is motivated by a four-dimensional gauge theory construction, involvinglocal operators sitting at a Y -shaped junction of three interfaces between GL-twisted N = 4 Super Yang Mills theories with gauge groups U ( L ), U ( M ), U ( N ). The conjec-tural S triality symmetry follows from a conjectural invariance of this system underpermutations of the interfaces combined with P SL (2 , Z ) S-duality transformations.Degenerate representations for the vertex algebra arise at the endpoints of topologicalline defects running along either of the three interfaces.The full derivation of the VOA from the gauge theory setup involves a certainextension of the beautiful results of [5, 6] relating Chern-Simons theory and GL-twisted N = 4 SYM. It extends and generalizes the results of [7] which give a gauge-theoryconstruction of W N conformal blocks where S-duality implies Feigin-Frenkel dualityand degenerate representations arise from boundary Wilson and ’t Hooft loops.The action of S-duality on the gauge theory setup involves both a small general-ization of the known action of S-duality on half-BPS interfaces discussed in [8–10] anda novel statement about the S-duality co-variance of the junctions we employ. We mo-tivate such statement by a string theory brane web construction, involving a junctionbetween an NS5 brane, a D5 brane and a (1 ,
1) fivebrane together with L , M and N D3branes filling in the three angular wedges between the fivebranes.In this paper we will only sketch the relation to the gauge theory and brane con-structions, mostly in order to produce instructive pictures. Instead, we will bringevidence for our conjecture from the VOA side, matching central charges, the structureof degenerate modules, etc. It would be interesting to fill in the gaps in our analysisand give a rigorous gauge theory derivation of our proposal.For various values of parameters the VOA Y L,M,N [Ψ] coincides with known and well-studied examples of W-algebras. Our conjecture unifies a large collection of knowndualities relating different constructions of these W-algebras and makes a variety ofpredictions about their representation theory.
In the process of computing the characters of the vacuum module and of degener-ate modules, we stumbled on a beautiful combinatorial conjecture: the characters arecounting functions of 3d partitions, possibly with semi-infinite ends of shapes λ , µ , ν ,restricted to lie in the difference between the standard positive octant and the positiveoctant with origin at z = L , x = M , y = N . If we send L , M or N to infinity, thecharacters are thus related to the topological vertex [11].– 4 –ecall that W N , through the AGT correspondence [12, 13], plays a role in local-ization calculations in N = 2 gauge theory. Mathematically, this appears as an actionof W N on the equivariant cohomology of U ( N ) instanton moduli spaces [14–16]. The (cid:15) and (cid:15) parameters appear as equivariant parameters on C . Physically, one expectsthe W N generators to appear as BPS local operators in five-dimensional maximallysupersymmetric U ( N ) gauge theory. The cohomology of instanton moduli spaces mayalso be interpreted in terms of BPS bound states of N D D C ,combined with some judicious string dualities, suggests that Y (cid:15) ,(cid:15) ,(cid:15) L,M,N may act on theequivariant cohomology of some generalization of instanton moduli spaces, involvingthree stacks of D C in C bound to any num-ber of D C ) have beenintroduced recently in [17].More general VOAs discussed at the end of this paper may be associated to modulispaces of D0 branes bound to D4 branes wrapping cycles in general toric Calabi-Yauthree-folds. The characters for these general VOA’s are conjecturally assembled fromthe characters of Y L,M,N in a manner akin to the composition of topological vertices. W ∞ The computation of the vacuum character of Y L,M,N [Ψ] strongly suggests that all theseVOA can be interpreted as truncations of a W ∞ algebra, such as the two-parameterfamily of algebras introduced in [18]. The W ∞ algebras have families of fully de-generate modules which are analogue to the degenerate modules we encounter, withcharacters associated to the topological vertex [19].These algebras admit truncations along certain families of lines in the parameterspace, where the vacuum module acquire a null vector [20]. It is tempting to speculatethat Y L,M,N [Ψ] coincide with such truncations.
The addition of O Y ± L,M,N [Ψ] and ˜ Y ± L,M,N [Ψ] associated to
OSp -type supergroups. These includeas a special case the N = 1 super-Virasoro algebra and many other known W-algebras.We conjecture that they enjoy similar properties as the Y L,M,N [Ψ].
The paper will be structured as follows. Sections 2, 3, 4 contain a quick review ofsome useful facts (some well-known, some conjectural) about interfaces and junctions– 5 –n four-dimensional gauge theory, their Chern-Simons interpretation and the relation toVOAs. A reader which is only interested in the definition of our VOAs can safely skipthese sections. Section 5 contains the actual definition of the VOAs. Section 6 discussesthe three sets of degenerate modules exchanged by triality. Section 7 presents in detailexamples which arise from U (1) gauge theory. Section 8 presents several exampleswhich involve U (2) gauge theory. Section 9 contains a computation of central chargesand anomalous dimensions of degenerate modules for general L , M , N . It also containsa computation of characters for vacuum modules and degenerate modules. Section10 discusses the ortho-syplectic generalization of our VOAs. Section 11 discusses thepossible definition of a general class of VOAs associated to more complicated fivebranejunctions. ( p, q ) -fivebrane interfaces and their junc-tions. Brane constructions in Type IIB string theory imply the existence of a family of half-BPS interfaces B ( p,q ) for 4d N = 4 SYM with unitary gauge groups, parameterized bytwo integers ( p, q ) defined up to an overall sign. The main property of these interfacesis that they are covariant under the action of P SL (2 , Z ) S-duality transformations,which act in the obvious way on the integers ( p, q ). Concretely, these interfaces ariseas the field theory limit of a setup involving two sets of D3 branes ending on a single( p, q )-fivebrane from opposite sides [8–10].Most of the B ( p,q ) interfaces do not admit a straightforward, weakly coupled defi-nition. Rather, they involve some intricate 3d SCFT coupled to the U ( N ) and U ( M )gauge theories on the two sides of the interface. The exceptions are B (0 , and B (1 ,q ) interfaces.The B (0 , interface, also denoted as a D5 interface, has a definition which dependson the relative value of N and M : • If N = M , a D5 interface breaks the U ( N ) L × U ( N ) R gauge symmetry of thebulk theories to a diagonal U ( N ). A set of 3d hypermultiplets transformingin a fundamental representation of U ( N ) is coupled to the U ( N ) gauge fields.Concretely, the 4d fields on the two sides of the interface are identified at theinterface, up to some discontinuities involving bilinears of the 3d fields. • If N > M , a D5 interface breaks the U ( M ) L × U ( N ) R gauge symmetry of thebulk theories to a block-diagonal U ( M ). Concretely, U ( N ) R is broken to a block-diagonal U ( N − M ) R × U ( M ) R and U ( N − M ) R × U ( M ) L × U ( M ) R is further– 6 –roken to the diagonal U ( M ). The breaking of U ( N − M ) R involves a Nahm poleboundary condition of rank N − M . No further matter fields are needed at theinterface. • If M > N , a D5 interface breaks the U ( M ) L × U ( N ) R gauge symmetry of thebulk theories to a U ( N ), including a Nahm pole of rank M − N .The B (1 , interface, also denoted as an NS5 interface, has a uniform definition forall N and M [8]: the gauge groups are unbroken at the interface and coupled to 3dhypermultiplets transforming in a bi-fundamental representation of U ( M ) × U ( N ). The B (1 ,q ) interface is obtained from a B (1 , interface by adding q units of Chern-Simonscoupling on one side of the interface, − q on the other side.A well known property of ( p, q )-fivebranes is that they can form quarter-BPS webs[21, 22], configurations with five-dimensional super-Poincare invariance involving five-brane segments and half-lines drawn on a common plane, with slope determined bythe phase of their central charge. For graphical purposes, the slope can be taken to be p/q , though the actual slope depends on the IIB string coupling τ and is the phase of pτ + q . The simplest example of brane web is the junction of three semi-infinite branesof type (1 , ,
1) and (1 , S triality symmetry acting simultaneously onthe branes and the IIB string coupling.Five-brane webs are compatible with the addition of extra D3 branes filling infaces of the web. These configurations preserve four super-charges, organized in a (0 , L , M , N D3 branesrespectively filling the faces of the junction in between the (1 ,
1) and (1 ,
0) fivebranes,the (1 ,
0) and (0 ,
1) fivebranes and the (0 ,
1) and (1 ,
1) fivebranes.The resulting configuration is invariant under S triality transformations, the com-bination of permutations of L , M , N and duality transformations. The field theory limitof such a configuration, from the point of view of the D3 brane worldvolume, is that ofa junction between B (1 , , B (0 , and B (1 , interfaces between U ( L ), U ( M ) and U ( N ) N = 4 SYM defined on three wedges in the plane of the junction. The junction will beinvariant under the triality transformations. Notice that this statement will only holdif we identify correctly the field theory description of the junction.We will next conjecture the field theory description of the junction. Our conjecturewill be motivated by some matching of 2d anomalies and consistency with the GL-twisted description in the next section.A field theory description in a given duality frame is naturally given in a very weakcoupling limit. In that limit, it is natural to take the (1 , q ) fivebranes to be essentiallyvertical in the plane, and the D5 brane to be horizontal. Thus we will describe a T -– 7 – N D3 M D3 L D3 x x x , x , x × C × R x , x x , x , x Figure 1 . The brane system engineering our Y-junction for four-dimensional N = 4 SYM.The three fivebranes extend along the 01456 directions together with a ray in the 23 plane.The stacks of D3 branes extend along the 01 directions and fill wedges in the 23 plane. Noticethe SO (3) × SO (3) isometry of the system, which becomes the R-symmetry of a (0 , shaped junction, with an U ( L ) gauge theory on the negative x half-plane, U ( N ) on thetop right quadrant and U ( M ) in the bottom right quadrant. L = 0 junctions2.1.1 The N = M case At first, we can set N = M and L = 0. That means we have a U ( N ) gauge theorydefined on the x > x = 0. Theboundary conditions are deformed by an unit of Chern-Simons boundary coupling onthe x < x = 0, where the U ( N )gauge theory is coupled to a set of N
3d hypermultiplets transforming in a fundamentalrepresentation of the gauge group.The interface meets the boundary at x = x = 0. The hypermultiplets must havesome boundary conditions at the origin of the plane, preserving (0 ,
4) supersymmetry,such as these described in Appendix E. There is a known example of such a boundarycondition, involving Neumann boundary conditions for all the scalar fields. We expectit to appear in the field theory limit of the junction setup. The choice of Neumannb.c. is natural for the following reasons: the relative motion of the D3 branes on thetwo sides of the D3 interface involves the 3d hypermultiplets acquiring a vev. Thejunction allows for such a relative motion to be fully unrestricted and thus the 3dhypermultiplets boundary conditions should be of Neumann type.– 8 – ,
1) ( − , − − ,
0) 1 − (1 ,
0) (0 , ,
1) Ψ
T S ( − , −
1) (1 , , − − Ψ T ST SNML NMLNML
Figure 2 . The dualities which motivate the identification (1.1) of the VOA Y L,M,N [Ψ], Y N,L,M [ − Ψ ] and Y M,N,L [1 − ]. The (0 ,
4) boundary conditions for the hypermultiplets have an important feature:they set to zero the left-moving half of the hypermultiplet fermions at the boundary.Such a chiral boundary condition has a 2d gauge anomaly which is cancelled by anomalyinflow from the boundary U ( N ) Chern-Simons coupling along the negative imaginaryaxis. This anomaly will reappear in a similar role in the next section. N (cid:54) = M cases Next, we can consider N = M + 1 and L = 0. Now we do not have 3d matter alongthe positive real axis, but the gauge group drops from U ( M + 1) to U ( M ) across theboundary. The four-dimensional gauginoes which belong to the U ( M + 1) Lie algebrabut not to the U ( M ) subalgebra live on the upper right quadrant of the junctionplane with non-trivial boundary conditions on the two sides. They may in principlecontribute a 2d U ( M ) gauge anomaly at the corner. It is a bit tricky to compute it, butwe will recover it from a vertex algebra computation in Section 3. Again, we expect itto cancel the anomaly inflow from the boundary U ( M ) Chern-Simons coupling alongthe negative imaginary axis. – 9 – igure 3 . The gauge theory image of a Y-junction on the 23 plane. We denote the specificjunction as Y L,M,N . The Y L,M,N [Ψ] VOA will arise as a deformation of the algebra of BPSlocal operators at the junction.
Similar considerations for general N (cid:54) = M and L = 0, though the positive real axisnow supports a partial Nahm pole boundary condition along with the reduction from U ( N ) to U ( M ) or vice-versa. Again, we will describe the corresponding anomalies andtheir cancellations in Section 3. L > junctions2.2.1 The N = M case Next, we can set N = M but take general L . That means we have an U ( N ) gaugetheory defined on the x > U ( L ) gauge theory defined on the x < x < x = 0, the gauge fields are coupled to 3d L × N bi-fundamental hypermultiplets. We also have an interface at x = 0, x > U ( N ) gauge theory is coupled to a set of N
3d hypermultiplets transformingin a fundamental representation of the gauge group.– 10 –he interfaces meet at x = x = 0. The fundamental hypermultiplets should begiven a boundary condition at the origin which preserve (0 ,
4) symmetry. The boundarycondition may involve the bi-fundamental hypermultiplets restricted to the origin and,potentially, extra 2d degrees of freedom defined at the junction only.We can attempt to define the boundary condition starting from the basic (0 , x = x = 0 of the bi-fundamental and fundamental hypersbehave as (0 ,
4) hypermultiplets and (0 ,
4) twisted hypermultiplets respectively. Thereis a known way to couple these types of fields in a (0 , ,
4) fields: Fermi multiplets transforming in the fun-damental representation of U ( L ) which can enter in a cubic fermionic superpotentialwith the hypermultiplets and twisted hypermultiplets [23].This coupling is known to occur in similar situations involving multiple D-branesending on an NS5 brane [24]. The Fermi multiplets should arise from D − D D − D D − D (cid:48) and D (cid:48) − D U ( L ) fundamental Fermi multiplets also play another role: they consist of 2dleft-moving fermions, whose anomaly compensates the inflow from the boundary U ( L )Chern-Simons coupling along the negative imaginary axis. N (cid:54) = M cases Next, we can consider N = M + 1 and general L . Now the number of hypermultipletsalong the imaginary axis drops from L × N to L × M across the origin of the junction’splane. We can glue together M × L of them according to the embedding of U ( M )in U ( N ) along the real axis, but we need a boundary condition for the remaining L hypermultiplets.Neumann boundary conditions for these L hypermultiplets would contribute ananomaly of the wrong side to cancel the inflow from the boundary U ( L ) Chern-Simonscoupling along the negative imaginary axis. The opposite choice of boundary condi-tions, i.e. Dirichlet b.c. for the scalar fields, imposes the opposite boundary conditionon the hypermultiplet’s fermions and seems a suitable choice. We will thus not need toadd extra 2d Fermi multiplets at the corner. Notice that one can obtain such boundary conditions starting from Neumann boundary conditionsand coupling them to (0 ,
4) 2d Fermi multiplets, which get eaten up in the process. It would be nice – 11 –imilar considerations for general N (cid:54) = M and general L , though the positive realaxis now supports a partial Nahm pole boundary condition along with the reductionfrom U ( N ) to U ( M ) or vice-versa. The boundary conditions at the corner for the | N − M | × L hypermultiplets which do not continue across the corner will be affectedby the Nahm pole. We will refrain from discussing them in detail here and focus onthe GL-twisted version in the next section. The analysis of [5] gives a prescription for how to embed calculations in (analyticallycontinued) Chern-Simons theory into GL-twisted four-dimensional N = 4 Super-Yang-Mills theory.Concretely, a Chern-Simons calculation on a three-manifold M maps to a four-dimensional gauge theory calculation on M × R + with a specific boundary conditionwhich deforms the standard supersymmetric Neumann boundary conditions. The (an-alytically continued) Chern-Simons level is related to the coupling Ψ of the GL-twisted N = 4 SYM as [25] k + h = Ψ (3.1)It is natural to wonder about the possible implications in Chern-Simons theory ofthe S-duality group of the four-dimensional gauge theory [26]. In order to do so, weneed to overcome a simple problem: supersymmetric Neumann boundary conditionsare not invariant under S-duality. For example, they are mapped to a regular Nahmpole boundary condition by the S element of P SL (2 , Z ).Assuming that the deformed Neumann boundary conditions transform in a manneranalogous to the undeformed ones, that means the S transformation will map theChern-Simons setup to a different setup involving a deformed Nahm pole boundarycondition. This was a basic step in the gauge-theory description of categorified knotinvariants in [25].In general, we expect the B ( p,q ) boundary conditions to admit deformations ˜ B ( p,q ) compatible with the GL twist, such that ˜ B (1 , coincides with deformed Neumannboundary conditions and P SL (2 , Z ) duality transformations act in the obvious wayon the integers ( p, q ). In Appendix F we discuss briefly the deformations ˜ B (1 , and˜ B (0 , . to follow in detail in the field theory the process of separating a D N = M system and flowing to the N = M + 1 system, by giving a vev to the fundamental hypermultipletswhich induces a bilinear coupling of the 2d Fermi multiplets – 12 –ome elements of P SL (2 , Z ) do leave Neumann b.c. invariant: the Nahm poleboundary conditions are invariant under T and thus Neumann b.c. are invariant under ST n S . This invariance “explains” why the partition function of Chern-Simons theoryis a function of q ≡ e πik + h = e πi Ψ : they are invariant under Ψ − → Ψ − + n . In order to broaden the set of interesting S-duality transformations and obtainfurther duality relations, one may consider configurations involving multiple boundaryconditions. That is the basic idea we pursue in this paper.
The general formalism of [5] relates a variety of analytically continued path integralsin d dimensions and topological field theory calculations in d + 1 dimensions, possiblyincluding local observables or defects. Intuitively, observables which are functions ofthe d -dimensional fields will map to the same functions applied to the boundary valuesof ( d + 1)-dimensional fields, but modifications of the d -dimensional path integral maypropagate to modifications of the ( d + 1)-dimensional bulk. Extra degrees of freedomadded in the d -dimensional setup may remain at the boundary of the ( d +1)-dimensionalbulk or analytically continued to extra degrees of freedom in the bulk.A simple, rather trivial example of this flexibility is the observation that one cansplit off a well-defined multiple of the Chern-Simons action before analytic continuation,giving rise to a bulk theory with coupling Ψ + q with a ˜ B (1 ,q ) boundary condition.A more important example is analytically continued Chern-Simons theory definedon a manifold with boundary, M = C × R + , with some boundary condition B d . Thissetup will map to a calculation involving four-dimensional gauge theory on a cornergeometry C × R + × R + . One of the two sides of the corner will have deformed Neumannboundary condition ˜ B (1 , . The other side will have some boundary condition B d whichcan be derived from the boundary condition B d in a systematic fashion. At the corner,the two boundary conditions will be intertwined by some interface which is also derivedfrom the boundary condition B d .The simplest possibility is to consider holomorphic Dirichlet boundary conditions D d in Chern-Simons theory, given by A ¯ z = 0 at the boundary. It is well known thatthese boundary conditions support WZW currents J = A z | ∂ of level Ψ − h , given bythe holomorphic part of the connection restricted to the boundary. These boundaryconditions will lift to a deformation of Dirichlet boundary conditions in SYM.A slightly more refined possibility is to consider a generalization of holomorphicDirichlet boundary conditions D dρ which is labelled by an su (2) embedding in the gauge This statement has to be slightly modified for gauge groups which are not their own Langlandsdual, so that the duality group is reduced to a subgroup of
P SL (2 , Z ). – 13 –roup [27, 28]. These boundary conditions require the boundary gauge field to be ageneralized oper of type ρ . They are expected to support the Vertex Operator Algebras W ρ [ G Ψ − h ] obtained from G Ψ − h WZW by a Quantum Drinfeld Sokolov reduction. Inparticular, for the regular su (2) embedding one obtains the standard W-algebras. Theseboundary conditions will lift to a deformation of the regular Nahm pole boundaryconditions in SYM. We describe the deformation briefly in Appendix F.The regular Nahm pole boundary condition in SYM is precisely B (0 , . That meansthe Chern-Simons setup leading to the standard W-algebras lifts to a corner geometry inSYM with ˜ B (1 , on one edge and a boundary condition we expect to coincide with ˜ B (0 , on the other edge. This is supported by the analysis of [7], which reduced the problemon a compact Riemann surface C and found conformal blocks for the correspondingW-algebras.In particular, the symmetry of the standard W-algebras under the Feigin-Frenkelduality, which maps Ψ → Ψ − , strongly suggests that the junction at the corner shouldbe S-duality invariant. We expect that for a U ( N ) gauge group the junction willtake the form of a deformation of the corner configuration in the previous section, for L = M = 0. At this point it is natural to seek configurations in Chern-Simons theory which couldbe uplifted to a deformation of the junctions in the previous section for general L , M , N , involving ˜ B (1 , , ˜ B (0 , and ˜ B (1 , interfaces.We take the same coupling Ψ uniformly in the whole plane of the Y L,M,N junctionand the T-shaped configuration of Figure 3: the construction of [5] applied along the x direction maps the four-dimensional gauge theory with ˜ B (1 , boundary conditions at x > k + h = Ψ and the four-dimensional gauge theorywith ˜ B (1 , boundary conditions at x < k + h = Ψ − x = 0 together with the junction will encode some two-dimensionalinterface between the two Chern-Simons theories, as described in the following. L = 0 and N > M cases
At first, we can take L = 0 and N > M . In order to re-produce the (deformation ofthe) bulk Nahm pole, we can consider the following interface between U ( N ) and U ( M )Chern-Simons theories at levels Ψ − N and Ψ − M −
1. First, we take the boundarycondition D dN − M, , ··· , for the former CS theory, defined by the same su (2) embedding in U ( N ) as the Nahm pole we need to realize, which decomposes the fundamental of U ( N )into a dimension N − M irrep together with M copies of the trivial representation. This– 14 –oundary condition preserves an U ( M ) subgroup of the U ( N ) gauge group, which wecouple to the U ( M ) gauge fields on the other side of the interface.Classically, the U ( N ) connection at the interface decomposes into blocks A U ( N ) | ∂ = (cid:18) ∗ ( N − M ) × ( N − M ) ∗ ( N − M ) × M ∗ M × ( N − M ) A U ( M ) | ∂ (cid:19) (3.2)with one block identified with the U ( M ) connection and the other blocks subject tothe oper boundary condition.In order for this interface to make sense quantum mechanically, the anomaly of the U ( M ) WZW currents in the VOA W N − M, , ··· , [ U ( N ) Ψ ] must be cancelled by anomalyinflow from the expected level Ψ − M − U ( M ) Chern-Simons theory. We willdemonstrate this fact for general N − M later on with a detailed Quantum DrinfeldSokolov reduction. Essentially, the naive level Ψ − N is shifted to Ψ − M − N = M + 1 it is almost obvious: the SU ( M )currents in U ( N ) Ψ currents have anomaly Ψ − N = Ψ − M −
1, just as expected. SeeAppendix A for further details. L = 0 and N = M case Next, we can take L = 0 and N = M . Recall that the bulk setup involves fundamentalhypermultiplets extended along the ˜ B (0 , interface. We show in Appendix E that thetopological twist of these 3d degrees of freedom implements an analytically continuedtwo-dimensional path integral for a theory of free chiral symplectic bosons . This isanother name for a βγ system here the dimension of both β and γ are 1 /
2, so that theycan be treated on the same footing. Each hypermultiplet provides a single copy of thesymplectic bosons VOA. See Appendix A for details on the symplectic boson VOA.Thus we will consider a simple interface between U ( N ) Ψ − N and U ( N ) Ψ − N − Chern-Simons theories: we identify the gauge fields across the interface, but couple themto the theory Sb U ( N ) of N chiral symplectic bosons transforming in a fundamentalrepresentation of U ( N ). This VOA includes U ( N ) WZW currents β a γ b whose anomaliesprecisely compensate the shift of CS levels. We refer the reader to Appendix A fordetails. This is just another manifestation of the corner anomaly cancellation discussedin the previous section. L > cases Next, we can consider general L . Now we will have ˜ B (1 , and ˜ B (1 , interfaces between U ( L ) and U ( N ) gauge theories. According to [6], a ˜ B (1 , interface between U ( L ) and We remind the reader again that the VOA we denote as U ( N ) Ψ has an SU ( N ) Ψ − N currentsubalgebra. – 15 – ( N ) GL-twisted gauge theories will map to a U ( N | L ) Chern-Simons theory at levelΨ − N + L . We can thus proceed as before and consider interfaces between U ( N | L )and U ( M | L ) Chern-Simons theories at levels Ψ − N + L and Ψ − M + L − N (cid:54) = M , the interface should be a super-group generalization D dN − M, , ··· , | , ··· , ofthe Nahm-pole-like boundary condition, preserving an U ( M | L ) subgroup of the gaugegroup which can be coupled to the Chern-Simons gauge fields on the other side of theinterface. The oper-like boundary conditions have an obvious generalization to super-groups, with su (2) embedding into the bosonic subgroup. It would be interesting todetermine the corresponding boundary condition on the bi-fundamental hypermulti-plets present on the ˜ B (1 , and ˜ B (1 , interfaces.If N = M , we need to generalize the symplectic boson VOA to something whichadmits an action of U ( N | L ) with appropriate anomalies. The obvious choice is toadd at the interface both N copies of the chiral symplectic bosons VOA and L chiralcomplex fermions. The fermions do not need to be uplifted to 3d fields and can insteadbe identified in four-dimensions with the (0 ,
4) Fermi multiplets at the origin of thejunction.The symplectic bosons and fermions combine into a fundamental representationof U ( N | L ) and define together a VOA Sb U ( N | L ) which includes the required U ( N | L )WZW currents. See Appendix A for details. In the gauge theory constructions of Section 3 we have encountered a variety of bound-ary conditions and interfaces for (analytically continued) Chern-Simons theory. In thissection we discuss the chiral VOA of local operators located at these boundaries orinterfaces.The best known example, of course, is the relation between Chern-Simons theoryand WZW models [29]: a Chern-Simons theory with gauge group G at level k definedon a half-space with appropriate orientation and an anti-chiral Dirichlet boundarycondition A ¯ z = 0 supports at the boundary a chiral WZW VOA G k based on the Liealgebra of G , with currents J of level k which are proportional the restriction of A z tothe boundary. Notice that the coupling of the Sb U ( N | L ) VOA to the 3d Chern Simons theory induces a discon-tinuity of A z across the interface proportional to the WZW currents in the VOA. In particular, thediscontinuity of the odd currents in U ( N | L ) is proportional to products of a 2d symplectic boson anda 2d fermion. This must correspond to the effect of the junction coupling between the (0 ,
4) Fermimultiplets and the restrictions of the fundamental and bi-fundamental hypermultiplets to the junction. The proportionality factor is k . – 16 –irichlet boundary conditions are associated to a full reduction of the gauge groupat the boundary: gauge transformations must go to the identity at the boundary andconstant gauge transformations at the boundary become a global symmetry of theboundary local operators. For our purposes, we need to consider a more general sit-uation, where the gauge group is only partially reduced and may be coupled at theboundary to extra two-dimensional degrees of freedom.First, we should ask if Neumann boundary conditions could be possible, so thatthe gauge group is fully preserved at the boundary. In the absence of extra 2d matterfields, this is not possible, because of the boundary gauge anomaly inflowing from thebulk Chern-Simons term. We would like to claim that Neumann boundary conditionsare possible if extra 2d matter fields are added, say a 2d chiral CFT T d equipped withchiral, G -valued WZW currents J d of level − k − h .Indeed, we can produce Neumann boundary conditions by coupling auxiliary two-dimensional chiral gauge fields to the combination of T d and standard Dirichlet bound-ary conditions. The level of T d is chosen in such a way to cancel the naive bulk anomalyinflow when combined with the ghost contribution to the boundary anomaly. The effectif coupling two-dimensional gauge fields to VOA is well understood from the study ofcoset conformal field theory [30, 31].The VOA of boundary local operators should be built from the combination of G k , T d and a bc ghost system bc g valued in the Lie algebra of G , taking the cohomologyof the BRST charge Q BRST = (cid:73) dz Tr (cid:20)
12 : b ( z )[ c ( z ) , c ( z )] : + c ( z )( J ( z ) + J ( z ) d ) (cid:21) + Q d BRST (4.1)which implements quantum-mechanically the expected boundary conditions J ( z ) + J ( z ) d = 0. We included Q d BRST to account for the possibility that T d itself wasdefined in a BV formalism. We will denote such procedure as a g -BRST reduction.The relation to coset constructions is related to the observation that the BRSTcohomology includes the sub-algebra of local operators in T d which are local with theWZW currents J d . In other words, the boundary VOA includes the current algebra coset T d G − k − h (4.2)which generalizes the idea that Neumann boundary conditions support local gauge-invariant operators in T d . One can envision the bc g ghosts as cancelling out both the G k and the G − k − h currents, in a sort of Koszul quartet or Chevalley complex, leaving– 17 –ehind precisely the coset. The BRST complex above can be thought of as a sort ofdifferential graded or derived version of a coset, perhaps better suited than the usualcomplex to non-unitary VOAs.Interfaces can be included in this discussion by a simple folding trick. The changein orientation maps k → − h − k . Thus we can consider a Neumann-type interfacebetween G k and G (cid:48) k (cid:48) Chern-Simons theories coupled to a 2d chiral CFT T d equippedwith chiral, G × G (cid:48) -valued WZW currents of levels − k − h and k (cid:48) .The interface VOA will be the g ⊕ g (cid:48) -BRST reduction of the combination of G k , G (cid:48)− k (cid:48) − h (cid:48) , T d and bc g ⊕ g (cid:48) . This implements a coset T d G − k − h ⊗ G (cid:48) k (cid:48) (4.6)The construction above has an obvious generalization to mixed boundary condi-tions, where the gauge group is reduced to a subgroup H at the boundary and coupledwith extra degrees of freedom T d equipped with chiral, H -valued WZW currents J d of level − k H − h H . The boundary VOA will consist of the h -BRST reduction of thecombination of G k , T d and bc h .The simplest example of this construction is a trivial interface between G k and G k Chern-Simons theories. The interface breaks the G × G gauge groups to the diagonalcombination, gluing together the gauge fields on the two sides. The VOA of localoperators should be the BRST cohomology of G k × ˆ G − k − h combined with one set bc g of bc ghosts valued in the Lie algebra of G . This BRST cohomology is trivial: the trivialinterface in Chern-Simons theory supports no local operators except for the identity.A more interesting example is an interface where the G k Chern-Simons theoryis coupled to some 2d degrees of freedom T d equipped with chiral, G -valued WZWcurrents J d of level k (cid:48) . Notice that the levels on the two sides of the interface shouldbe k and k + k (cid:48) . For example, the final central charge is the expected c d + c G k + c bc g = c d + d G kk + h − d G = c d − d G ( − k − h ) − k − h + h = c d − c G − k − h (4.3)as expected from the coset. Because[ Q BRST , k + h Tr b ( J − J ( z ) d ) = T G k + T G − h − k − Tr b∂c (4.4)the total stress tensor is indeed equivalent to the coset stress tensor T T d + T G k − Tr b∂c = T T d − T G − h − k (4.5) – 18 –hen the interface VOA will be given by the BRST cohomology of G k × T d × G − k − k (cid:48) − h combined with one set of b c ghosts valued in the Lie algebra of G . This canbe interpreted as either of two conjecturally equivalent cosets G k × T d G k + k (cid:48) ? = G − k − k (cid:48) − h × T d G − k − h (4.7)An example of this was discussed in [32] with T d taken to be a set of chiral fermionstransforming in the fundamental representation of U ( N ), resulting in the coset SU ( N ) k × SU ( N ) SU ( N ) k +1 (4.8)which is a well-known realization of a W N VOA. That construction was a source ofinspiration for this project.A second important topic we need to discuss is the Quantum Drinfeld-Sokolovreduction W ρ [ G k ] of G k , the VOA which appear at “oper-like” boundary conditions fora G k Chern-Simons theory, labelled by an su (2) embedding ρ .As a starting point, we may recall the construction for SU (2) gauge group and theregular su (2) embedding [27]. The classical boundary condition takes the schematicform A ¯ z = (cid:18) a K ¯ z ∗ − a K ¯ z (cid:19) A z = (cid:18) ∗ ∗ ∗ (cid:19) (4.9)where the ∗ denotes elements which are not fixed by the boundary condition and a K ¯ z is the connection on the canonical bundle.Gauge-transformations can be used to locally gauge-fix the holomorphic connectionto A z = (cid:18) t ( z ) 0 (cid:19) (4.10)with t ( z ) behaving as a classical stress tensor.Quantum mechanically, one proceeds as follows [33–35]. The stress tensor of theusual SU (2) k WZW currents is shifted by ∂J , in such a way that J + ( z ) acquiresconformal dimension 0 and J − ( z ) acquires conformal dimension 2. Furthermore, asingle pair of b c ghosts is added, allowing us to define a BRST charge Q BRST = (cid:73) dzc ( z )( J + ( z ) −
1) (4.11)enforcing the J + ( z ) = 1 constraint. The total stress tensor T = T SU (2) k − ∂J − b∂c (4.12)– 19 –s in the BRST cohomology and generates it. It has central charge3 kk + 2 − k − − k + 2 − k + 2) = 1 + 6( b + b − ) (4.13)with b = − ( k + 2).The construction generalizes as follows [35–37]. Take the t element in the su (2)embedding ρ . The Lie algebra g decomposes into eigenspaces of t as g = ⊕ i g i/ (4.14)The raising generator t + of ρ is an element in g . Naively, we want to set to zeroall currents of positive degree under t except for the one along t + , which should beset to 1. We cannot quite do so because if we set to zero all currents in g / we willalso set to zero their commutator, including the current along the t + direction. Thecommutator together with the projection to t + gives a symplectic form on g / and weare instructed to only set to zero some Lagrangian subspace g +1 / in g / .Then W ρ [ G k ] is defined as the BRST cohomology of a complex which is almost thesame as the one we would use to gauge the triangular sub-group n = g +1 / ⊕ (cid:77) i> g i/ (4.15)In particular, we add to G k a set of b c ghosts valued respectively in n and n ∗ .The main difference is that we will shift the stress tensor by the t component of ∂J and by a similar ghost contribution [ b, t ] · c in such a way that currents and b -ghostsin g i/ have conformal dimension 1 − i/
2. This allows us to add the crucial extra termsetting the t + component of J to 1: Q qDS BRST = (cid:73) dz Tr (cid:20)
12 : b ( z )[ c ( z ) , c ( z )] : + c ( z ) J ( z ) (cid:21) − t + · c ( z ) (4.16)In general, if the su (2) embedding ρ commutes with some subgroup H of G , theWZW currents in H can be corrected by ghost contributions to give H WZW currentsin W ρ [ G k ]. The ghost contributions will shift the level away from the value inheritedfrom G k .The oper-like boundary conditions can be further modified by gauging subgroupsof H coupled to appropriate 2d degrees of freedom and/or promoted to interfaces byidentifying the H subgroup of the G connection with an H connection on the otherside of the interface. This will lead to further h -BRST cosets involving W ρ [ G k ] as aningredient. – 20 – From Chern-Simons interfaces to Y L,M,N [Ψ]
We now have all ingredients we need in order to provide a definition of Y L,M,N [Ψ].We can start from the case N = M and L = 0. Recall that we have a U ( N ) ChernSimons theory with an interface supporting a two-dimensional theory Sb U ( N ) consistingof N pairs of symplectic bosons transforming in a fundamental representation of U ( N ).See Appendix A for details of the corresponding VOA.The SU ( N ) level of the CS theory is Ψ − N on one side of the interface, Ψ − N − Y ,N,N [Ψ]is the u ( N )-BRST reduction of the product U ( N ) Ψ × Sb U ( N ) × U ( N ) − Ψ+1 × bc u ( N ) (5.1)Recall our conventions that U ( N ) Ψ contains SU ( N ) Ψ − N WZW currents. The level ofthe SU ( N ) currents in Sb U ( N ) is −
1. The anomalies of the U ( N ) WZW currents inSb U ( N ) are precisely such that they can be added to U ( N ) Ψ currents to give U ( N ) Ψ − currents. We refer the reader to Appendix A for details.Thus the VOA Y ,N,N [Ψ] can be identified with either of the two cosets Y ,N,N [Ψ] = U ( N ) Ψ × Sb U ( N ) U ( N ) Ψ − = Sb U ( N ) × U ( N ) − Ψ+1 U ( N ) − Ψ (5.2)Notice that the BRST definition is symmetric under Ψ ↔ − Ψ.Next, we can consider the case N = M + 1 and L = 0. Recall that we havean interface between a U ( N ) Chern Simons theory and a U ( M ) Chern-Simons theorydefined simply by reducing the U ( N ) gauge symmetry to U ( M ) at the boundary andidentifying with the gauge symmetry on the other side.The SU ( N ) level of the CS theory is Ψ − N on one side of the interface, Ψ − M − − N on the other side. According to the prescription in Section 4, the interface VOA Y ,N − ,N [Ψ] is the u ( N − U ( N ) Ψ × U ( N − − Ψ+1 × bc u ( N − (5.3)Notice that the U ( N ) Ψ currents precisely contain a block-diagonal U ( N − Ψ − WZWsubalgebra. See Appendix A for details.Thus the VOA Y ,N − ,N [Ψ] can be identified with the coset Y ,N − ,N [Ψ] = U ( N ) Ψ U ( N − Ψ − (5.4)Similarly, for N = M − L = 0 we would define Y ,N +1 ,N [Ψ] as the u ( N )-BRSTreduction of the product U ( N ) Ψ × U ( N + 1) − Ψ+1 × bc u ( N ) (5.5)– 21 –quivalently, the coset Y ,N +1 ,N [Ψ] = U ( N + 1) − Ψ+1 U ( N ) − Ψ (5.6)For general N > M and L = 0, we need to reduce the U ( N ) gauge symmetry atthe interface by the oper-like boundary condition involving the su (2) embedding whichdecomposes the fundamental representation of U ( N ) into an ( N − M )-dimensional irrepand M copies of the trivial irrep. The residual U ( M ) symmetry can be identified withthe gauge symmetry on the other side.First, we need to make sure that the Chern-Simons levels work out. The block-diagonal U ( M ) Ψ − N + M subalgebra of U ( N ) Ψ should be combined with the ghost contri-butions to give BRST-closed total U ( M ) currents. It is easy to see that the triangularsubalgebra n includes N − M − U ( M )and several copies of the trivial representation. Each set of ghosts transforming in thefundamental representation of U ( M ) will shift by 1 unit the level of the U ( M ) WZWcurrents. Thus W N − M, , ··· , [ U ( N ) Ψ ] has an U ( M ) Ψ − subalgebra. See Appendix A fordetails.We are ready to define Y ,M,N [Ψ] as the u ( M ) BRST reduction of the product W N − M, , ··· , [ U ( N ) Ψ ] × U ( M ) − Ψ+1 × bc u ( M ) (5.7)i.e. the coset Y ,M,N [Ψ] = W N − M, , ··· , [ U ( N ) Ψ ] U ( M ) Ψ − (5.8)We can combine the quantum DS reduction and the u ( M )-BRST coset into a singledeformed ( n ⊕ u ( M ))-BRST quotient of the product U ( N ) Ψ × U ( M ) − Ψ+1 × bc n ⊕ u ( M ) (5.9)Similarly, for N < M and L = 0 we would get a BRST coset of the form Y ,M,N [Ψ] = W M − N, , ··· , [ U ( M ) − Ψ+1 ] U ( N ) − Ψ (5.10)Notice that these definitions are trivially symmetric under N ↔ M together withΨ ↔ − Ψ.For general L we need to upgrade all the constructions described above to super-groups. The quantum DS reduction for WZW VOA’s based on super-Lie algebrasworks in the same way as for standard Lie algebras, except that fermionic ghosts arereplaced by bosonic ghosts in the odd components of n [38].– 22 –ccording to the prescription in Section 4, the interface VOA Y L,N,N [Ψ] is the u ( N | L )-BRST reduction of the product U ( N | L ) Ψ × Sb U ( N | L ) × U ( N | L ) − Ψ+1 × bc u ( N | L ) (5.11)The Sb U ( N | L ) consists of N sets of symplectic bosons and L complex fermions. The U ( N ) WZW currents J Sb U ( N | L ) are precisely such that they can be added to U ( N | L ) Ψ currents to give U ( N | L ) Ψ − currents. We refer the reader to Appendix A for details.Thus the VOA Y L,N,N [Ψ] can be identified with either of the two cosets Y L,N,N [Ψ] = U ( N | L ) Ψ × Sb U ( N | L ) U ( N | L ) Ψ − = Sb U ( N | L ) × U ( N | L ) − Ψ+1 U ( N | L ) − Ψ (5.12)Notice that the BRST definition is symmetric under Ψ ↔ − Ψ.For
N > M we will use a quantum DS reduction of the super-groups. We willdefine Y L,M,N [Ψ] as the u ( M | L )-BRST reduction of the product W N − M, , ··· , | , ··· , [ U ( N | L ) Ψ ] × U ( M | L ) − Ψ+1 × bc u ( M | L ) (5.13)i.e. the coset Y L,M,N [Ψ] = W N − M, , ··· , | , ··· , [ U ( N | L ) Ψ ] U ( M | L ) Ψ − (5.14)We can combine the quantum DS reduction and the u ( M | L )-BRST coset into a singledeformed ( n ⊕ u ( M | L ))-BRST quotient of the product U ( N | L ) Ψ × U ( M | L ) − Ψ+1 × bc n ⊕ u ( M | L ) (5.15)Similarly, for N < M we would get a BRST coset of the form Y L,M,N [Ψ] = W M − N, , ··· , | , ··· , [ U ( M | L ) − Ψ+1 ] U ( N | L ) − Ψ (5.16)Notice that these definitions are trivially symmetric under N ↔ M together withΨ ↔ − Ψ. The standard W N algebras have maximally degenerate modules M λ,µ labelled by a pairof dominant weights of su ( N ). The Feigin-Frenkel duality b → b − exchanges the roleof the two weights. – 23 –hese modules are expected to arise in the gauge theory construction from localoperators at the corner which are attached to a boundary Wilson line of weight λ alongthe NS5 boundary and a boundary ’t Hooft line of weight µ along the D5 boundary.These two line defects are correspondingly exchanged by S-duality.If we denote W λ = M λ, and H µ = M ,µ , then the following facts hold true: • The W λ have the same fusion rules W λ × W λ (cid:48) ∼ (cid:88) λ (cid:48)(cid:48) c λ (cid:48)(cid:48) λ,λ (cid:48) W λ (cid:48)(cid:48) (6.1)as finite-dimensional SU ( N ) irreps. They have non-trivial braiding and fusionmatrices which are closely related to these of SU ( N ) Ψ − N . Conformal blocks with W λ insertions satisfy BPZ differential equations. • The H µ also have the same fusion rules H µ × H µ (cid:48) ∼ (cid:88) µ (cid:48)(cid:48) c µ (cid:48)(cid:48) µ,µ (cid:48) H µ (cid:48)(cid:48) (6.2)as finite-dimensional SU ( N ) irreps. They have non-trivial braiding and fusionmatrices which are closely related to these of SU ( N ) Ψ − − N . Conformal blockswith H µ insertions satisfy BPZ differential equations • The W λ and H µ vertex operators are almost mutually local. They are local ifwe restricts the weights to those of GL-dual groups. The fuse in a single channel M λ,µ .We expect analogous statements for maximally degenerate modules of Y L,M,N [Ψ],involving local operators sitting at the end to three boundary lines, one for each com-ponent of the gauge theory junction. These modules should thus carry three labels,permuted by the S triality symmetry, corresponding to the possible labels of BPS linedefects living on the ˜ B ( p,q ) boundary conditions. It is known [6] that such line defectsinclude analogues of Wilson lines, labelled by data akin to dominant weights of U ( N | L ), U ( L | M ), U ( M | N ) respectively.In particular, we expect the following to be true: if we denote W λ , H µ and D σ themodules associated to either type of boundary lines • The W λ should have the same fusion rules as finite-dimensional U ( N | L ) irreps,with appropriate non-trivial braiding and fusion matrices and BPZ-like differen-tial equations. – 24 – The H µ should have the same fusion rules as finite-dimensional U ( M | N ) irreps,with appropriate non-trivial braiding and fusion matrices and BPZ-like differen-tial equations. • The D σ should have the same fusion rules as finite-dimensional U ( L | M ) irreps,with appropriate non-trivial braiding and fusion matrices and BPZ-like differen-tial equations. • The W λ , H µ and D σ vertex operators should be mutually local and fuse togetherinto a single channel M λ,µ,ν U ( L ) U ( N ) U ( M ) W λ H µ D σ Figure 4 . Modules W λ , H µ , D σ associated to the three classes of boundary lines. In the coset constructions for Y L,M,N [Ψ], the data for U ( N | L ) and U ( L | M ) repre-sentations appears rather naturally, as one may implement the BRST reduction start-ing from Weyl modules of the current algebras built from irreducible representationsof the zeromode algebra, up to subtleties in relating weights and representations forsupergroups.The data of U ( M | N ) is much harder to uncover, though in principle it can bedone with the help of the gauge theory description in [6]. In general, the line defectalong the D5 interface will map to some disorder local operator at the interface betweenChern-Simons theories.We will analyze some basic examples through the rest of the paper and then comeback to the general story in Section 9. Another natural enrichment of the four-dimensional gauge theory setup is to includesurface defects which fill the whole wedge between two interfaces. Gukov-Witten sur-face defects are labelled by a Levi subgroup of the gauge group and have non-trivialcouplings. In the GL-twisted theory the couplings are essentially valued on productsof elliptic curves with modular parameter Ψ, up to some discrete identifications.– 25 –pon reduction to 3d, these GW surface defects are known to implement theinsertion of analytically continued versions of Wilson loops in the Chern-Simons theoryaway from integral weights.At the intersection with the junction, the surface defects will produce a variety ofmodules for the Y L,M,N [Ψ] algebras. We leave a general analysis to future work.
In this section we discuss the VOA associated to junctions in U (1) gauge theory. Thebuilding blocks of the corresponding vertex algebras will be U (1) and U (1 |
1) currentalgebras together with symplectic bosons. U (1) Ψ The simplest example is a U (1) gauge theory defined on the upper right quadrant ofthe plane, with deformed Neumann and Dirichlet boundary conditions at the two sides. U (1) This four-dimensional setup can be related first to three-dimensional analytically continued U (1) Chern-Simons theory atlevel Ψ with standard boundary conditions.In turns, these boundary conditions support a U (1) Ψ currentalgebra, with OPE J Ψ ( z ) J Ψ ( w ) ∼ Ψ( z − w ) (7.1)and Sugawara stress tensor T U (1) Ψ = 12Ψ J Ψ J Ψ (7.2)of central charge 1.Thus we define Y , , [Ψ] ≡ U (1) Ψ (7.3)Notice that the level of a U (1) current is a mere formality. The actual effect of thebulk CS coupling is to determine which boundary vertex operators arise at the end ofa bulk Wilson loop of charge n : they will be vertex operators of charge n under J Ψ .We will come back to that momentarily.There are two other inequivalent definition of the junction vertex algebra whichmust give us the same answer as Y , , [Ψ]: Y , , [Ψ − ] and Y , , [ − Ψ ]. The other threeconfigurations Y , , [ ΨΨ − ], Y , , [1 − Ψ], Y , , [1 − Ψ − ] do not produce a new definition.– 26 –he second definition, Y , , [Ψ − ] gives obviously a U (1) Ψ − current algebra. Wecan identify it with Y , , [Ψ] by the trivial rescaling J Ψ = Ψ J Ψ − (7.4) U (1) The third definition Y , , [ − Ψ ] is more intricate. Recall that thethree-dimensional setup involves a U (1) Chern-Simons theory cou-pled to a single complex free fermion at a two-dimensional interface.According to our prescription, the resulting VOA is the u (1)-BRSTcoset of U (1) − − Ψ × U (1) − ΨΨ − × Ff U (1) × bc , (7.5)with Ff U (1) being the VOA of a single complex free fermion with generators ( ψ, χ ≡ ψ † ).The level are such that total U (1) current J tot = J − − Ψ + J − ΨΨ − + J ψχ (7.6)has level 0. The BRST charge is Q BRST = (cid:73) cJ tot (7.7)In a more conventional language, we would express Y , , [ − Ψ ] ≡ Y , , [ ΨΨ − ] as either ofthe two cosets Y , , [ 11 − Ψ ] ≡ U (1) − − Ψ × Ff U (1) U (1) − Ψ1 − Ψ = U (1) Ψ1 − Ψ × Ff U (1) U (1) − (7.8)The linear combination J Ψ = Ψ J − − Ψ + J − ΨΨ − (7.9)is BRST close, as it has trivial OPE with the total U (1) current. It has level Ψ.We expect it to generate the BRST cohomology and coincide with the current whichappears in the Y , , [Ψ] definition.For completeness, we can compute the vacuum character in the different descrip-tions. The character for a single U (1) current is χ Y , , ( q ) ≡ χ U (1) ( q ) = 1 (cid:81) n> (1 − q n ) (7.10)– 27 –he character for the u (1)-BRST coset can be computed as a Witten index, thoughone has to deal separately with the c ghost zeromode and implement “by hand” theprojection on the global gauge singlets by a contour integral: χ Y , , ( q ) ≡ (cid:73) dz πiz χ U (1) ( q ) χ (cid:48) bc ( q ) χ Ff U (1) ( q ; z ) (7.11)The ghosts (excluding the c zeromode) cancel precisely the U (1) currents contributions.Because of the well-known relation χ Ff U (1) ( q ; z ) ≡ (cid:89) n ≥ (1 − zq n + )(1 − z − q n + ) = 1 (cid:81) n> (1 − q n ) (cid:88) n ( − z ) n q n (7.12)then we recover the desired χ Y , , ( q ) ≡ (cid:73) dz πiz (cid:80) n ( − z ) n q n (cid:81) n> (1 − q n ) = 1 (cid:81) n> (1 − q n ) (7.13) Going back to Y , , [Ψ], the local operators which sit at the end of a boundary Wilsonline in the Neumann boundary correspond to endpoints of charge n Wilson lines in the3d CS theory and thus to charge n “electric” vertex operators for J Ψ : W n ( z ) = V Ψ n ≡ e i n Ψ φ ( z ) (7.14)of conformal dimension ∆ W n = n . Here φ is the bosonization of the current J Ψ = − i∂φ .The dyonic operators are absent in this example. On the other hand, the gaugetheory description of Abelian boundary ’t Hooft operators is simple enough that we canattempt to identify the corresponding three-dimensional configuration and then definedirectly in Y , , [Ψ] the corresponding junction local operators H m .We expect the boundary ’t Hooft lines on the Dirichlet boundary to map to bound-ary local operators in the Chern-Simons theory defined by a Hecke modification of theboundary condition on A ¯ z = 0. In turn, these should map to “magnetic” vertex oper-ators H m ( z ) = V Ψ m Ψ ≡ e imφ ( z ) (7.15)of conformal dimension ∆ H m = Ψ2 m and charge m Ψ under J Ψ .The identification is motivated by the observation that these operators induce thecorrect classical singularity in the boundary value of the connection A z | ∂ = Ψ − J Ψ and that they induce zeroes or poles of order nm in the expectation values of vertexoperators of electric charge n . – 28 –his answer is perfectly compatible with the S-dual description Y , , [Ψ − ]: underthe identification J Ψ = Ψ J Ψ − we see that H m are electric vertex operators of charge m for J Ψ − and W n are magnetic vertex operators of charge n for J Ψ − .It is a bit more interesting to look at the realization of these vertex operators in Y , , [ − Ψ ]. Both sets of boundary line defects in four dimensions map to Wilson linesin the Chern Simons theories on either sides of the interface. Because the U (1) gaugesymmetry is unbroken at the interface, a Wilson line of charge n ending on the interfacefrom either side will need to end on a local operator of charge n in the free fermioninterface theory.In the BRST construction, that is the BRST close combination of a charge n vertexoperator for either U (1) theory and and the simplest charge − n operator O − n built fromthe fermions: W n ≡ V − ΨΨ − n O − n H m ≡ V − − Ψ m O − m (7.16)The dimensions of these BRST close representatives are indeed∆ W n = n − −
1) + n H m = m −
1) + m W n fuse as W n × W n (cid:48) ∼ W n + n (cid:48) and have OPE singularities controlledby nn (cid:48) Ψ − . Similarly, H m fuse as H m × H m (cid:48) ∼ H m + m (cid:48) and have OPE singularitiescontrolled by mm (cid:48) Ψ. On the other hand, W n and H m are mutually local and fuse to M m,n ( z ) = e i ( m + n Ψ ) φ ( z ) (7.18)of conformal dimension ∆ n,m = Ψ2 m + n + nm .Notice as well that these vertex operators define perfectly normal modules for the U (1) Ψ VOA, with no null vectors. The moniker “degenerate” here only indicates thatthey play an analogous role as the degenerate modules for the W N algebra. General vertex operators for U (1) Ψ arise from 4d gauge theory configurations involvinga Gukow-Witten surface defect, which descends to a monodromy defect in the 3dChern-Simons theory and then to the generic vertex operator S p ( z ) = e i p Ψ φ ( z ) (7.19)– 29 –ith the momentum p being the complex combination of the GW defect parameterswhich survives the GL twist. This has dimension ∆ S p = p .Notice that the p parameter is not periodic: although the parameters of the GWdefect are valued in a torus of modular parameter Ψ, p encodes an extra choice ofboundary conditions on the two sides of the corner. We can change these boundaryconditions by fusing the surface defect boundary with boundary Wilson or ’t Hooftlines, which results in shifts of p by n or m Ψ: W n × S p ∼ S p + n H m × S p ∼ S p + m Ψ (7.20)The general vertex operator reduces to the “degenerate” ones when we set p = n Ψ + m ,the values at which the surface defect in the gauge theory description disappears.Under Ψ → Ψ − the parameter p transforms as p → p Ψ − , as expected from theduality properties of the surface defect.In order to realize analogous vertex operators in Y , , [ − Ψ ] we can combine anoperator of momentum ˜ p for U (1) − − Ψ and − ˜ p for U (1) − ΨΨ − . This gives a BRST closedoperator of momentum p = (Ψ − p for J Ψ . The 3d picture is that of a monodromydefect of parameter ˜ p crossing the interface. U (1) corners This is a very instructive example. The final answer for the junction VOA is simple,but it is realized in a very non-trivial manner in all duality frames. For clarity, we willanticipate here the final answer and then detail the derivation in the three alternativeduality frames.We claim that the junction VOA is the product Y , , [Ψ] = U (1) Ψ − − × Sf (7.21)where Sf is a vertex algebra which can be compactly defined as the charge 0 subalgebraof a vertex algebra Sf defined by two “fermionic currents” x ( z ), y ( z ) of dimension 1,OPE x ( z ) y ( w ) ∼ z − w ) (7.22)and charges ± U (1) o symmetry. The vertex algebra Sf appears in a variety of bosonization constructions, includingthe bosonization of symplectic bosons and the bosonization of U (1 |
1) WZW models.That is how it will appear in our construction. We refer the reader to Appendix A fordefinitions and references.The VOA Sf has two natural classes of modules: Equivalently, Sf can be defined as a
P SU (1 |
1) current algebra. – 30 –
The other charge sectors Sf n in Sf. They have highest weight vectors of conformaldimension n + | n | . • The charge 0 sector V xyλ of the twisted modules for Sf. They have highest weightvectors of conformal dimension λ − λ which induce singularities z − λ in x ( z ) and z λ − in y ( z ).with fusion rules Sf n × Sf m ∼ Sf n + m Sf n × V xyλ ∼ V xyλ + n V xyλ × V xyλ (cid:48) ∼ V xyλ + λ (cid:48) + V xyλ + λ (cid:48) − (7.23)We refer the reader to Appendix A for a few more details and references.The W n and D n degenerate modules will result from dressing Sf n respectively witha magnetic operator of charge n or an electric operator of charge − n for U (1) Ψ − − : W n = V U (1) Ψ − − n (Ψ − − O n D n = V U (1) Ψ − − − n O n (7.24)They fuse and braid as expected, with the symplectic fermion operators going alongfor the ride. They have dimensions ∆ W n = n | n | D n = n − Ψ + | n | H s,t will instead involve V xyλ =Ψ s + t : H s,t = V U (1) Ψ − − (1 − Ψ) s V xy Ψ s + t (7.26)Locality with W n and D n is the result of a delicate cancellation between the two ingre-dients of the VOA: the U (1) vertex operator induces singularities of order z ns (1 − Ψ) in W n and z − ns Ψ in D n which cancel the non-integral part of the singularities induced by V xy Ψ s + t on operators in Sf n .Furthermore, we can compare the fusion rules H s,t × H s (cid:48) ,t (cid:48) ∼ H s + s (cid:48) ,t + t (cid:48) + H s + s (cid:48) ,t + t (cid:48) − (7.27)– 31 –ith the fusion rules of u (1 |
1) irreps: typical finite-dimensional representations of u (1 | e, n ) [39], with non-zero e . Under tensor product,the e label is additive. If e + e (cid:48) (cid:54) = 0, the representations multiply as( e, n ) ⊗ ( e (cid:48) , n (cid:48) ) = ( e + e (cid:48) , n + n (cid:48) ) ⊕ ( e + e (cid:48) , n + n (cid:48) −
1) (7.28)If e + e (cid:48) =0 the tensor product is a single decomposable representation. The H s,t thusfuse as u (1 |
1) representations, with t and ˜ t = s + t being the weights of the irrep.The conformal dimension of H s,t is∆ H s,t = (1 − Ψ)Ψ s s + t )(Ψ s + t − s (2 t + s − t ( t − ss (cid:48) + st (cid:48) + s (cid:48) t )Ψ. It is perhaps more naturalto label the operators by t and ˜ t = s + t , so that the pairing is the natural pairing˜ t ˜ t (cid:48) − tt (cid:48) for weights of u (1 | S p,p (cid:48) = V Ψ − − − − p − p (cid:48) V xyp + p (cid:48) (7.30)with fusion rules W n × S p,p (cid:48) ∼ S p + n,p (cid:48) D n × S p,p (cid:48) ∼ S p,p (cid:48) + n H s,t × S p,p (cid:48) ∼ S p +Ψ( s + t ) ,p (cid:48) +(1 − Ψ) t + S p +Ψ( s + t − ,p (cid:48) +(1 − Ψ)( t − (7.31)and dimension ∆ S p,p (cid:48) = p p (cid:48) ) − Ψ − p + p (cid:48) p = n , p (cid:48) = 0 will gives a module which is not quite the same as W n , as the limit λ → − n of V xyλ is a non-trivial extension of two modules, one of whichis W n . Similar considerations apply to the p = 0 and p (cid:48) = m specialization and D m .On the other hand, p = t Ψ and p (cid:48) = Ψ s + (1 − Ψ) t gives directly H s,t .This is as expected from gauge theory: these special values of p and p (cid:48) are all suchthat the surface defect becomes transparent, disappearing away from the interfaces andleaving behind some interface line defects.For completeness, we can present some relevant characters. The character of Sfadmits a useful expansion: χ xy = ∞ (cid:89) n =0 (1 − q n +1 t )(1 − q n +1 t − ) = 1 (cid:81) ∞ n =0 (1 − q n +1 ) ∞ (cid:88) n =0 n (cid:88) m = − n t m ( − n − m q n ( n +1)2 (7.33)– 32 –here t is the U (1) o fugacity. Thus we can write χ Y , , ( q ) = 1 (cid:81) ∞ n =0 (1 − q n +1 ) ∞ (cid:88) n =0 ( − n q n ( n +1)2 (7.34)and χ W m Y , , ( q ) = q ∆ Wm (cid:81) ∞ n =0 (1 − q n +1 ) ∞ (cid:88) n = | m | ( − n − m q n ( n +1)2 χ W m Y , , ( q ) = q ∆ Dm (cid:81) ∞ n =0 (1 − q n +1 ) ∞ (cid:88) n = | m | ( − n − m q n ( n +1)2 (7.35)The character of V xyλ for the full xy VOA is even simpler χ xyλ = 1 (cid:81) ∞ n =0 (1 − q n +1 ) ∞ (cid:88) n = −∞ t n ( − n q ( n − λ )( n − λ +1)2 (7.36)so that χ H s,t Y , , = q ∆ Hs,t (cid:81) ∞ n =0 (1 − q n +1 ) (7.37)and similarly χ S p,p (cid:48) Y , , = q ∆ Hs,t (cid:81) ∞ n =0 (1 − q n +1 ) (7.38)Next, we can derive these facts from the various dual images of the junction. Y , , [Ψ] = Y , , [Ψ − ] = Y , , [ 11 − Ψ ] = Y , , [ ΨΨ − Y , , [1 − Ψ] = Y , , [1 − Ψ − ] (7.39) Y , , [Ψ] description U (1) U (1) The first configuration involves a single set of symplectic bosons(
X, Y ) with OPE X ( z ) Y ( w ) ∼ z − w (7.40)coupled at an interface a U (1) Chern-Simons theory. The defini-tion of Y , , [Ψ] is a u (1) BRST quotient of the product U (1) Ψ × U (1) − Ψ × Sb × bc.We can propose an explicit description of Y , , [Ψ] with the help of the bosonizationrelation between the symplectic boson VOA and the symplectic fermion VOA. The– 33 –osonization relation can be understood as follows: we bosonize the level − J XY =: XY := ∂ϕ XY and write X ( z ) = e ϕ XY ( z ) x ( z ) Y ( z ) = e − ϕ XY ( z ) y ( z ) (7.41)Then the symplectic bosons VOA decomposes as a sum of products of modules ofcharge n for J XY and charge n sectors Sf n in the symplectic fermions VOA.Sb = ⊕ n ∈ Z V U (1) − n ⊗ Sf n (7.42)of the symplectic boson VOA into modules of a U (1) − × Sf subalgebra. We refer thereader to Appendix A for details and references.The BRST quotient only affects the U (1) − current sub-algebra, reducing the prod-uct U (1) Ψ × U (1) − Ψ × U (1) − × bc to a single U (1) current, which can be taken to bethe BRST-closed representative U (1) Ψ − − = J − Ψ − − ΨΨ J Ψ (7.43)or, equivalently, U (1) Ψ1 − Ψ = − J Ψ + Ψ1 − Ψ J − Ψ . (7.44)Thus we arrive to the anticipated claim Y , , [Ψ] = Sf × U (1) Ψ − − (7.45)We can see the bosonization in action in the vacuum characters. We begin fromthe following relation for the character of symplectic bosons: χ XY = 1 (cid:81) ∞ n =0 (1 − q n +1 ) ∞ (cid:88) n =0 n (cid:88) m = − n z m ( − n − m q n ( n +1) − m (7.46)Here z is the fugacity for the U (1) current J XY . The U (1) currents for U (1) Ψ and U (1) − Ψ and the ghosts contributions (except the c zeromode) cancel each other andthe projection to charge 0 leads to the expected character χ Y , , = 1 (cid:81) ∞ n =0 (1 − q n +1 ) ∞ (cid:88) n =0 ( − n q n ( n +1)2 (7.47)– 34 – .2.2 “Degenerate” modules This description of the junction VOA makes it easy to identify W n and D n .A line defect along the NS5 interface ending on the junction maps to a Chern-Simons Wilson loop ending at the interface from the direction of level Ψ. At theinterface it should be attached to a symplectic boson vertex operator of the correctgauge charge. That maps to a charge − n vertex operator for J Ψ combined with asymplectic boson vertex operator of charge n to give a BRST closed candidate for W n .With the help of bosonization, W n can be described as the product of a charge − n magnetic vertex operator for U (1) Ψ − − times a charge n vertex operator in thesymplectic fermions VOA, an element of Sf n , as anticipated. The characters can bereadily matched as well.Similarly, a line defect along ˜ B , ending on the junction maps to a Chern-SimonsWilson loop ending at the interface from the direction of level Ψ −
1. This leads to acharge m vertex operator for J − Ψ combined with a symplectic boson vertex operatorof charge m to give a BRST closed candidate for D m . After bosonization, this is anelectric vertex operator of charge m for U (1) Ψ − − , as anticipated. The characters canbe readily matched as well.In order to produce H s,t as a BRST closed operator in the original Y , , [Ψ] de-scription we can employ a Ramond vertex operator R s Ψ+ t for the symplectic bosons.See Appendix A for a definition. This has U (1) − charge ( s + t + )Ψ + (1 − Ψ)( t + ).If we dress it with a J Ψ vertex operator of charge − ( s + t + )Ψ and a J − Ψ vertexoperator of charge − ( t + )(1 − Ψ) we will get a BRST-closed representative for H s,t .The 4d gauge theory interpretation of these modules seems to be a generalized’t Hooft line defect along the D5 interface, which has magnetic charges ˜ t + and t + in the two half-spaces and involves some non-trivial line defect for the interfacehypermultiplets which somehow produces the R s Ψ+ t module. Y , , [Ψ − ] and relatives U (1) U (1) The second, third, fourth and sixth descriptions in (7.39) involvean interface between an U (1 |
1) and an U (1) Chern-Simons the-ories. In the second and sixth descriptions, we have some U (1)BRST reduction of U (1 | Ψ − × U (1) Ψ − − . In the third and fourthwe have some U (1) BRST reduction of U (1 | (1 − Ψ) − × U (1) Ψ1 − Ψ .In our conventions, a U (1 | κ VOA has currents J = J + 12 κ ( I + J ) J = A J = B J = − I + 12 κ ( I + J ) (7.48)– 35 –ith OPE J ( z ) J ( w ) ∼ κ ( z − w ) J ( z ) A ( w ) ∼ A ( w ) z − wJ ( z ) B ( w ) ∼ − B ( w ) z − wI ( z ) I ( w ) ∼ − κ ( z − w ) I ( z ) A ( w ) ∼ − A ( w ) z − wI ( z ) B ( w ) ∼ B ( w ) z − wA ( z ) B ( w ) ∼ κ ( z − w ) + J ( w ) + I ( w ) z − w (7.49)The central charge is 0. We refer to Appendix A for more details.The BRST reduction employs the current J , whose level 1 − Ψ − cancels theanomaly of U (1) Ψ − − . The BRST close bosonic current surviving the coset can betaken to be J Ψ − − ( z ) = J ( z ) − Ψ2 ( J ( z ) + I ( z )) (7.50)which is local with J , matching what we found in Y , , [Ψ].We can recover the anticipated form of the junction VOA by employing the bosoniza-tion of the U (1 |
1) WZW model, which decomposes it into a sum of products of modulesfor the I and J currents and Sf : U (1 | Ψ − = ⊕ n V J,In, − n ⊗ Sf n (7.51)The u (1)-BRST quotient remodels the U (1) into a single BRST closed currents J Ψ − − ( z )and leaves Sf unaffected, leading to Y , , [Ψ − ] = U (1) Ψ − − × Sf (7.52) The gauge theory description suggests that D m should arise from a U (1) Wilsonloop ending on an operator of appropriate degree built from the boundary value ofa fermionic U (1 |
1) generator and its derivatives. This means a charge − m operator for U (1) Ψ − − combined with an element of the U (1 |
1) VOA of charge m under J ( z ). Ithas charge m under the BRST closed J Ψ − − ( z ) as well and involves a charge m vertexoperator in the ( x, y ) VOA. This agrees with the description of D m in the S-dual frame.– 36 –imilarly, the Wilson loop of the U (1 |
1) Chern-Simons theory maps to a vertexoperator V U (1 | s,t described in Appendix A . The corresponding module contains a de-scendant of the form V J,Is − Ψ2 s, Ψ2 s V xys Ψ+ t which is BRST close and gives the anticipated formof H s,t .Finally, in the S-dual frame we have identified W n as a charge n vertex operatorfor U (1) Ψ1 − Ψ times a charge n vertex operator in the ( x, y ) VOA. That means it shouldhave charge (Ψ − − n under J Ψ − − ( z ).We can engineer this from a nice U (1 | Ψ − module, generated from a bosonizedvertex operator of charge n Ψ − − n under J ( z ) and n under I ( z ), times a charge n vertex operator in the ( x, y ) VOA. This is a descendant of V J,In Ψ − + n , − n and gives asimple BRST closed representative of W n . In 3d, this must correspond to an interfacevortex of some kind. We would like to identify in this context the general vertex operators S p,p (cid:48) . We need torecover the product of a vertex operator of momentum − ΨΨ p (cid:48) − p for J (cid:48) ( z ) and V xy − p − p (cid:48) .We can simply take V U (1 | Ψ − p (cid:48) Ψ − , − p Ψ − and dress it with a U (1) Ψ − − vertex operator ofmomentum Ψ − p to get BRST invariance. This is perfectly reasonable in the gaugetheory. U (1) corners The most symmetric configuration comes from three U (1) factors as in the figure on theleft. In all duality frames the construction of the algebra is the same, up to differentchoices of levels: Y , , [Ψ] = Y , , [Ψ − ] = Y , , [ 11 − Ψ ] == Y , , [ ΨΨ − Y , , [1 − Ψ] = Y , , [1 − Ψ − ] (7.53)In the first duality frame, we need to consider a U (1 |
1) BRST quotient of U (1 | Ψ × Sb | × U (1 | − Ψ+1 (7.54)This setup is rather more intricate than the previous two examples. It is hardto describe BRST closed vertex operators and even harder to make sure they are notBRST exact. The central charge of all ingredients vanishes independently of the valueof Ψ and so does the central charge of Y , , [Ψ]. It is likely that some of these features are linked to the observation that the three D3 brane wedgesin the brane setup can recombine to a single D3 brane and move away from the junction. This maymean that the BRST charge of Y , , [Ψ] could be deformed to make the VOA trivial. – 37 –ne reason to believe that Y , , [Ψ] itself should be non-trivial is that it shouldadmit three sets of degenerate modules W s,t , H s,t and D s,t with non-trivial fusion andbraiding properties.The modules W s,t and D s,t should simply arise from operators built from the sym-plectic bosons and fermions, attached to Chern-Simons Wilson loops which carry thecorresponding irreducible representations of u (1 | V U (1 | Ψ s,t or V U (1 | − Ψ s,t dressed by appropriate combinations of fermions andsymplectic bosons. The overall conformal dimension of these vertex operators cannotvanish.Although a full analysis of the U (1 |
1) BRST quotient goes beyond the scope of thispaper, we can sketch a simpler procedure which we expect to be equivalent to it andto give a conjectural free field realization of Y , , [Ψ]. Intuitively, we bosonize all the U (1 |
1) WZWs and the fermions and symplectic bosons in Sb | and execute the cosetof BRST reduction in two stages: we first deal with the bosonic currents in U (1 |
1) andthen with the leftover fermionic currents.The bosonic reductions are identical to these considered for Y , , [Ψ] and Y , , [Ψ].They leave us with currents I ( z ) and J ( z ) of levels 1 − Ψ − and Ψ − −
1, together withthe xy currents from the symplectic boson and the xy currents from the bosonizedWZW models.From the perspective of a coset, the total fermionic currents A tot ( z ) = A U (1 | Ψ ( z ) + ψX ( z ) B tot ( z ) = B U (1 | Ψ ( z ) + χY ( z ) (7.55)map after the bosonic coset to some combination of the rough form x tot ( z ) = V IJ − Ψ − , Ψ − ( z ) x Ψ ( z ) + V IJ , − ( z ) x ( z ) y tot ( z ) = V IJ Ψ − , − Ψ − ( z ) y Ψ ( z ) + V IJ − , ( z ) y ( z ) (7.56)Here x Ψ , y Ψ denote the symplectic fermions which arise from bosonization of U (1 | Ψ ,with OPE proportional to Ψ.The composite fields x tot ( z ) and y tot ( z ) have the same OPE as free fermionic cur-rents, thanks to a cancellation between the I + J terms in the OPE. Thus we canconsider the coset VOA generated by the currents in the bosonic coset which are localwith x tot and y tot .From the perspective of BRST reduction, one would consider the combination of I , J and three xy systems, together with a BRST charge built with the help of twoauxiliary βγ ghost systems. – 38 – Examples with U (2) gauge groups. × U (1) Y , , [Ψ] U (2) The simplest example involves a U (2) gauge theory in the corner,i.e. Y , , [Ψ]. We already essentially analyzed this setup whenlooking at the three realizations of the Virasoro algebra, but it isinstructive to add the U (1) current algebra in order to get a full U (2) gauge group.Recall that according to our conventions, spelled out in Ap-pendix A, the diagonal current J ( z ) + J ( z ) in U (2) Ψ has level2Ψ while the SU (2) currents have level Ψ −
2. The OPE between J aa and J aa goes as Ψ − z − w ) . The OPEs between Cartan generators J and J take the form J ( z ) J (0) ∼ Ψ − z − w ) J ( z ) J (0) ∼ z − w ) J ( z ) J (0) ∼ Ψ − z − w ) . (8.1)The definition of Y , , [Ψ] involves the quantum DS reduction of U (2) Ψ by theregular su (2) embedding. It produces the combination of the Virasoro VOA with b = − Ψ and a U (1) current.It is instructive to follow this at the level of vacuum characters. We begin with the U (2) vacuum character χ U (2) Ψ ( z , z ; q ) = 1 (cid:81) n> (1 − q n ) (1 − z z q n )(1 − z z q n ) . (8.2)We add the ghost contribution and then adjust the Cartan fugacities z i to z = qz = q z in order to account for the shift of the stress tensor which gives dimension 0 to J and the symmetry breaking enforced by J = 1: χ W [ U (2) Ψ ] ( z ; q ) = 1 − z z (cid:81) n> (1 − q n ) = 1 − q (cid:81) n> (1 − q n ) (8.3)which is the expected vacuum character for Virasoro times U (1). Notice that the crucialfactor of 1 − q arises from the c zeromode which does not cancel against the off-diagonalcurrent contributions.The central charge of Y , , [Ψ] is, as expected, c = 3 Ψ −
2Ψ + 1 − − −
2) = 14 − − − (8.4)– 39 – .1.2 Modules A spin j vertex operator for SU (2) Ψ − can be combined with momentum p vertexoperators for U (1) to give a vertex operator of dimension j ( j +1)+ p / . It is naturalto define j = ( µ − µ ) / p = µ + µ to get a vertex operator V ( µ ,µ ) labelled bythe U (2) highest weight ( µ , µ ) with µ ≥ µ . The conformal dimension of the highestweight vector is controlled by the U (2) Casimir:∆ µ ,µ = µ + µ
2Ψ + µ − µ . (8.5)A Wilson loop ending at the boundary in the 3d Chern-Simons theory will give risein 2d to the DS reduction of the module generated by V ( µ ,µ ) . This is our definition of W µ . The resulting degenerate modules have conformal dimension∆ W µ = µ + µ − µ µ − µ
2Ψ + µ χ U (2) Ψ µ ,µ ( z , z ; q ) = q µ µ + µ − µ z µ +11 z µ − z µ +12 z µ ( z − z ) (cid:81) n> (1 − q n ) (1 − z z q n )(1 − z z q n ) (8.7)As before, we add the ghost contribution and adjust the fugacities to z = qz = q z in order to account for the shift of the stress tensor which gives dimension 0 to J andthe symmetry breaking enforced by J = 1: χ W [ U (2) Ψ ] µ ,µ ( z ; q ) = q µ µ + µ − µ z µ z µ − z µ +12 z µ − (cid:81) n> (1 − q n ) = z µ + µ q ∆ Wµ − q µ − µ +1 (cid:81) n> (1 − q n ) (8.8)which is the expected degenerate character for Virasoro times U (1), with a null vectorat level µ − µ + 1.The realization of the second family of degenerate modules H ν is less obvious,but still straightforward. We propose to combine spectral flow images of the vacuummodule of SU (2) Ψ − and magnetic vertex operators (possibly with half-integral charge)for U (1) . The former are simply vertex operators for the bosonized Cartan currentin SU (2) Ψ − , with momenta multiple of Ψ −
2. These have charges − ν (Ψ − − ν and − ν (Ψ − − ν under J and J . At the level of characters, we have˜ χ U (2) Ψ ν ,ν ( z , z ; q ) = z − ν (Ψ − − ν z − ν (Ψ − − ν q Ψ ν ν − ( ν − ν (cid:81) n> (1 − q n ) (1 − z z q n + ν − ν )(1 − z z q n + ν − ν ) (8.9)– 40 –ext, we implement the DS reduction, which is well defined for ν > ν . The resultis ˜ χ W [ U (2) Ψ ] ν ,ν ( z ; q ) = ( − ν − ν z − Ψ( ν + ν ) q ∆ Hν − q ν − ν +1 (cid:81) n> (1 − q n ) (8.10)where we simplified ν − ν ratios of the form − z z q k − z z q − k .We recognize the expected character for the degenerate operators H µ of dimension∆ H ν = ν + ν − ν ν − ν ν W µ and H µ (cid:48) vertex operators are mutually local.We expect that general modules S p can be obtained from the DS reduction ofhighest weight modules for U (2) Ψ with generic, non-integral weights p and p . Inparticular, they are associated to infinite highest weight representations of the zeromodealgebra. At the level of characters,ˆ χ U (2) Ψ p ,p ( z , z ; q ) = q p p + p − p z p z p (1 − z z ) (cid:81) n> (1 − q n ) (1 − z z q n )(1 − z z q n ) (8.12)The usual manipulations lead to the obviousˆ χ W [ U (2) Ψ ] p ,p ( z ; q ) = z p + p q p p + p − p + p − p (cid:81) n> (1 − q n ) (8.13)of dimension p + p
2Ψ + p − p − −
1) (8.14) Y , , [Ψ] U (2) In this duality frame we have a U (2) BRST quotient of U (2) − × U (2) − − × Ff U (2) (8.15)where Ff U (2) denotes a pair of complex fermions. The coset of thenon-Abelian part is essentially the analytic continuation of the an-alytically continued Virasoro minimal model coset SU (2) − − × SU (2) SU (2) − − . (8.16)– 41 –hus we expect the BRST coset to give again the product of Virasoro and an a U (1)current with the correct total central charge c = 3 (cid:18) − − (cid:19) (Ψ −
1) + 1 + 2 − (cid:18) − − (cid:19) (1 − Ψ − ) − − − − (8.17)The U (1) current can be taken to be the combination of diagonal currents J ( z ) = Ψ J − + J − − (8.18)It is instructive to follow this at the level of vacuum characters. We start from theproduct of characters χ U (2) ( z , z ; q ) χ Ff ( z , z ; q ) = (cid:81) n> (1 − q n + z )(1 − q n + z − )(1 − q n + z )(1 − q n + z − ) (cid:81) n> (1 − q n ) (1 − z z q n ) (1 − z z q n ) (8.19)The u (2) ghosts cancel out the whole denominator. The c ghost zeromodes contributea Vandermonde determinant for the projection to gauge-invariant operators χ Y , , ( q ) = (cid:73) dz dz z z (1 − z z )(1 − z z ) (cid:89) n> (1 − q n + z )(1 − q n + z − )(1 − q n + z )(1 − q n + z − )(8.20)Expanding the product through a basic theta function identity gives the desired answer χ Y , , ( q ) = (cid:73) dz dz z z (1 − z z )(1 − z z ) (cid:80) n ,n ( − n + n z n z n q n n (cid:81) n> (1 − q n ) (8.21)i.e. χ Y , , ( q ) = 1 − q (cid:81) n> (1 − q n ) (8.22) The W µ ,µ and H ν ,ν vertex operators descend from the corresponding electric vertexoperators in either U (2) VOA, dressed appropriately with the free fermions to makethem gauge invariant.For example,∆ W µ = µ + µ − −
1) + µ − µ − −
1) + µ + µ U (2) − − – 42 –t the level of characters, χ Y , , W µ ( q ) = q ∆ µ (cid:73) dz dz z z (1 − z z )( z µ z µ − z µ +12 z µ − ) (cid:80) n ,n ( − n + n z n z n q n n (cid:81) n> (1 − q n ) (8.24)i.e. χ Y , , W µ ( q ) = q ∆ µ q µ µ − q ( µ µ − (cid:81) n> (1 − q n ) = q ∆ Wµ − q µ − µ (cid:81) n> (1 − q n ) (8.25)In a similar manner, general modules S p arise from a combination of modules ofgeneral non-integral weights for both U (2)’s. In order to get a gauge-invariant combi-nation, we need to combine Weyl modules induced from a highest weight representationof one U (2) and a lowest weight representation of the other U (2), with the same weight(˜ p , ˜ p ).The resulting conformal dimension is˜ p + ˜ p − −
1) + ˜ p − ˜ p − −
1) + ˜ p + ˜ p − −
1) + ˜ p − ˜ p − −
1) (8.26)i.e. in terms of p i = (1 − Ψ)˜ p i p + p
2Ψ + p − p − −
1) (8.27) × U (1)Next, we look at the VOA realized by Y , , [Ψ] = Y , , [Ψ − ] = Y , , [ 11 − Ψ ] == Y , , [ ΨΨ − Y , , [1 − Ψ] = Y , , [1 − Ψ − ] (8.28)The result will be a combination of a U (1) current at level Ψ − (Ψ − −
2) andthe analytic continuation Pf Ψ − of a well known VOA Pf k of Z k parafermions.We will encounter two well-known coset constructions of parafermions and a lesswell-known one. In the process, we will define three families of modules which combinewith U (1) vertex operators into the degenerate modules W µ ,µ , D s and H ν ,ν ,ν . Y , , and parafermions. – 43 – (2) U (1) The first realization involves a U (1) BRST quotient of U (2) Ψ × U (1) − Ψ (8.29)by the sum of the J , which has level Ψ −
1, and the U (1) − Ψ current.The BRST cohomology of the vacuum module can only involvevertex operators in U (2) Ψ of J charge 0. That sub-algebra factor-izes as U (1) J × U (1) J × Pf Ψ − (8.30)where Pf κ is by definition the coset VOAPf κ ≡ SU (2) κ U (1) κ (8.31)For integral κ , this is known as the Z κ -parafermion VOA.The BRST quotient, as usual, reduces the tree U (1) currents U (1) J × U (1) J × U (1) − Ψ to a single U (1) current. A convenient choice J c ( z ) = (1 − Ψ − ) J ( z ) − Ψ − J ( z ) (8.32)has level Ψ − (Ψ − −
2) and gives integral charges to the off-diagonal WZW currents.The Pf Ψ − VOA goes along for the ride and thus we can write Y , , [Ψ] = U (1) Ψ − (Ψ − − × Pf Ψ − (8.33) c [ Y , , [Ψ]] = 3 − − (8.34)Computing the vacuum character requires some judicious manipulations of the U (2) character: χ [ Y , , ] = (cid:73) dzz (cid:81) n> (1 − q n )(1 − zq n )(1 − z − q n ) == (cid:73) dzz − z (cid:81) ∞ n =0 (1 − q n +1 ) ∞ (cid:88) n =0 n (cid:88) m = − n z m ( − n − m q n ( n +1) − m ( m +1)2 == 1 (cid:81) ∞ n =0 (1 − q n +1 ) (cid:32) ∞ (cid:88) n =1 ( − n q n ( n +1)2 (cid:33) (8.35) Recall the conventions U (2) Ψ = U (1) × SU (2) Ψ − and see appendix A. – 44 – .2.2 Degenerate modules The W µ modules will be the BRST reduction of the V µ modules for U (2) Ψ times thevacuum module of U (1) − Ψ .The basic BRST closed representative will involve a vector of weight (0 , µ + µ ) in V µ ,µ . It has charge (1 − Ψ − )( µ + µ ) under J c . The dimension of the highest weightvector is different depending on (0 , µ + µ ) being an element of ( µ , µ ) irrep of the u (2) current zeromodes or not. Recall the dimension of the U (2) Ψ primaries∆ µ ,µ = µ + µ
2Ψ + µ − µ . (8.36)We can compute the character as before. χ W µ [ Y , , ]( y ; q ) = q ∆ µ ,µ (cid:73) dzz z µ − z µ +1 (1 − z ) (cid:81) n> (1 − q n )(1 − zq n )(1 − z − q n ) == q ∆ µ ,µ (cid:73) dzz z µ − z µ +1 (cid:81) ∞ n =0 (1 − q n +1 ) ∞ (cid:88) n =0 n (cid:88) m = − n z m ( − n − m q n ( n +1) − m ( m +1)2 == q ∆ µ ,µ (cid:80) ∞ n = | µ | ( − n + µ q n ( n +1) − µ µ + (cid:80) ∞ n = | µ | ( − n + µ q n ( n +1) − µ µ − (cid:81) ∞ n =0 (1 − q n +1 ) (8.37)The SU (2) /U (1) parafermion VOA has modules M j,m which arise from vertexoperators of weight m in the SU (2) module of spin j . Here we take such a modulewith j = µ − µ and m = − µ + µ and dress it with a U (1) vertex operator of charge(1 − Ψ − )( µ + µ ).The D s modules will be the BRST reduction of the vacuum modules for U (2) Ψ − times the charge s module of U (1) − Ψ . It has charge s under J c . The highest weightvector arises from s powers of an off-diagonal current and thus∆ D s = s − Ψ + | s | (8.38)We can compute the character as before: χ D s [ Y , , ] = q s
22 11 − Ψ (cid:73) dzz z s (cid:81) n> (1 − q n )(1 − zq n )(1 − z − q n ) == q s
22 11 − Ψ (cid:73) dzz z s − z s +1 (cid:81) ∞ n =0 (1 − q n +1 ) ∞ (cid:88) n =0 n (cid:88) m = − n z m ( − n − m q n ( n +1) − m ( m +1)2 == q s
22 11 − Ψ (cid:16)(cid:80) ∞ n = s ( − n + s q n ( n +1) − s ( s − + (cid:80) ∞ n = s +1 ( − n + s q n ( n +1) − s ( s +1)2 (cid:17)(cid:81) ∞ n =0 (1 − q n +1 ) (8.39)– 45 –otice that the sums are economical for s ≥
0. For negative s the first 2 s terms in eachsum cancel pairwise and the summations can start from − s and − s − s . Changing the first sumin that manner, we can rewrite the sum as χ D s [ Y , , ] = q s
22 11 − Ψ (cid:16)(cid:80) ∞ n = s ( − n + s q n ( n +1) − s ( s − + (cid:80) ∞ n = s +1 ( − n + s q n ( n +1) − s ( s +1)2 (cid:17)(cid:81) ∞ n =0 (1 − q n +1 ) = q s
22 11 − Ψ (cid:16)(cid:80) ∞ n =0 ( − n q n ( n − s +1)2 + (cid:80) ∞ n =1 ( − n q n ( n +2 s +1)2 (cid:17)(cid:81) ∞ n =0 (1 − q n +1 ) (8.40)The SU (2) /U (1) parafermion VOA has modules M s which arise from vertex oper-ators of weight s in the SU (2) vacuum module. These are essentially the parafermionsthemselves. Here we take such a module and dress it with a J c vertex operator ofelectric charge s .The mutual locality between W µ and D s is obvious before the BRST coset. In termsof parafermion and U (1) modules, it follows from a conspiracy between the braidingphases of individual factors.We will not attempt a direct construction of the H ν modules here. We will find acandidate S-dual description of H ν in the next section. Y , , and parafermions. U (2) U (1) The second realization gives a construction that combines DS-reduction with U (1) coset Y , , [Ψ − ] ≡ W | [ U (2 | Ψ − ] U (1) − Ψ − : c = 3 −
6Ψ (8.41)The DS reduction step is well-understood: it is known to give theproduct of a U (1) current and a N = 2 super Virasoro algebra,generated by the stress-energy tensor T , two fermionic generators G ± of conformaldimension and a U (1) current [40]. The super Virasoro algebra has central charge c = 3 − .Thus we can write a simplified definition Y , , [Ψ − ] = sVir N =2 [ c = 3 − ] × U (1) Ψ − U (1) − Ψ − (8.42)– 46 –n turn, the BRST cohomology of the vacuum module can only involve vertex operatorsin sVir N =2 of U (1) charge 0. The parafermion algebra is known to arise as a coset [41]Pf κ = sVir N =2 [ c = 3 − κ +2 ] U (1) − κ +2 (8.43)and thus we recover the expected Y , , [Ψ − ] = U (1) Ψ − (Ψ − − × Pf Ψ − (8.44)A direct calculation of the vacuum character of Y , , is also reassuring. The vacuumcharacter for U (2 |
1) is χ [ U (2 | z , z , w ; q ) = (cid:81) n> (1 − q n z w )(1 − q n z w )(1 − q n wz )(1 − q n wz ) (cid:81) n> (1 − q n ) (1 − z z q n )(1 − z z q n ) (8.45)Adding the bc ghosts for J and the βγ ghosts for J we get a simpler product χ [ U (2 | × DS ghosts]( z , z , w ; q ) = (1 − z z ) (cid:81) n> (1 − q n z w )(1 − q n wz )(1 − wz ) (cid:81) n> (1 − q n ) (8.46)The shift of the stress tensor which makes J of dimension 0 and J of dimension and the reduction of symmetry enforced by J = 1 are implemented in the characterby the specialization of fugacities z = qz = q z : χ [ W | U (2 | z, w ; q ) = (1 − q ) (cid:81) n> (1 − q n + zw )(1 − q n + wz ) (cid:81) n> (1 − q n ) (8.47)which is the vacuum character for N = 2 super-Virasoro times U (1).Notice that the current J has initial level 2 − Ψ − in our conventions. The βγ ghostsystem contributes an extra − I after the DS reductionof level 1 − Ψ − . The current J + J has initial level 2Ψ − + 2. The βγ ghost systemcontributes an extra − J after the DS reduction of level2Ψ − + 1. The OPE coefficient in ˜ I ˜ J is 1, shifted from 2 by the βγ contribution.The next step is the quotient by U (1) − Ψ − given by ˜ I . We need to compute a– 47 –ontour integral χ [ Y , , ]( z, w ; q ) = (1 − q ) (cid:81) n> (1 − q n ) (cid:73) dzz (cid:89) n> (1 − q n + z )(1 − q n + z − )= (1 − q ) (cid:81) n> (1 − q n ) (cid:73) dzz (cid:80) ∞ n = −∞ ( − n z n q n (1 − q z )(1 − q z − )= (1 − q ) (cid:81) n> (1 − q n ) (cid:73) dzz (cid:88) a,b ≥ ∞ (cid:88) n = −∞ q a + b z a − b ( − n z n q n = (1 − q ) (cid:81) n> (1 − q n ) (cid:88) a,b ≥ q a + b ( − a + b q ( a − b )22 = 1 (cid:81) ∞ n =0 (1 − q n +1 ) (cid:32) ∞ (cid:88) n =1 ( − n q n ( n +1)2 (cid:33) (8.48)Adding up the contributions for fixed a − b one gets a geometric series which cancelsthe 1 − q prefactor, leaving the expected answer on the last row, the same as χ [ Y , , ]The current (1 − Ψ − ) ˜ J − ˜ I is local with ˜ I and survives the quotient. It has level(1 − Ψ − ) (2Ψ − + 1) − (1 − Ψ − ) = Ψ − (Ψ − − J c = (Ψ −
1) ˜ J − Ψ ˜ I (8.49)to match with the current which appears in Y , , . In this realization, H ρ and D s are simply given by the BRST reduction of U (2 | U (1) Weyl modules built from irreducible finite-dimensional representations of thecorresponding Lie algebras.Notice that the charge s module for U (1) Ψ − − dressed by appropriate powers of– 48 –he off-diagonal currents will have charge s under J c . The characters are computed as χ D s [ Y , , ]( z, w ; q ) = q s
22 Ψ1 − Ψ (1 − q ) (cid:81) n> (1 − q n ) (cid:73) dzz z s (cid:89) n> (1 − q n + z )(1 − q n + z − )= q s
22 Ψ1 − Ψ (1 − q ) (cid:81) n> (1 − q n ) (cid:73) dzz z s (cid:80) ∞ n = −∞ ( − n z n q n (1 − q z )(1 − q z − )= q s
22 Ψ1 − Ψ (1 − q ) (cid:81) n> (1 − q n ) (cid:73) dzz z s (cid:88) a,b ≥ ∞ (cid:88) n = −∞ q a + b z a − b ( − n z n q n = q s
22 Ψ1 − Ψ (1 − q ) (cid:81) n> (1 − q n ) (cid:88) a,b ≥ q a + b ( − a + b + s q ( a − b + s )22 = q s
22 Ψ1 − Ψ (cid:81) ∞ n =0 (1 − q n +1 ) (cid:32) ∞ (cid:88) n =0 ( − n + s q ( n + s )2+ n + ∞ (cid:88) n =1 ( − n + s q ( n − s )2+ n (cid:33) (8.50)and match the S-dual description.The finite-dimensional irreducible representation of u (2 |
1) we will use to definethe H ν modules are Kac modules labelled by a weight ( ν , ν , ν ). They are familiarin physics: one splits the odd generators in two halves, pick an irrep of the bosonicsubalgebra and declare it annihilated by half of the odd generators. The rest of themodule is built by acting with the other half of the odd generators.The character of a Weyl module V ν of this type should take the form χ V ν [ U (2 | z , z , w ; q ) = q ∆ ν w ν z ν z ν − z ν +12 z ν − − z z ·· (1 − z w )(1 − wz ) (cid:81) n> (1 − q n z w )(1 − q n z w )(1 − q n wz )(1 − q n wz ) (cid:81) n> (1 − q n ) (1 − z z q n )(1 − z z q n ) (8.51)Adding the ghosts and specializing the fugacities for the DS reduction gives χ V ν [ W | U (2 | z, w ; q ) = q ∆ (cid:48) ν w ν z ν + ν (1 − q ν − ν +1 ) (cid:81) n ≥ (1 − q n + zw )(1 − q n + wz ) (cid:81) n> (1 − q n ) (8.52)This appears to be the character of a degenerate module for N = 2 super-Virasorotimes U (1), with a highest weight vector of generic U (1) charges and a single nullvector at level ν − ν + 1.Taking next the U (1) quotient we get a very simple character χ [ Y , , ] H ν ( z ; q ) = ( − ν q ∆ (cid:48) ν + ν z ν + ν + ν − q ν − ν +1 (cid:81) n> (1 − q n ) (8.53)– 49 –his description of the VOA makes W µ into a possibly intricate magnetic object.It would be interesting to reconstruct the ancestor U (2 | Ψ − module and give a gaugetheory interpretation. Y , , and parafermions. U (1) U (2) The last realization of the VOA leads to an interface between U (2 |
1) supergroup Chern-Simons theory and U (2) Chern-Simonstheory, with with U (2) embedded inside U (2 |
1) in the obvious block-diagonal way. This configuration leads to the BRST coset Y , , [ 11 − Ψ ] = U (2 | − − Ψ U (2) − − Ψ +1 : c = −
6Ψ + 3 (8.54)and we can see that central charge of the theory matches the previous two realizations.The emergence of the parafermion VOA from such a coset is less familiar than theprevious constructions, but it known [42]. Essentially, the para-fermions are used todress spin modules for U (2) in order to assemble the odd currents of U (2 | J has level − Ψ + 2 = − − Ψ and OPE coefficient 2 with J + J ,which has level − − Ψ + 2 = − − Ψ . The BRST-close combination Ψ1 − Ψ J + J + J haslevel Ψ (2Ψ − − − ΨΨ − = Ψ(Ψ − − . Hence we can identify tentatively J c = (Ψ − J + (Ψ − Ψ ( J + J ) (8.55)The match of vacuum characters is striking. We begin with the familiar χ [ U (2 | z , z , w ; q ) = (cid:81) n> (1 − q n z w )(1 − q n z w )(1 − q n wz )(1 − q n wz ) (cid:81) n> (1 − q n ) (1 − z z q n )(1 − z z q n ) (8.56)In order to execute the (BRST) coset we multiply by the U (2) character and the ghosts,take the contour integral χ [ V , , ]( w ; q ) = (cid:73) dz dz z z (1 − z z )(1 − z z ) (cid:81) n> (1 − q n z w )(1 − q n z w )(1 − q n wz )(1 − q n wz ) (cid:81) n> (1 − q n ) == (cid:73) dz dz z z (1 − z z )(1 − z z ) (cid:80) n,m ≥ (cid:80) ns = − n (cid:80) mt = − m ( − n + m q n ( n +1)+ m ( m +1)2 w s + t z − s z − t (cid:81) n> (1 − q n ) == (cid:80) n,m ≥ ( − n + m q n ( n +1)+ m ( m +1)2 − (cid:80) n,m ≥ ( − n + m q n ( n +1)+ m ( m +1)2 (cid:81) n> (1 − q n ) == 1 + 2 (cid:80) n> ( − n q n ( n +1)2 (cid:81) n> (1 − q n ) (8.57)– 50 –nd obtain the expected vacuum character. This realization of the VOA should gives a simple description of H ν and W µ in termsof standard representations of U (2 |
1) and U (2).For example, we can compute the character for W µ : χ W µ [ V , , ]( w ; q ) = q ∆ µ (cid:73) dz dz z z (1 − z z )( z µ z µ − z µ +12 z µ − ) ·· (cid:81) n> (1 − q n z w )(1 − q n z w )(1 − q n wz )(1 − q n wz ) (cid:81) n> (1 − q n ) == q ∆ µ (cid:73) dz dz z z (1 − z z )( z µ z µ − z µ +12 z µ − ) ·· (cid:80) n,m ≥ (cid:80) ns = − n (cid:80) mt = − m ( − n + m q n ( n +1)+ m ( m +1)2 w s + t z − s z − t (cid:81) n> (1 − q n ) == q ∆ µ w µ + µ (cid:16)(cid:80) n ≥| µ | ,m ≥| µ | − (cid:80) n ≥| µ +1 | ,m ≥| µ − | (cid:17) ( − n + m q n ( n +1)+ m ( m +1)2 (cid:81) n> (1 − q n ) (8.58)It is not hard to match this with the dual calculation.We can also compute again the character for H ν : χ H ν [ V , , ]( w ; q ) = q ∆ ν w ν (cid:73) dz dz z z (1 − z z )( z ν z ν − z ν +12 z ν − )(1 − z w )(1 − wz ) ·· (cid:81) n> (1 − q n z w )(1 − q n z w )(1 − q n wz )(1 − q n wz ) (cid:81) n> (1 − q n ) == q ∆ ν w ν (cid:73) dz dz z z (1 − z z )( z ν z ν − z ν +12 z ν − ) ·· (cid:80) ∞ n,m = −∞ ( − n + m q n ( n +1)+ m ( m +1)2 w n + m z − n z − m (cid:81) n> (1 − q n ) == ( − ν + ν q ∆ ν w ν + ν + ν q ν ν ν ν − q ( ν ν ν ν − (cid:81) n> (1 − q n ) = ( − ν + ν q ∆ ν + ν ν ν ν w ν + ν + ν − q ν − ν +1 (cid:81) n> (1 − q n ) (8.59)which reduces again to the dual result – 51 – Central charges, characters and 3d partitions
We are now ready for some preliminary investigation of the general L , M , N setup. Webegin by computing the central charge of the VOA and checking its duality invariance. The definition of Y L,M,N [Ψ] is somewhat different depending on the relative magnitudeof N and M . The final expression for the corresponding central charge c L,M,N [Ψ] willhold uniformly for all cases: c L,M,N [Ψ] = 12 1Ψ ( L − N ) (cid:0) ( L − N ) − (cid:1) + 12 (1 −
1Ψ )( N − L ) (cid:0) ( N − L ) − (cid:1) ++ 12 Ψ( M − N ) (cid:0) ( M − N ) − (cid:1) + 12 (1 − Ψ)( N − M ) (cid:0) ( N − M ) − (cid:1) + 12 11 − Ψ ( L − M )(( L − M ) −
1) + 12 ΨΨ − M − L )(( M − L ) − L − N − M )(2 M − N − L )(2 N − L − M ) (9.1)which is manifestly S -symmetric.The calculation is straightforward, but the details are somewhat tedious. Wepresent them in Appendix B. Notice that the answer only depends on the differencesbetween L , M , N . Concretely, this happens because the central charge of U ( N | M ) Ψ only depends on | N − M | : c U ( N | M ) Ψ = 1 + Ψ − N + M Ψ (cid:0) ( N − M ) − (cid:1) (9.2)More conceptually, it is likely a consequence of the fact that full D3 branes can becontinuously added or removed from the system without breaking supersymmetry. Next, we can look at vacuum characters. For simplicity, we will focus at first on thesituation where at least one of the three labels L , M , N vanishes. This allows us toavoid dealing with the subtleties of superghost zeromodes in the BRST reductions. Atthe end, we will give some conjectural statements about general L , M , N . Y ,M,N . At first, we can consider the N = M subcase.In order to compute the character, we start with the product of the vacuum char-acters of two U ( N ) WZWs and the N symplectic bosons, as a function of fugacities x i – 52 –or the Cartan generators. The u ( N )-valued ghost non-zeromodes precisely cancel thecontributions to the character of the two sets of WZW currents.We trade the c zeromode contributions for a contour integral projecting on u ( N )invariants: χ [ V ,N,N ]( q ) = 1 N ! (cid:73) N (cid:89) i =1 dx i x i (cid:81) i
1. Now the vacuumcharacter becomes χ [ V ,M,M +1 ]( q ) = 1 M ! 1 (cid:81) n> (1 − q n ) (cid:73) M (cid:89) i =1 dx i x i (cid:81) i
Next, we want to count 3d partitions restricted to lie in the difference between thepositive octant and the shifted positive octant with origin at z = 0, x = M , y = N , M < N .As we take a diagonal slicing, we get a sequences µ n of interlaced 2d partitionswhich are restricted to lie in the vertical 2d slab 0 ≤ z , 0 ≤ w ≤ N for n ≥ ≤ z , 0 ≤ w ≤ N + n for M − N ≤ n ≤ ≤ z , 0 ≤ w ≤ M for n ≤ M − N .– 58 –he counting function e λ ( q ) for the sequence µ n for n ≥ M − N is (cid:88) λ e λ ( q ) χ U ( M ) λ t ( x i ) = (cid:88) λ d U ( N ) µ ( q ) χ U ( N ) µ t ( y i = q N − M x i , y M + j = q N − M − j + ) == 1 (cid:81) n ≥ (cid:81) Mi =1 (1 − x i q n + N − M + ) (cid:81) N − Mi =1 (1 − q n + i ) (9.23)The inner product with the counting function d U ( M ) λ ( q ) for the sequence µ n for n ≤ M − N gives precisely the contour integral representation 9.9 of χ [ V ,M,N ]( q )! Y L, ,N Now we are ready to slice the previous 3d partitions along a different axis. We want tocount 3d partitions restricted to lie in the difference between the positive octant andthe shifted positive octant with origin at z = L , x = 0, y = N , M < N .The sequence of 2d partitions now includes partition restricted to the 2d L-hook R n,L : the difference between the positive quadrant and the shifted positive quadrantwith origin at z = L , w = n . These can be identified with characters χ λ ( x i ; y a ) ofirreducible representations of U ( n | L ). We will assume now some simple combinatorialrelations which generalize the relations we used until now for U ( n ) characters.In this particular setup, the slicing of a restricted 3d partition gives a sequence ofinterlaced 2d partitions which are restricted to lie in R N,L for n ≥ R N + n,L for − N ≤ n ≤ ≤ z ≤ L , 0 ≤ w for n ≤ − N .We expect the following combinatorial statement to be true χ µ ( x i ; y a ) (cid:81) La =1 (1 + y a ) (cid:81) Nj =1 (1 − x j ) = (cid:88) ν (cid:31) µ χ ν ( x i ; y a ) (9.24)We also expect χ U ( n | L ) µ ( x i = x (cid:48) i , x n = 1; y a ) = (cid:88) ν ≺ µ χ U ( n − | L ) ν ( x (cid:48) i ; y a ) (9.25)These statements then imply that the coefficients in (cid:81) n ≥ (cid:81) Ni =1 (1 + y a q n + ) (cid:81) n ≥ (cid:81) Ni =1 (1 − x i q n + ) = (cid:88) λ d U ( n | L ) λ ( q ) χ λ ( x i ; y a ) (9.26)counts as before the sequences of interlaced 2d partitions µ n , n ≥ µ = λ , butrestricted to R N,L . – 59 –hey also imply that the coefficients in (cid:81) n ≥ (cid:81) Ni =1 (1 + y a q N + n + ) (cid:81) n ≥ (cid:81) Ni =1 (1 − q n + i ) = (cid:88) λ f λ ( q ) χ U ( L ) λ t ( y a ) (9.27)counts as before the sequences of interlaced 2d partitions µ n , n ≥ − N , µ − N = λ , to liein R N,L for n ≥ R N + n,L for − N ≤ n ≤ Y L, ,N then coincides with the inner product χ [ V L, ,N ]( q ) = (cid:88) λ c λ ( q ) f λ ( q ) (9.28)i.e. the counting function of the restricted 3d partitions. Y L,M,N
We want to count 3d partitions restricted to lie in the region R L,M,N , defined as thedifference between the positive octant and the shifted positive octant with origin at z = L , x = M , y = N , M ≤ N . x yz Figure 6 . Example of a 3d partition for algebra Y , , [Ψ]. All the boxes of all allowedpartitions are constrained to lie between the corner with a vertex at the origin and shifted(red) corner with vertex at (2 , , We can proceed as before. The slicing of a restricted 3d partition gives a sequenceof interlaced 2d partitions which are restricted to lie in R N,L for n ≥ R N + n,L for M − N ≤ n ≤ R M,L for n ≤ − N .The coefficients in (cid:81) n ≥ (cid:81) Ni =1 (1 + y a q N − M + n + ) (cid:81) n ≥ (cid:81) Mi =1 (1 − x i q N − M + n + ) (cid:81) N − Mi =1 (1 − q n + i ) = (cid:88) λ f U ( M | L ) λ ( q ) χ U ( M | L ) λ ( x i ; y a ) (9.29)– 60 –ounts as before the sequences of interlaced 2d partitions µ n , n ≥ M − N , µ M − N = λ ,to lie in R N,L for n ≥ R N + n,L for M − N ≤ n ≤ (cid:88) λ d U ( M | L ) λ ( q ) f U ( M | L ) λ ( q ) (9.30)is the natural projection to U ( M | L ) invariants of the character for the ingredients ofthe BRST reduction defining Y L,M,N . We expect it to be the correct vacuum characterfor Y L,M,N . It is straightforward to modify the vacuum character calculations in order to computethe characters of degenerate modules of type W or D : essentially, one just insertscharacters of finite-dimensional irreducible representations in the contour integrals, withfugacities associated to DS-reduced directions specialized to the appropriate powers of q . There is an obvious extension of the 3d partition counting problem we associatedto the vacuum characters of Y L,M,N : one may consider 3d partitions with semi-infinitecylindrical ends modelled on 2d partitions λ , µ , ν , as in the definition of the topologicalstring vertex [11].The crucial observation is that the restriction for the 3d partition to lie in theregion R L,M,N forces λ , µ , ν to lie respectively in R M,N , R N,L and R L,M . Thus λ , µ , ν have precisely the same form as the data labelling our degenerate modules M λ,µ,ν = W µ × H λ × D ν .It is nice to observe that the 3d counting for λ = 0 is particularly simple, in thesame way as the computation of characters of W µ × D ν is particularly simple.Up to “framing factors”, the computation simply inserts extra factors of χ U ( N | L ) µ and χ U ( M | L ) µ in order to implement the boundary conditions on the 2d partitions at largepositive and negative n . This is precisely the same as what we would do to computethe characters of W µ × D ν .We are thus lead to the conjecture that the counting of 3d partitions with semi-infinite ends restricted to R L,M,N computes the character of M λ,µ,ν for Y L,M,N .
10 Ortho-symplectic Y -algebras In this section, we describe the generalization of the above construction to a Y -junctionof defects in N = 4 SYM with orthogonal and symplectic gauge groups. Theories with– 61 –hese gauge groups can be realized by D O O − planes give an SO (2 n ) gauge theory, ˜ O − planes give an SO (2 n + 1) gauge theory, O + planes givean Sp (2 n ) gauge theory and ˜ O + planes give a gauge theory denoted as Sp (2 n ) (cid:48) , whichis the same as Sp (2 n ) but has a different convention for the θ angle, so that θ = 0 in Sp (2 n ) (cid:48) is the same as θ = π in Sp (2 n ).The O − plane is unaffected by duality transformations. Correspondingly, SO (2 n ) N = 4 SYM has a P SL (2 , Z ) S-duality group. The remaining three types of O T transformation clearly maps Sp (2 n ) ↔ Sp (2 n ) (cid:48) and relates O + and ˜ O + . It leaves ˜ O − invariant. On the otherhand, an S transformation exchanges the Sp (2 n ) and SO (2 n + 1) gauge groups andthe ˜ O − and O + planes, while it maps Sp (2 n ) (cid:48) to itself and leaves ˜ O + invariant.The story is further complicated by the fact that the elementary interfaces in thepresence of O3 planes are associated to “half-fivebranes” that are Z projections ofordinary fivebranes. The type of O3 planes jumps across these interfaces. As a conse-quence, half-NS5 interfaces must interpolate between SO (2 n ) and Sp (2 m ) or between SO (2 n + 1) and Sp (2 m ) (cid:48) : N S N S O − O + ˜ O − ˜ O + SO (2 n ) Sp (2 m ) SO (2 n + 1) Sp (2 m ) while half-D5 interfaces must interpolate between SO (2 n ) and SO (2 m + 1) or Sp (2 n )and Sp (2 m ) (cid:48) : D D O − ˜ O − O + ˜ O + SO (2 n ) SO (2 m + 1) Sp (2 n ) Sp (2 m ) The gauge theory description of the interfaces is very similar to the unitary cases,except that the orbifold projection cuts in half the interface degrees of freedom. Half-– 62 –S5 interfaces support “half-hypermultiplets” transforming as bi-fundamentals of SO × Sp . Half-D5 interfaces between orthogonal groups involve a Nahm pole of odd rank .Half-D5 interfaces between symplectic groups involve a Nahm pole of even rank or ahalf-hypermultiplet in the fundamental representation of Sp . The half-(1 , Sp (cid:48) and Sp is reversed because of the extra interface Chern-Simons terms.The relation between the four-dimensional gauge theory setup and analyticallycontinued Chern-Simons theory works in the same manner as in the unitary case, upto matter of conventions for the levels of the corresponding Chern-Simons theories.We will employ OSp ( n | m ) κ WZW models. We use conventions where κ is thelevel of the SO currents and − κ/ SO ( n ) is n − Sp (2 m ) is m + 1. The critical level for OSp ( n | m )is 2 − n + 2 m . A half-NS5 interface in the presence of gauge theory parameter Ψ willresult in an OSp ( n | m ) ± Ψ − n +2 m +2 theory, depending on which side of the interface the SO and Sp or Sp (cid:48) groups lie. Notice that the level of the Sp WZW currents differ byan half-integral amount from ± Ψ if n is odd, which is when we have an Sp (2 m ) (cid:48) gaugegroup in four dimensions.The relation between Nahm poles and DS reductions will be the same as before.Furthermore, half-hypermultiplets in the fundamental representation of Sp (2 m ) willmap to symplectic bosons which support Sp (2 m ) − WZW currents. Adding n Majo-rana chiral fermions will promote that to
OSp ( n | m ) WZW currents. See AppendixA for details. Y -algebras Depending on the choice of O3 plane in the top right corner, the Y-junction setup fororthogonal and symplectic gauge groups gives rise to four classes of ortho-symplectic Y -algebras: Y ± L,M,N [Ψ] and ˜ Y ± L,M,N [Ψ]. Notice that half-hypermultiplets must transform in a symplectic representation, precluding suchelementary interfaces for SO × SO or Sp × Sp . Furthermore, half-hypermultiplets have a potentialanomaly which has to be cancelled by inflow from the bulk, constraining the choice of Sp (2 n ) vs Sp (2 n ) (cid:48) as predicted by string theory. Notice that the rank of the Nahm pole must be odd for the su ( ) embedding to exist in anorthogonal group Notice that the rank of the Nahm pole must be even for the su ( ) embedding to exist in anorthogonal group. Also, the type of Sp theory must jump across the interface for the same anomalyinflow constraint mentioned in the previous footnote. – 63 –ecause of the duality properties of O3 planes, ˜ Y + L,M,N [Ψ] will have the same tri-ality properties as Y L,M,N [Ψ]. Instead, triality will map into each other Y ± L,M,N [Ψ] and˜ Y − L,M,N [Ψ], up to the usual S action on labels and coupling.In particular, the definition of the algebras will imply Y + L,M,N [Ψ] = Y + L,N,M [1 − Ψ] Y − L,M,N [Ψ] = ˜ Y − L,N,M [1 − Ψ] (10.1)and the non-trivial S-duality conjecture is Y + L,M,N [Ψ] = ˜ Y − M,L,N [ 1Ψ ] Y − L,M,N [Ψ] = Y − M,L,N [ 1Ψ ] (10.2)etcetera. SO (2 N ) SO (2 M + 1) Sp (2 L ) SO (2 N + 1) SO (2 M ) Sp (2 L ) Y − L,M,N [Ψ] ˜ Y − L,M,N [Ψ] Sp (2 N ) Sp (2 M ) SO (2 L ) Sp (2 N ) Sp (2 M ) SO (2 L + 1) Y + L,M, [Ψ] ˜ Y + L,M,N [Ψ]
Figure 7 . Configurations defining ortho-symplectic Y -algebras. We will give now a brief definition of these vertex algebras.The VOAs Y − L,M,N [Ψ] corresponding to the first figure in 7 are defined as follows.There are a super Chern-Simons theory with gauge groups
OSp (2 N, L ) and OSp (2 M +1 , L ) induced at the NS5 interfaces. For L = 0, N = M or N = M + 1, there is noNahm-pole present and corresponding Y -algebra is a BRST reduction of SO (2 M ) Ψ − M +2 × SO (2 M + 1) − Ψ − M +2 SO (2 M + 2) Ψ − M × SO (2 M + 1) − Ψ − M +2 (10.3)– 64 –hat lead to cosets Y − ,M,M [Ψ] = SO (2 M + 1) − Ψ − M +2 SO (2 M ) − Ψ − M +2 Y − ,M,M +1 [Ψ] = SO (2 M + 2) Ψ − M SO (2 M + 1) Ψ − M . (10.4)For L = 0 and N > M + 1 , the VOA is defined as a BRST reduction of theDS-reduction by the (2 N − M − × (2 N − M −
1) block W N − M − [ SO (2 N ) Ψ − N +2 ] × SO (2 M + 1) − Ψ − M +2 (10.5)i.e. coset Y − ,M,N [Ψ] = W N − M − [ SO (2 N ) Ψ − N +2 ] SO (2 M + 1) Ψ − M . (10.6)and similary for N < MY − ,M,N [Ψ] = W M +1 − N [ SO (2 M + 1) Ψ − M ] SO (2 N ) Ψ − N +2 . (10.7)For L (cid:54) = 0, levels of the super Chern-Simons theories are Ψ − N + 2 L + 2 and − Ψ − M +2 L respectively. In the four cases described above, one gets BRST reductionsof similar combinations of DS-reduced and not reduced theory leading to Y − L,M,M [Ψ] =
OSp (2 M + 1 | L ) − Ψ − M +2+2 L OSp (2 M | L ) − Ψ − M +2+2 L ,Y − L,M,M +1 [Ψ] = OSp (2 M + 2 | L ) Ψ − M +2 L OSp (2 M + 1 | L ) Ψ − M +2 L ,Y − L,M,N [Ψ] = W N − M − [ OSp (2 N | L ) Ψ − N +2 L +2 ] OSp (2 M + 1 | L ) Ψ − M +2 L N > M + 1 ,Y − L,M,N [Ψ] = W M +1 − N [ OSp (2 M + 1 | L ) Ψ − M +2 L ] OSp (2 N | L ) Ψ − N +2+2 L N < M. (10.8)The VOA ˜ Y − L,M,N [Ψ] corresponding to the second configuration in 7 are definedsimply as ˜ Y − L,M,N [Ψ] = Y − L,N,M [1 − Ψ] . (10.9)Let us now define the VOAs Y + L,M,N [Ψ] corresponding to the bottom left diagramin 7. Let L = 0 and N = M . An Sp (2 N ) Chern-Simons theory is induced at thevertical boundary with shift in the level by . The anomaly mismatch compensated– 65 –y a (half)-symplectic boson in a fundamental representation of Sp (2 N ). The VOA isthen identified with the BRST reduction of Sp (2 N ) Ψ2 − N − × Sb Sp (2 N ) × Sp (2 N ) − Ψ2 − N − (10.10)i.e. the coset Y +0 ,N,N [Ψ] = Sp (2 N ) Ψ2 − N − × Sb Sp (2 N ) Sp (2 N ) Ψ2 − N − . (10.11)If M (cid:54) = N , there are no symplectic bosons present but Nahm-pole boundary con-ditions appears leading for N > M to Y +0 ,M,N [Ψ] = W N − M Sp (2 N ) Ψ2 − N − Sp (2 M ) Ψ2 − M − (10.12)and for N < M to Y +0 ,M,N [Ψ] = W M − N [ Sp (2 M ) − Ψ2 − M − ] Sp (2 N ) − Ψ2 − N − . (10.13)If L (cid:54) = 0, one gets analogous expression with super-groups and dual super-Coxeternumbers: Y + L,N,N [Ψ] =
OSp (2 L | N ) − Ψ+2 N − L +2 × Sb OSp (2 L | N ) OSp (2 L | N ) − Ψ+2 N − L +3 ,N > M Y L,M,N [Ψ] = W N − M [ OSp (2 L | N ) − Ψ+2 N − L +2 ] OSp (2 L | M ) − Ψ+2 M − L +3 (10.14) N < M Y
L,M,N [Ψ] = W M − N [ OSp (2 L | M ) − Ψ+2 M − L +1 ] OSp (2 L | N ) − Ψ+2 M − L +2 . (10.15)The last diagram of 7 gives rise to ˜ Y + L,M,N [Ψ]:˜ Y + L,N,N [Ψ] =
OSp (2 L + 1 | N ) − Ψ+2 N − L +1 × Sb OSp (2 L +1 | N ) OSp (2 L + 1 | N ) − Ψ+2 N − L +2 ,N > M ˜ Y + L,M,N [Ψ] = W N − M [ OSp (2 L + 1 | N ) − Ψ+2 N − L +1 ] OSp (2 L + 1 | N ) − Ψ+2 M − L +2 (10.16) N < M ˜ Y + L,M,N [Ψ] = W M − N [ OSp (2 L + 1 | M ) − Ψ+2 M − L ] OSp (2 L + 1 | N ) − Ψ+2 M − L +1 . (10.17)where Sb OSp ( n | N ) denotes a combination of N symplectic bosons and n real fermionswhich supports bilinear OSp ( n | N ) currents. – 66 – p (cid:48) (2) SO (1) Sp (0) We will quickly look at the simplest example of OY algebra:˜ Y +1 , , [Ψ]. This turns out to coincide with the N = 1 super Virasorovertex algebra. The analysis is completely parallel to the Virasorocase and triality manifests itself in the same manner: the firsttwo realizations lead to two descriptions related by Feigin-Frenkelduality and the third one to the coset model realization.The triality is then of the form W | [ OSp (1 | − Ψ ] ↔ W | [ OSp (1 | − Ψ − ] ↔ SO (3) − Ψ − × Ff SO (3) SO (3) − Ψ − . (10.18)The DS construction that produces N = 1 super Virasoro algebra from DS-reduction of OSp (1 |
2) can be found in the appendix of [40]. The central charge ofthe VOA is in our conventions c = 152 − − . (10.19)We can see that this expression is indeed invariant under the Feigin-Frenkel dualityΨ ↔ This coset realization is also known to lead to N = 1 super-Virasoro algebra: itis the analytic continuation of the well-known construction of N = 1 super Virasorominimal models. It produces the N = 1 super Virasoro algebra with central chargeindicated above. Central charge of orthosymplectic Y -algebras are given by (see appendix B for detailedcalculation) Central charge of ˜ Y − L,M,N [Ψ] can then be identified as ˜ c − L,M,N [Ψ] = c − L,N,M [1 − Ψ]. Recall that central charge of
OSp (2 N | L ) Ψ − N +2 L +2 is c − L,M,N [Ψ] = ˜ c − L,N,M [1 − Ψ]= − (2( L − M ) − L − M ) + 1)( L − M )Ψ −
1+ 2(2( L − N ) + 1)( L − N + 1)( L − N )Ψ+2Ψ(2( M − N ) + 1)( M − N + 1)( M − N ) − L (1 + 6 M + M (6 − N ) − N + 6 N )+4 M − M (1 − N ) + N (5 − N + 8 N ) (10.20)– 67 –nd c + L,M,N [Ψ] = ˜ c + L − ,M,N [Ψ]= − M − L )(2( M − L ) + 1)( M − L + 1)1 − Ψ − N − L )(2( N − L ) + 1)( N − L + 1)Ψ+Ψ(2( M − N ) − M − N ) + 1)( M − N )+ L (1 − M − N ) ) − N + 2( M − N ) (3 + 2 M + 4 N ) . (10.21)One can check that the expressions above are indeed invariant under transformations10.2. Note also that S action preserves ˜ Y + -algebras and we can indeed write theircentral charge ˜ c + L,M,N [Ψ] in S invariant way˜ c + L,M,N [Ψ] = 12 1Ψ ( L − N )(4( L − N ) −
1) + 12 (1 −
1Ψ )( N − L )(4( N − L ) − M − N )(4( M − N ) −
1) + 12 (1 − Ψ)( N − M )(4( N − M ) − − Ψ ( L − M )(4( L − M ) −
1) + 12 ΨΨ − M − L )(4( M − L ) − − L + M − N )( L − M + N )( − L + M + N ) + 12 . (10.22)
11 From Junctions to Webs
It is natural to consider brane or gauge theory configurations involving a more intricatejunction, perhaps involving several semi-infinite interfaces converging to a single two-plane.It is also natural to consider intricate webs, involving finite interface segments aswell as semi-infinite ones. Web configurations would break scale invariance. In the IR,they would approach a single junction.Conversely, one may consider webs with several simpler junction as a regularizationof an intricate junction. If all junctions are dual to our basic Y-junctions, this maybecome a computational tool to determine the VOA’s at generic junctions.There is a precedent to this: complicated half-BPS interfaces in N = 4 SYM canoften be decomposed as a sequence of simpler interfaces, with a smooth limit sending tozero the relative distances between the interfaces. This is an important computationaltool, as it allows one to apply S-duality transformations to well-understood individualpieces and then assemble them to the S-dual of the original, intricate interface.– 68 – concrete example could be a Nahm pole associated to a generic su (2) embedding ρ , realized as a sequence of individual simple Nahm pole interfaces. This is a smoothresolution, as long as the individual interfaces are ordered in a specific way [10]. The S-dual configuration is a sequence of bi-fundamental interfaces building up a complicatedthree-dimensional interface gauge theory with a good IR limit [8].One may want to follow that example for junctions, say to decompose a Y-junctionof complicated interfaces into a web of simpler Y-junctions.This idea raises a variety of hard questions, starting from figuring out criteria for asmooth IR limit of an interface web. Furthermore, the same configuration may be thelimit of many different inequivalent webs.Any of these questions would bring us far from the scope of this paper. In thissection we will limit ourselves to a few judicious speculations.In general, local operators at the final junction may arise either from local operatorsat each elementary junction in the web or from extended operators, such as a finiteline defect segment joining two consecutive junctions. Thus we may hope that thefinal VOA will be an extension of the product of the VOA’s at the vertices of the web,including products of degenerate modules associate to the finite line defect segment.This picture is supported by the observation that although the dimensions of de-generate modules are not integral, the sum of the dimensions of the local operatorsat the two ends of a finite line defect segment will be integral. For example, a finiteWilson line W µ on a finite segment of NS5 interface supports two local operators atthe endpoints which have dimensions which differ by integral amounts from ∆ µ [Ψ] and∆ µ [ − Ψ] = − ∆ µ [Ψ] respectively, where ∆ µ [Ψ] is the dimension of the µ vertex operatorin the U ( N | L ) Ψ WZW model.This is not quite a full definition of the final interface VOA, but it strongly restrictsits form. A striking observation is the formal resemblance between this idea and the waythe topological vertex is used to assemble the topological string partition function ofgeneral toric Calabi-Yau, by summing up over a choice of partition µ for each internalleg of the toric diagram [11]. Perhaps one may use this analogy to determine whichproducts of degenerate modules to included in the extended VOA.The simplest possible situation for us is a web which can be interpreted as a col-lection of D5 branes ending on a NS5 brane: a sequence of (1 , q i ) fivebrane segmentswith Y-junctions to semi-infinite D5 branes coming from the left or the right. Sucha configuration can be lifted directly to a sequence of interfaces in 3d Chern-Simons It may be possible to formalize this procedure as a sort of tensor product of VOAs over a commonbraided monoidal category. – 69 –heory. If the 3d interfaces have a good collision limit, one can derive directly thejunction VOA.This situation also allows one to start probing questions about the extension struc-ture of the final VOA and the equivalence between different web resolutions of the sameinterface.
The simplest possibility we can discuss is that of an infinite D5 interface crossing aninfinite NS5 interface. The four-way junction has two obvious resolutions, akin tothe toric diagram of the conifold, involving either a (1 ,
1) or a (1 , −
1) finite interfacesegment. ← →
KL MNKL MN KL MN
Figure 8 . Two possible resolutions of the configuration of D5-brane (horisontal) crossingNS5-brane (vertical). First resolution includes a finite segment of (1 , , − K, L, M, N
D3-branes are attached tofivebranes leading to webs of interfaces between U ( K ) , U ( L ) , U ( M ) , U ( N ) theories. We can denote the choices of gauge group in the four quadrants as
K, L, M, N ,counterclockwise from the top left quadrant.For some values of
K, L, M, N , the two resolutions produce obviously the same 3dinterface in the scaling limit and then the same VOA. For example, if K = L and N = M then the CS theory interface results from the collision of interfaces whichsupport some 2d matter coupled to the U ( N | K ) CS gauge fields. The two resolutionsgive the same two interfaces in different order, and the collision/scaling limit is obviouslythe same: an interface which supports both 2d matter fields at the same location.On the other hand, in other configurations the two resolutions give clearly differentanswers orproduce pairs of interfaces which do not have an obvious collision limit.In any case, the resolved web enjoys a non-trivial S-duality symmetry, exchanging,say, K and M while mapping Ψ → Ψ − .We will consider a single entertaining example, a small variation of the parafermionexample. N = 2 super-Virasoro – 70 – (2) U (1) Consider a four-way junction with K = 0, L = 0, N = 2, M = 1,resolved by a (1 ,
1) segment.In the 3d description, we have a U (2) CS theory which is firstreduced to a block-diagonal U (1) and then coupled to a single com-plex fermion. There is no obstruction to bringing the interfacestogether.The opposite resolution, involving a (1 , −
1) segment, wouldhave coupled a complex fermion doublet to U (2) and then reduced the gauge sym-metry to U (1). In a scaling limit, this would differ from the original resolution by anextra spurious complex fermion decoupled from the 3d gauge theory.The system engineers a u (1)-BRST coset VOA U (2) Ψ × Ff U (1) U (1) Ψ (11.1)We can recast this as the product U (1) × sVir N =2 (11.2)using the Kazama-Suzuki coset descriptionsVir N =2 = SU (2) κ × Ff U (1) U (1) κ +2 (11.3) U (2) U (1) The S-dual description involves a four-way junction with K = 1, L = 0, N = 2, M = 0, resolved by a (1 ,
1) segment.That gives simply the DS reduction of U (2 | U (1) × sVir N =2 (11.4)as well [40]. Another instructive example is a trivalent junction between a (1 , , k ) and k coincident (0 ,
1) fivebranes. We can resolve the stack of fivebranes into k parallel D5branes.If N − M = kS for integer S , one may let the number of D3 branes drop by S acrosseach D5 brane. This configuration is expected to preserve a global SU ( k ) symmetryon the limit of coincident fivebranes.This symmetry manifests itself in the junction VOA: the vertex algebra involves aDS reduction, say, of U ( N | L ) associated to an su (2) embedding with k blocks of size– 71 – and M blocks of size 1. The result of that reduction has a SU ( k ) WZW subalgebrawhich survives the coset by U ( M | L ).If N = M , instead, we have k copies of the symplectic bosons for U ( N | L ), whichalso have an SU ( k ) WZW subalgebra which survives the coset by U ( N | L ).It is natural to conjecture relations between this VOAs and some kind of W k + ∞ algebra and the topological vertex and D-brane counting in C / Z k [49]. Acknowledgements
We thank Kevin Costello, Thomas Creutzig, Tomaˇs Proch´azka, and Edward Witten forinteresting discussions. The research of DG and MR was supported by the PerimeterInstitute for Theoretical Physics. Research at Perimeter Institute is supported bythe Government of Canada through Industry Canada and by the Province of Ontariothrough the Ministry of Economic Development & Innovation.
A Conventions for current algebras
A.1 Free fermionsA.1.1 A single real fermion
A single chiral Majorana free fermion ψ ( z ) has OPE ψ ( z ) ψ ( w ) ∼ z − w (A.1)It has dimension with stress tensor T ψ = − ψ∂ψ (A.2)of central charge c Ff = .We will denote the corresponding (spin-)VOA as Ff. A.1.2 SO ( n ) fermions If we take n chiral Majorana free fermions ψ i ( z ) we get a VOA Ff SO ( n ) . It includes SO ( n ) WZW currents which we can sloppily normalize as J ij = ψ i ψ j (A.3)Indeed, J ij ( z ) J kt (0) ∼ δ it δ jk − δ ik δ jt z + δ jk J it (0) − δ jt J ik (0) − δ ik J jt (0) + δ it J jk (0) z (A.4)– 72 –his is a conformal embedding: the dimension of an SO ( n ) WZW model at level1 is c SO ( n ) = 1 × n ( n − n − n nc Ff (A.5)and : J ij J kt := δ jk ∂ψ i ψ t − δ jt ∂ψ i ψ k − δ ik ∂ψ j ψ t + δ it ∂ψ j ψ k + ψ i ψ j ψ k ψ t (A.6)hence the stress tensor coincides with the Sugawara stress tensor. T ≡ − ψ i ∂ψ i = 11 + n − ×
14 : J ij J ji (A.7) A.1.3 A single complex fermion and the bc system Two Majorana free fermions can be combined into complex fermions ψ ( z ) and χ ( z ) ≡ ( ψ ) † ( z ) with OPE ψ ( z ) χ ( w ) ∼ z − w (A.8)The fermions have dimension with stress tensor T ψχ = − ψ∂χ − χ∂ψ (A.9)of central charge c Ff = .The VOA includes a U (1) WZW current J = ψχ of level 1. Indeed J ( z ) J (0) ∼ z − w ) (A.10)This is a conformal embedding: T = 12 : J J : (A.11)We will denote the corresponding (spin-)VOA as Ff U (1) .The bc ghost system is the same as a complex fermion, except that the stress tensoris shifted by ∂J to T bc = − b∂c (A.12)so that the dimension of c ( z ) is 0 and of b ( z ) is 1. The central charge of the shiftedstress tensor is − δ∂J we obtain a ghost system suchthat c has dimension − δ , b has dimension + δ and the central charge is 1 − δ .– 73 – .1.4 U ( n ) fermions If we take 2 n chiral Majorana free fermions we can combine them into n pairs of complexgenerators ψ a ( z ) and χ a ( z ) ≡ ( ψ a ) † ( z ) with OPE ψ a ( z ) χ b ( w ) ∼ δ ab z − w (A.13)We denote this VOA as Ff U ( n ) . It includes U ( n ) WZW currents J ab = ψ a χ b (A.14)with OPE J ab ( z ) J cd ( w ) ∼ δ ad δ cb ( z − w ) + δ cb J ad ( w ) − δ ad J cb ( w ) z − w (A.15)The level of the SU ( n ) current sub-algebra J ab − n δ ab J cc is 1. The diagonal U (1) J cc haslevel n .This is a conformal embedding: c SU ( n ) + c U (1) = 1 × ( n − n + 1 = n = 2 nc Ff (A.16) A.2 Symplectic fermions VOAs Sf and Sf The vertex algebra Sf is generated by two “symplectic fermions” x ( z ), y ( z ), which arefermionic currents of dimension 1 and OPE x ( z ) y ( w ) ∼ z − w ) (A.17)The stress tensor is simply T = xy (A.18)with central charge −
2. The vertex algebra Sf has an SU (2) o global symmetry rotatingthe symplectic fermions as a doublet, which is not promoted to an affine symmetry.It is useful to consider the Cartan subgroup U (1) o acting on the the two fermioniccurrents with charge ± e consisting of bosonic vertex operators in Sf and of the “singlet algebra” Sf consisting of vertex operators in Sf of U (1) o charge 0. We refer to [50] for the definitionof Sf and to [51] and references therein for a detailed discussion of these subalgebrasand their modules.The VOA Sf has two natural classes of modules: Equivalently, Sf can be defined as a
P SU (1 |
1) current algebra. – 74 –
The other charge sectors Sf n in Sf. They have highest weight vectors of conformaldimension n + | n | . • The charge 0 sector V xyλ of the twisted modules for Sf. They have highest weightvectors of conformal dimension λ − λ which induce singularities z − λ in x ( z ) and z λ − in y ( z ).In order to define V xyλ we consider a vector | λ ; xy (cid:105) which satisfies x λ + n | λ ; xy (cid:105) = 0 n ≥ y − λ + n | λ ; xy (cid:105) = 0 n ≥ x λ − | λ ; xy (cid:105) = | λ − xy (cid:105) y − λ | λ ; xy (cid:105) = | λ + 1; xy (cid:105) (A.20)so that V xyλ generates a module for the xy VOA which is defined modulo λ → λ + 1and includes the Sf modules V xyλ + n for all n .Notice that as we take λ to 0 we do not get the vacuum module, but rather adecomposable module. Similar considerations apply for other integer values of λ . Ingeneral we will assume λ to be a non-integer complex number.Some of the fusion rules between these modules are obvious. For example, almostby definition we have Sf n × Sf m ∼ Sf n + m Sf n × V xyλ ∼ V xyλ + n (A.21)The OPE of two twisted modules is more subtle. As they are twist fields for the xy currents, the tensor product is likely to contain modules of the form V xyλ + λ (cid:48) + k with integer k . As x ( z ) has a pole z λ and y ( z ) a pole z − λ near V xyλ , they can potentially have poles z λ + λ (cid:48) and z − λ − λ (cid:48) near the result of the OPE. i It is thus not unreasonable to expect V xyλ × V xyλ (cid:48) ∼ V xyλ + λ (cid:48) + V xyλ + λ (cid:48) − (A.22)This is indeed the case, see e.g. [50]. – 75 – .3 Symplectic bosonsA.3.1 Symplectic bosons, symplectic fermions and βγ systems. The vertex algebra Sb of a single symplectic boson has two bosonic generators, X ( z )and Y ( z ), with OPE X ( z ) Y ( w ) ∼ z − w (A.23)and conformal dimension 1 /
2. We can also denote the generators as a doublet Z α withOPE Z α ( z ) Z β ( w ) ∼ (cid:15) αβ z − w (A.24)Several of the features we discuss below can be found discussed at length in [52].The stress tensor can be written as T = 12 X∂Y − Y ∂X = 12 (cid:15) αβ Z α Z β (A.25)and gives a central charge of c Sb = − SU (2) WZW currents J αβ = Z α Z β (A.26)of level − : J αβ ( z ) J γδ (0) ∼ (cid:15) αγ (cid:15) βδ + (cid:15) αδ (cid:15) βγ z + (cid:15) αγ J βδ + (cid:15) αδ J βγ + J αγ (cid:15) βδ + (cid:15) βγ J αδ z (A.27)This is likely to be a conformal embedding: c SU (2) − = 3 / / − − U (1) − current subalgebra generated by the current XY is also important. Itplays an important role in the bosonization relation between symplectic bosons andsymplectic fermions. In particular, Sf is the result of the u (1)-BRST quotientSf = (cid:110) Sb × Ff U (1) × bc , Q BRST (cid:111) (A.29)with BRST charge Q BRST = (cid:72) c ( XY + ψχ ). The fermionic currents are the BRST-closed generators x ( z ) = X ( z ) ψ ( z ) y ( z ) = Y ( z ) χ ( z ) (A.30) This is just the bosonization of βγ system familiar in superstring perturbation theory, with currents η and ∂ξ . – 76 –f we restrict ourselves to the U (1) subalgebra in Ff U (1) we get the VOA Sf , thecharge 0 subalgebra of Sf. In other words, there is a coset relationSf = Sb U (1) − (A.31)Conversely, one can reconstruct the symplectic boson VOA by dressing the fermioniccurrents with charge ± U (1) − current algebra.The symplectic boson VOA has a variety of interesting modules, which have intri-cate relations with modules for SU (2) − and Sf .In particular, there is a family of Ramond modules R λ , with non-integral λ ∈ C / Z defined by relations X | n (cid:105) = ( λ + n ) | n − (cid:105) Y | n (cid:105) = | n + 1 (cid:105) X k | n (cid:105) = 0 k > Y k | n (cid:105) = 0 k > n ∈ Z . We denote as R λ the vertex operator associated to the vector | (cid:105) as well.It has conformal dimension − and U (1) charge λ + .This modules give Weyl modules for SU (2) − associated to infinite dimensionalprincipal series representations of the algebra of zeromodes.In the context of bosonization, R λ can be combined with a vertex operator of charge λ for U (1) to produce the vertex operators and modules V xyλ for Sf .Finally, symplectic bosons are a special example of βγ systems. Indeed, if we shiftthe stress tensor by appropriate multiples of ∂ ( XY ) we can get either the standard βγ system of conformal dimensions 1 and 0 or more general ones of dimensions ± δ , withcentral charge is − δ . A.3.2 Sp (2 n ) and U ( n ) symplectic bosons If we take n pairs of symplectic bosons Z α ( z ) with OPE Z α ( z ) Z β ( w ) ∼ ω αβ z − w (A.33)we get a VOA Sb Sp (2 n ) . It includes Sp (2 n ) − WZW currents J αβ = Z α Z β (A.34)with OPE J αβ ( z ) J γδ (0) ∼ ω αγ ω βδ + ω αδ ω βγ z + ω αγ J βδ + ω αδ J βγ + ω βδ J αγ + ω βγ J αδ z (A.35)– 77 –his is likely to be a conformal embedding: the dimension of an Sp (2 n ) WZWmodel at level − is c Sp (2 n ) − = − × n (2 n + 1) − + n + 1 = − n = nc Sb (A.36)If we separate the symplectic bosons in two dual sets X a ( z ) and Y a ( z ) we get U ( n )WZW currents J ab = X a Y b (A.37)such that the level of the SU ( n ) subalgebra is − J ab ( z ) J cd (0) ∼ − δ ad δ cb z + δ ad J cb (0) − δ cb J ad (0) z (A.38)For n (cid:54) = 1 this is likely to be a conformal embedding: c SU ( n ) − + 1 = − × ( n − − n + 1 = − n = nc Sb (A.39) A.3.3
OSp ( n | m ) fermions If we combine n Majorana fermions ψ i and m pairs of symplectic bosons Z α ( z ) we geta VOA Ff OSp ( n | m ) . We can combine them into fields u a = ( ψ i , Z α ) with OPE u A ( z ) u B ( w ) ∼ η AB z − w (A.40)where η AB is Koszul-antisymmetric: (cid:15) BA = ( − p ( A ) p ( B ) (cid:15) AB with p ( i ) = 1 and p ( α ) =0. It includes OSp ( n | m ) WZW currents J = (cid:18) ψ i ψ j ψ i Z β Z α ψ j Z α Z β (cid:19) (A.41)i.e. J AB = u A u B , with J BA = ( − p ( A ) p ( B ) J BA and OPE J AB ( z ) J CD (0) ∼ η BC η AD + ( − p ( A ) p ( B ) η AC η BD z (A.42)+ η BC J AD (0) + ( − p ( A ) p ( B ) η AC J BD (0)) z (A.43)+ ( − p ( C ) p ( D ) η BD J AC (0) + ( − p ( A ) p ( B )+ p ( C ) p ( D ) η AD J BC (0) z (A.44)This is likely to be a conformal embedding: the dimension of an OSp ( n | m ) WZWmodel at level 1 is c OSp ( n | m ) = 1 × ( n − m )( n − m − − n − m − n − m (A.45)– 78 – .3.4 U ( n | m ) symplectic bosons If we combine n pairs of symplectic bosons X a ( z ) and Y b ( z ) and m complex fermions ψ i , χ j we get a VOA Sb U ( n | m ) . we get U ( n ) WZW currents J = (cid:18) X a Y b X a ψ i χ j Y b χ j ψ i (cid:19) (A.46)such that the level of the SU ( n | m ) subalgebra is −
1. For n − m (cid:54) = 1 this is likely to bea conformal embedding: c SU ( n | m ) − + 1 = − × (( n − m ) − − n − m + 1 = − n + m (A.47)We can collect the fermions and symplectic bosons in super-vectors u A = ( X a , χ i )and v A = ( Y a , ψ i ) with p ( i ) = 1 and p ( α ) = 0, write the OPE as u A ( z ) v B ( w ) ∼ δ AB z − w v B ( z ) u A ( w ) ∼ − ( − p ( A ) p ( B ) δ AB z − w (A.48)and the currents as J AB = u A v B (A.49)with OPE J AB ( z ) J CD (0) ∼ − ( − p ( B ) p ( C ) δ AD δ CB z ++ ( − p ( A ) p ( B )+ p ( C ) p ( D )+ p ( C ) p ( B ) δ AD J CB (0) − ( − p ( B ) p ( C ) δ CB J AD z (A.50)Notice that in these conventions the overall U (1) is the super-trace J = (cid:80) A ( − p ( A ) J AA : J ( z ) J CD (0) ∼ − δ CD z J ( z ) J (0) ∼ − n − mz (A.51)We can obtain alternative normalizations of the overall U (1) by defining ˆ J AB = J AB + cδ AB J so that ˆ J AB ( z ) ˆ J CD (0) ∼ − ( − p ( B ) p ( C ) δ AD δ CB + (2 c + c ( n − m )) δ AB δ CD z ++ ( − p ( A ) p ( B )+ p ( C ) p ( D )+ p ( C ) p ( B ) δ AD ˆ J CB (0) − ( − p ( B ) p ( C ) δ CB ˆ J AD z (A.52)Notice that we we can also exchange the role of fermions and symplectic bosons,with some ˜ u A = ( χ a , X i ) and ˜ v A = ( ψ a , Y i ). If we use the same p ( i ) = 1 and p ( α ) = 0convention, we have OPE˜ u A ( z )˜ v B ( w ) ∼ δ AB z − w ˜ v B ( z )˜ u A ( w ) ∼ ( − p ( A ) p ( B ) δ AB z − w (A.53)– 79 –nd currents ˜ J AB = − ˜ u A ˜ v B (A.54)with OPE˜ J AB ( z ) ˜ J CD (0) ∼ ( − p ( B ) p ( C ) δ AD δ CB z ++ ( − p ( A ) p ( B )+ p ( C ) p ( D )+ p ( C ) p ( B ) δ AD J CB (0) − ( − p ( B ) p ( C ) δ CB J AD z (A.55)of SU ( n | m ) at level 1 A.4 U ( N ) κ currents Thorughout the paper, we use following notation for U ( N ) κ VOA U ( N ) κ = U (1) Nκ × SU ( N ) κ − N . (A.56)The specific combination of levels of SU ( N ) and U (1) is natural [53] and correspondsto the Sugawara stress tensor and conformal dimensions being given by the standard U ( N ) Casimir. The notation for the level is unusual, but very convenient for this paper.If J ab are the U ( N ) κ currents, the coresponding OPE is given by J ab ( z ) J a (cid:48) b (cid:48) ( w ) ∼ ( κ − N ) δ ab (cid:48) δ a (cid:48) b + δ ab δ a (cid:48) b (cid:48) ( z − w ) + δ a (cid:48) b J ab (cid:48) − δ ab (cid:48) J a (cid:48) b z − w (A.57)One can indeed check that U (1) Nκ element J Nκ given by J = N (cid:88) i =1 J ii (A.58)satisfy J ( z ) J ( w ) ∼ N κ ( z − w ) (A.59)and elements in the cartan of SU ( N ) κ − N such as H ij = J ii − J jj satisfy H ij ( z ) H ij ( w ) ∼ κ − N )( z − w ) (A.60)which is consistent with OPE of the off-diagonal components.Notice that if we have WZW currents ˜ J ab with OPE˜ J ab ( z ) ˜ J a (cid:48) b (cid:48) ( w ) ∼ kδ ab (cid:48) δ a (cid:48) b ( z − w ) + δ a (cid:48) b ˜ J ab (cid:48) − δ ab (cid:48) ˜ J a (cid:48) b z − w (A.61)– 80 –hen J ab + ˜ J ab are U ( N ) κ + k currents.This is the case, in particular, if ˜ J ab are bilinears ψ ai ψ ib or X ai · Y ib of kN complexfermions (or bc ghosts) or − kN symplectic bosons (or βγ ghosts) transforming in thefundamental representation of U ( N ).Furthermore, notice that a block-diagonal U ( N −
1) subalgebra of U ( N ) κ has thecorrect OPE to me identified with U ( N − κ − .Finally, consider a u ( N )-valued ghost system with currents I ij = b ik c kj − b kj c ik (A.62)then I ij ( z ) I st (0) = 2 N δ it δ sj − δ st δ ij z + δ sj I it − δ it I sj z (A.63)which is precisely what is needed for the sum of U ( N ) κ currents, U ( N ) − κ currents and I ij currents to have no z − term in the OPE, as needed for a u ( N )-BRST reduction. A.5 U ( M | N ) κ currents The currents are labeled as components of supermatrix J ba where a, b = 1 , . . . , M arefermionic bosonic and a, b = M + 1 , . . . , M + N are fermionic. The two diagonalblocks consist of bosonic generators and the two off-diagonal blocks are fermionic. By U ( M | N ) κ , we really mean U ( M | N ) κ = U (1) ( M − N ) κ × SU ( M | N ) κ − M + N (A.64)As in the case of U ( N ) currents, OPE of super-currents components is J AB ( z ) J CD (0) ∼ ( − p ( B ) p ( C ) ( κ − M + N ) δ AD δ CB + δ AB δ CD z ++ ( − p ( A ) p ( B )+ p ( C ) p ( D )+ p ( C ) p ( B ) δ AD J CB (0) − ( − p ( B ) p ( C ) δ CB J AD z (A.65)where p ( a ) = 0 for a = 1 , . . . , M and p ( a ) = 1 otherwise.Notice that if we have WZW currents ˜ J ab with OPE˜ J AB ( z ) ˜ J CD (0) ∼ k ( − p ( B ) p ( C ) δ AD δ CB z ++ ( − p ( A ) p ( B )+ p ( C ) p ( D )+ p ( C ) p ( B ) δ AD J CB (0) − ( − p ( B ) p ( C ) δ CB J AD z (A.66)then J AB + ˜ J AB are U ( N | M ) κ + k currents. – 81 –his is the case, in particular, if ˜ J AB are bilinears of complex fermions (or bc ghosts)and symplectic bosons (or βγ ghosts) transforming in the fundamental representationof U ( N | M ).Furthermore, notice that a block-diagonal U ( N − | M ) subalgebra of U ( N | M ) κ has the correct OPE to me identified with U ( N − | M ) κ − . A.6 The bosonization of U (1 | κ The typical convention for the U (1 |
1) OPE’s at level κ is J ( z ) J ( w ) ∼ κ ( z − w ) J ( z ) A ( w ) ∼ A ( w ) z − wJ ( z ) B ( w ) ∼ − B ( w ) z − wI ( z ) I ( w ) ∼ − κ ( z − w ) I ( z ) A ( w ) ∼ − A ( w ) z − wI ( z ) B ( w ) ∼ B ( w ) z − wA ( z ) B ( w ) ∼ κ ( z − w ) + J ( w ) + I ( w ) z − w (A.67)In order to match our conventions for U (1 | κ , we can define J = J + 12 κ ( I + J ) J = A J = B J = − I + 12 κ ( I + J ) (A.68)We get the correct diagonal OPE’s J ( z ) J ( w ) ∼ κ + 1( z − w ) J ( z ) J ( w ) ∼ z − w ) J ( z ) J ( w ) ∼ − κ + 1( z − w ) (A.69)The bosonization of the U (1 |
1) WZW model [54] is obtained by writing the oddcurrents A = V κ, − κ , − x B = κV κ, − κ − , y (A.70)– 82 –s the product of vertex operators for the U (1) κ × U (1) − κ currents J and I . Then theOPE of x and y is the free OPE of symplectic fermions.Accordingly, we can decompose the WZW vacuum module into products of modulesfor U (1) κ × U (1) − κ × Sf U (1 | κ = ⊕ n V κ, − κn, − n ⊗ Sf n (A.71)We can give a bosonized description of several other important modules for U (1 | κ .A nice discussion of this VOA and its modules can be found in [55] In particular, thereare Weyl modules built from finite-dimensional irreducible representations of u (1 | r, r ) under J and J can be obtained from the highest weight vector V κ, − κr, − r of conformal dimension0 and decompose as V U (1 | κ r = ⊕ n V κ, − κr + n, − r − n ⊗ Sf n (A.72)Typical modules associated to two-dimensional representations ( t + s, t ) of weights( t + s, t ) under J and J can be obtained from V κ, − κ (1 − κ ) s + t, κ s − t ⊗ Sf κ − s of conformaldimension s κ − s κ + 12 κ ( s + s (2 t − sκ )) = 12 κ s ( s + 2 t −
1) = 12 κ (˜ t − ˜ t − t + t ) (A.73)with ˜ t = s + t and decompose as V U (1 | κ s,t = ⊕ n V κ, − κ (1 − κ ) s + t − n, κ s − t + n ⊗ Sf κ − s + n (A.74) B General central charge
B.1 Unitary case
Recall the central charge of U ( N | M ) Ψ . c U ( N | M ) Ψ = 1 + Ψ − N + M Ψ (cid:0) ( N − M ) − (cid:1) (B.1)That means that if M = N , c L,N,N [Ψ] = 1 + Ψ − N + L Ψ (cid:0) ( N − L ) − (cid:1) + − N + L − − Ψ − N + L − − (cid:0) ( N − L ) − (cid:1) == ( N − L )(( N − L ) − (cid:18) − − (cid:19) − N + L (B.2)– 83 –n the other hand, if N = M + 1 we have c L,N − ,N [Ψ] = 1 + Ψ − N + L Ψ (cid:0) ( N − L ) − (cid:1) + − − Ψ − N + L Ψ − (cid:0) ( N − L − − (cid:1) == ( L − N ) (cid:0) ( N − L ) − (cid:1)
1Ψ ++ ( N − L − (cid:0) ( N − L − − (cid:1) − N − L − N = M − c L,N − ,N [Ψ] = − − Ψ + N − L Ψ (cid:0) ( N − L ) − (cid:1) + 1++ Ψ − N − L + 1Ψ − (cid:0) ( N − L + 1) − (cid:1) == ( L − N ) (cid:0) ( N − L ) − (cid:1)
1Ψ ++ ( N − L + 1) (cid:0) ( N − L + 1) − (cid:1) − N − L + 1 (B.4)The general case requires a bit more work. A simplifying feature is that at all stepsonly the differences between L , M , N will matter. Lets first set N > M + 1.Let us first analyze the DS-reduction part. We need to both add the ghosts valuedin n and then shift the stress tensor by the derivative of the t component of the totalcurrents.The su (2) embedding in u ( N | L ) is given by decomposing the fundamental repre-sentation of u ( N | L ) as the dimension N − M irrep plus M + L copies of the trivialrepresentation. Thus t is the Cartan generator t = ( N − M − , · · · , − N − M − , , · · · , | , · · ·
0) (B.5)the level of t · J U ( N | L ) Ψ is easily computed to be κ t · J U ( N | L )Ψ = (Ψ − N + L ) N − M − (cid:88) i =0 ( N − M − − i ) − N + L )( N − M ) (cid:0) ( N − M ) − (cid:1) (B.6)Thus the central charge of the WZW part is shifted to˜ c U ( N | L ) Ψ = 1+ Ψ − N + L Ψ (cid:0) ( N − L ) − (cid:1) − (Ψ − N + L )( N − M ) (cid:0) ( N − M ) − (cid:1) (B.7)– 84 –he u ( N | L ) generators can be decomposed into blocks accordingly as (cid:18) D CB A (cid:19) (B.8)Then n consists of the upper triangular part of D , with ( N − M )( N − M − / N − M − M even and ( N − M − L odd elements in C ⊕ B .After the shift of the stress tensor, the c and γ ghosts end up with dimension equalto the t charge q . The corresponding central charge is c bc = − q − + 1 . (B.9)Of the ghosts in D , N − M − N − M − c gh D = N − M − (cid:88) n =1 ( N − M − n ) (cid:0) − n − + 1 (cid:1) = − ( N − M )( N − M −
1) (( N − M )( N − M − − . (B.10)On the other hand, of the bc ghosts in C ⊕ B , for even N − M we have M of charge 1 / M of charge 3 /
2, etc, while for odd N − M we have 2 M of charge 1, etc. Combinedwith the βγ ghosts we get for even N − Mc gh C ⊕ B = 2 M ( N − M ) / (cid:88) n =2 ( − n − + 1) + M = ( L − M )( N − M − (cid:0) ( N − M − − (cid:1) . (B.11)and the same expression for odd N − M Thus the DS reduction has central charge c W N − M, ··· U ( N | L ) Ψ = 1 − N − L Ψ (cid:0) ( N − L ) − (cid:1) − Ψ( N − M ) (cid:0) ( N − M ) − (cid:1) + − ( N − M )( N − M −
1) (( N − M )( N − M − −
1) ++ ( L − M )( N − M − (cid:0) ( N − M − − (cid:1) ++ (cid:0) ( N − L ) − (cid:1) + ( N − L )( N − M ) (cid:0) ( N − M ) − (cid:1) (B.12)i.e. c W N − M, ··· U ( N | L ) Ψ = 1 − N − L Ψ (cid:0) ( N − L ) − (cid:1) + − Ψ( N − M ) (cid:0) ( N − M ) − (cid:1) ++ (2 N + M − L )( N − M − N − M )++ ( N − L )( N − M −
1) + ( N − L ) − U ( M | L ) currents in W N − M, ··· U ( N | L ) Ψ have indeed level Ψ −
1, as we expected. In order to verify this fact, we need to takeinto account the contribution of the ghosts: before the DS reduction the currents inthe block A in U ( N | L ) Ψ form an U ( M | L ) Ψ − N + M current subalgebra. The ghosts whichcontribute to the level shift are these in the C ⊕ B blocks: N − M − Sb L | M system. They shift the level ofthe U ( M | L ) currents by precisely N − M − −
1, as itshould.Doing the coset by U ( M | L ) Ψ − we get c L,M,N [Ψ] = 1Ψ ( L − N ) (cid:0) ( L − N ) − (cid:1) + Ψ( M − N ) (cid:0) ( M − N ) − (cid:1) ++ 1Ψ − M − L )(( M − L ) −
1) + (2 N + M − L )( N − M ) + L − N (B.14)We can make the symmetries manifest by some simple manipulations: c L,M,N [Ψ] = 12 1Ψ ( L − N ) (cid:0) ( L − N ) − (cid:1) + 12 (1 −
1Ψ )( N − L ) (cid:0) ( N − L ) − (cid:1) ++ 12 Ψ( M − N ) (cid:0) ( M − N ) − (cid:1) + 12 (1 − Ψ)( N − M ) (cid:0) ( N − M ) − (cid:1) ++ 12 11 − Ψ ( L − M )(( L − M ) −
1) + 12 ΨΨ − M − L )(( M − L ) − L − N − M )(2 M − N − L )(2 N − L − M ) (B.15)We see a sum of all the S images of the first term plus a symmetric function of N , M , L . The calculation for M > N gives the same answer.
B.2 Ortho-symplectic case
This appendix gives some details of the calculation of central charges for ortho-symplecticalgebras.Let us denote central charge of Y − L,M,N [Ψ] as c − L,M,N [Ψ]. By definition, the centralcharge of ˜ Y − L,M,N [Ψ] can be identified as ˜ c − L,M,N [Ψ] = c − L,N,M [1 − Ψ].Recall that the central charge of
OSp (2 N | L ) Ψ − N +2 L +2 is c OSp (2 N | L ) Ψ − N +2 L +2 = ( L − N )(2( L − N ) + 1)(Ψ + 2( L − N ) + 2)Ψ . (B.16)Assuming N > M + 1, one needs to perform a DS-reduction in the O (2( N − M ) − OSp (2 N | L ). The principal embedding of the su (2) algebra inside– 86 – (2( N − M ) −
1) can be identified with the 2( N − M ) − su (2). The stress-energy tensor modification term leads to a contribution to the centralcharge given by c ∂H = − L − N ) + 2) N − M − (cid:88) n = − N + M +1 n = 2(2 + 2( L − N ) + Ψ)(1 + 2( M − N ))( M − N )(1 + M − N ) . (B.17)There are againt two kinds of contributions coming from ghosts in different blocks. Inthe block where the DS-reduction is performed, different components of the currentalgebra decompose as SO (2( N − M ) − (cid:39) ⊕ ⊕ ⊕ · · · ⊕ N − M ) − c gh D = N − M − (cid:88) n =1 2 n − (cid:88) m =1 (1 − m − )= − M − N ) (1 + 4 M − M ( − N ) + 4( − N ) N ) . (B.19)Off-diagonal blocks contain (2( M − N ) − × (2 L + 2 M + 1) components that are infundamental representation of both SO (2( N − M ) −
1) in the D-block and
OSp (2 M +1 | L ) in the A-block. To fix the fermionic components, one needs also to introducebosonic ghosts of appropriate dimension. The central charge of such bosonic ghostsequals minus the central charge of fermionic ghosts and the contribution from thisblock can be identified with c B ⊕ C = (2( M − L ) + 1) N − M − (cid:88) n =1 (1 − n − ) (B.20)= − M − L ) + 1)(1 + M − N )(1 + 2 M − M ( N −
1) + 2( N − N ) . Putting everythig together and subtracting the contribution coming from the cosetpart, one gets c − L,M,N [Ψ] = − (2( L − M ) − L − M ) + 1)( L − M )Ψ −
1+ 2(2( L − N ) + 1)( L − N + 1)( L − N )Ψ+2Ψ(2( M − N ) + 1)( M − N + 1)( M − N ) − L (1 + 6 M + M (6 − N ) − N + 6 N )+4 M − M (1 − N ) + N (5 − N + 8 N ) (B.21)– 87 –he central charge for Y + L,M,N [Ψ] will be denoted as c + L,M,N [Ψ]. It can be calculatedin the same way as the one for ˜ Y + L,M,N [Ψ] since the whole construction is independentof the value of L and one can simply set ˜ c + L,M,N [Ψ] = c + L + ,M,N [Ψ].The central charge of OSp (2 L | N ) − Ψ+2( N − L )+2 equals c OSp (2 L | N ) − Ψ+2( N − L )+2 = ( N − L )(2( N − L ) + 1)( − Ψ + 2( N − L ) + 2) − Ψ (B.22)Now, we need to perform DS-reduction in the Sp (2( N − M )) block of OSp (2 L | N ). Theprincipal embedding of su (2) inside Sp (2( N − M )) can be identified with the 2( N − M )dimensional representation of su (2) and modification term contributes to the centralcharge by c ∂H = 12( − Ψ + 2( N − L ) + 2) N − M (cid:88) n =1 (cid:18) n − (cid:19) = (Ψ + 2( N − L ) − M − N )(2( M − N ) + 1)(2( M − N ) −
1) (B.23)The decomposition of the currents in D block is again of the form Sp (2( N − M )) = 3 ⊕ ⊕ ⊕ · · · ⊕ N − M ) − . (B.24)The corresponding ghosts contribute to the central charge as c D = N − M (cid:88) n =1 2 n − (cid:88) m =1 (1 − m − ) = − M − N ) ( − M − N ) ) . (B.25)Finally, the currents in the off-diagonal block are in the product of fundamental rep-resentations of Sp (2( N − M )) and OSp (2 L | M ). Similar arguments as in the case of Y − applies here with only exception that fields of weight half are now present and weneed to fix only half of the corresponding currents. One gets a contribution c B ⊕ C = 2( M − L ) N − M (cid:88) n =2 (1 − n − ) + M − L = ( M − L )(1 + 2( M − N ))( − − N + 4( M + M − M N + N )) . (B.26)Putting everything together and subtracting the contribution coming from the cosetpart leads to c + L,M,N [Ψ] = − M − L )(2( M − L ) + 1)( M − L + 1)1 − Ψ − N − L )(2( N − L ) + 1)( N − L + 1)Ψ+Ψ(2( M − N ) − M − N ) + 1)( M − N )+ L (1 − M − N ) ) − N + 2( M − N ) (3 + 2 M + 4 N ) (B.27)– 88 – Series, products and contour integrals
A useful contour integral identity with symplectic boson denominators (cid:73) N (cid:89) i =1 dx i x i x s i i (cid:81) i,n (1 − q n + x i )(1 − q n + x − i ) == 1 (cid:81) n> (1 − q n ) N (cid:73) N (cid:89) i =1 dx i x i x s i i ∞ (cid:88) n i =0 n i (cid:88) m i = − n i (cid:89) i x m i i ( − (cid:80) i ( n i − m i ) q (cid:80) i ni ( ni +1) − m i == 1 (cid:81) n> (1 − q n ) N ∞ (cid:88) n i = | s i | (cid:89) i ( − (cid:80) i ( n i + s i ) q (cid:80) i ni ( ni +1) − s i == 1 (cid:81) n> (1 − q n ) N ∞ (cid:88) n i =0 (cid:89) i ( − (cid:80) i n i q (cid:80) i ni ( ni +1)+(2 ni +1) | si | == 1 (cid:81) n> (1 − q n ) N ∞ (cid:88) n i =0 (cid:89) i ( − (cid:80) i n i q (cid:80) i ni ( ni +1)2 q ( n i + ) s i (C.1)The only non-trivial step is the removal of the absolute value | s i | → s i : if s i is negativethe sum without the absolute value has the first 2 s i terms cancelling out in pairs. Thefinal result is that of a sum over residues of the contour integral at x i = q ( n i + ) .In a similar manner, a contour integral with current denominators (cid:73) N (cid:89) i =1 dx i x i x s i i (cid:81) i,n (1 − q n +1 x i )(1 − q n +1 x − i ) = (cid:73) N (cid:89) i =1 dx i x i q si x s i i (cid:81) i (1 − q x i ) (cid:81) i,n (1 − q n + x i )(1 − q n + x − i ) == 1 (cid:81) n> (1 − q n ) N ∞ (cid:88) n i =0 (cid:89) i ( − (cid:80) i n i (1 − q n i +1 ) q (cid:80) i ni ( ni +1)2 q ( n i +1) s i (C.2) D Characters for W N − M, , ··· , U ( N ) The contribution to the index from the U ( N ) currents can be split it to the contributionscoming from different blocks as χ DS M U ( N ) = χ A χ B χ C χ D . (D.1)The fields in the A -sector do not have a modified conformal weight but they are gradedwith respect to the currents J { h } preserved by the DS-reduction. The corresponding– 89 –haracter is then χ A = ∞ (cid:89) n =1 N − M (cid:89) i,j =1 − x i x − j q n (D.2)where x i is a fugacity for the current J { h i } . In the D -block, there are no factors of x i but the conformal weights of the fields are non-trivially shifted. One gets the character χ D = ∞ (cid:89) n =1 N − M − (cid:89) j =1 − q n + j ) N − M − j − q n ) N − M − q n − j ) N − M − j . (D.3)Both B - and C - blocks give the same contribution. The fields in these blocks arecharged under J { h } (with opposite charges in the two blocks) and have dimensionsshifted by the the stress-energy tensor modification. The characters are then χ B = ∞ (cid:89) n =1 N − M (cid:89) i =1 M (cid:89) j =1 − x i q n + M +12 − j ,χ C = ∞ (cid:89) n =1 N − M (cid:89) i =1 M (cid:89) j =1 − x − i q n + M +12 − j . (D.4)The contributions to the character from the ghost sector can be again divided intocontributions from different blocks. There are no ghosts associated to the A -block. Thecontribution from the bc -ghosts in the D -sector is χ bcD = ∞ (cid:89) n =1 N − M − (cid:89) i =1 (1 − q n + i − ) N − M − i (1 − q n − i ) N − M − i . (D.5)The contributions from the bc -ghosts in the B - and C -sectors contains fugacities x i since they are charged under the total U ( M ) currents, again with opposite charges inthe two blocks, χ bcB = ∞ (cid:89) n =1 N − M (cid:89) i =1 M (cid:89) j =1 (1 − x j q n + i − )(1 − x j q n − i + ) ,χ bcC = ∞ (cid:89) n =1 N − M (cid:89) i =1 M (cid:89) j =1 (1 − x − j q n + i − )(1 − x − j q n − i + ) . (D.6)with some extra correction depending on N − M being odd or even to account correctlyfor the ghosts with weight 1 /
2. – 90 –
Boundary conditions for hypermultiplets
The hypermultiplet SUSY transformations take the schematic form δ A ˙ Aα q Ba = (cid:15) AB ρ ˙ Aαa δ A ˙ Aα ρ ˙ Bβa = (cid:15) ˙ A ˙ B ∂ αβ q Aa (E.1)where A, · · · , ˙ A, · · · respectively denote indices for SU (2) H and SU (2) C , α, · · · spinorindices and a is a flavor index.The supercurrents are S A ˙ Aαβγ = ω ab ∂ ( αβ q Aa ρ ˙ Aγ ) b (E.2)We seek boundary conditions which preserve (0 ,
4) supersymmetry at the boundary.Correspondingly, the normal components of the supercharges which are right-chiral onthe boundary must vanish: S A ˙ A + −− = 0 (E.3)There are two natural Lorentz-invariant boundary conditions for the fermions whichpreserve the SU (2) C R-symmetry and flavor groups: ρ ˙ A + a = 0 ( N ) ρ ˙ A − a = 0 ( D ) (E.4)These conditions then require respectively ∂ + − q Aa = 0 ( N ) ∂ −− q Aa = 0 ( D ) (E.5)which explain our monikers: the first possibility requires Neumann boundary conditionsfor all hypermultiplet scalars, while the second possibility (together with the CPTconjugate relation) requires Dirichlet boundary conditions for all hypermultiplet scalars: q Aa = 0 ( D ) (E.6)Next, we need to consider some deformations of these boundary conditions whichbreak Lorentz symmetry, but preserve a twisted Lorentz group which is defined eitherwith the help of the Cartan of SU (2) H or the Cartan of SU (2) C .If we twist by SU (2) H then we have scalar supercharges Q + ˙ A + and Q − ˙ A − . We mayseek a boundary condition which preserves Q − ˙ A − + ζQ + ˙ A + . That means S − ˙ A + −− + ζS + ˙ A ++ − = 0 (E.7)– 91 –he natural boundary conditions on the fermions are unchanged ρ ˙ A + a = 0 ( N ) ρ ˙ A − a = 0 ( D ) (E.8)which imply ∂ + − q − a + ζ∂ ++ q + a = 0 ( N ) ∂ −− q − a + ζ∂ + − q + a = 0 ( D ) (E.9)The first choice is an interesting deformation of the standard Neumann boundary con-ditions. It will be important for us. The second choice is not a deformation of Dirichletboundary conditions. Rather, it gives the parity conjugate of the deformed Neumannboundary conditions. Thus Dirichlet boundary conditions do not admit a deformationof this type.There is a useful way to think about the deformation of boundary conditions in-duced by a deformation of the preserved supersymmetry. If we add some extra termto the boundary action which breaks Q − ˙ A − , the variation of the term under Q − ˙ A − willappear as the boundary value S − ˙ A + −− of the corresponding supercurrent. Thus we canfind a deformation which preserves the deformed SUSY if we can write the boundaryvalue S + ˙ A ++ − of the other supercurrent as a Q − ˙ A − variation of some boundary action O ++++ .For example, for Neumann b.c. we have S + ˙ A ++ − = ω ab ∂ ++ q + a ρ ˙ A − b = δ − ˙ A − (cid:0) ω ab ∂ ++ q + a q + b (cid:1) (E.10)which is the variation of a natural boundary action which is equal to the action forsymplectic bosons. On the other hand, for Dirichlet b.c. we have S + ˙ A ++ − = ω ab ∂ + − q + a ρ ˙ A + b (E.11)which is not a δ − ˙ A − variation.If we twist by SU (2) C then we have scalar supercharges Q A ˙++ and Q A ˙ −− . We mayseek a boundary condition which preserves Q A ˙ −− + ˙ ζQ A ˙++ . That means S A ˙ − + −− + ˙ ζS A ˙+++ − = 0 (E.12)The boundary conditions on the fermions can now be twisted as well ρ ˙++ a = ηρ ˙ −− a ( N ) ρ ˙ − + a = 0 ( N ) ρ ˙+ − a = 0 ( D ) ρ ˙ −− a = ηρ ˙++ a ( D ) (E.13)– 92 –hich imply (1 + 2 η ˙ ζ ) ω ab ∂ + − q Aa ρ ˙ −− b + ˙ ζω ab ∂ ++ q Aa ρ ˙+ − b = 0 ( N )2 ηω ab ∂ + − q Aa ρ ˙++ b + ω ab ∂ −− q Aa ρ ˙ − + b + 2 ˙ ζω ab ∂ + − q Aa ρ ˙++ b = 0 ( D ) (E.14)The first choice is inconsistent. There is no linear boundary condition on the scalarfields which can satisfy this constraint. Thus Neumann boundary conditions do notadmit this type of deformation.On the other hand, standard Dirichlet b.c. for the hypermultiplet scalars, togetherwith η = − ˙ ζ , give a useful deformation of Dirichlet boundary conditions. It will beimportant for us.Again, the deformability or lack thereof is related to the observation that for Dirich-let b.c. we have S A ˙+++ − = ω ab ∂ + − q Aa ρ ˙++ b = δ A ˙ −− ( ω ab ρ ˙++ a ρ ˙++ b ) (E.15)while for Neumann S A ˙+++ − = ω ab ∂ ++ q Aa ρ ˙+ − b (E.16)cannot be written as an δ A ˙ −− variation. E.1 Neumann boundary VOA
Deformed Neumann boundary conditions support supersymmetric boundary local op-erators: the bulk SUSY transformations (cid:16) δ − ˙ A − + ζδ + ˙ A + (cid:17) q Ba = (cid:15) − B ρ ˙ A − a + ζ(cid:15) + B ρ ˙ A + a (cid:16) δ − ˙ A − + ζδ + ˙ A + (cid:17) ρ ˙ Bβa = (cid:15) ˙ A ˙ B ∂ − β q − a + ζ(cid:15) ˙ A ˙ B ∂ + β q + a (E.17)restricted to the boundary give (cid:16) δ − ˙ A − + ζδ + ˙ A + (cid:17) q Ba = (cid:15) − B ρ ˙ A − a (cid:16) δ − ˙ A − + ζδ + ˙ A + (cid:17) ρ ˙ B − a = (1 + ζ ¯ ζ ) (cid:15) ˙ A ˙ B ∂ −− q − a (E.18)showing that the q − a are supersymmetric and holomorphic modulo operators in theimage of the supercharges.Notice that for non-zero ζ the q − a are supersymmetric only at the boundary. Thisfact allows them to have non-trivial holomorphic OPE. A simple calculation of boundary-to-boundary propagators recovers the symplectic boson OPE with coefficient propor-tional to ζ . – 93 –et us introduce real fields q i such that q − = q + iq ,q + = q − iq . (E.19)where we supressed the flavor indices s . in terms of the real fields, we can write aboveboundary conditions as ∂q + ζ ( ∂ q − ∂ q ) = 0 ,∂q − ζ ( ∂ q + ∂ q ) = 0 ,∂q − ζ ( ∂ q − ∂ q ) = 0 ,∂q + ζ ( ∂ q + ∂ q ) = 0 . (E.20)All the componens further satisfy bulk equations of motion ∆ q i = 0. Going to momen-tum space and introducing boundary source, and substituting k → k + k , we canexpress boundary-to-boundary propagator as (cid:104) q i ( k ) q j (0) (cid:105) = 11 − ζ k + k k ζk − ζk k − ζk − ζk ζk − ζk k − ζk − ζk k . The only non-vanishing boundary to boundary propagator of Q -closed operators is then (cid:104) ¯ q + ( x ) q − (0) (cid:105) = 2 ζ − ζ (cid:90) dk dk e ik x + ik x k − ik k + k . (E.21)Note that this correlation function can be expressed as= 2 ζζ + 1 ¯ ∂ (cid:90) dk dk e i ( x k + x k ) ( k + k ) = 12 π ζζ + 1 ¯ ∂ ln | z | = 1 π ζζ + 1 1 z . (E.22)This is the propagator of the symplectic boson. E.2 Dirichlet boundary VOA
Deformed Dirichlet boundary conditions support supersymmetric boundary local op-erators: the bulk SUSY transformations are (cid:16) δ A ˙ −− + ˙ ζδ A ˙++ (cid:17) q Ba = (cid:15) AB ρ ˙ −− a + ˙ ζ(cid:15) AB ρ ˙++ a (cid:16) δ A ˙ −− + ˙ ζδ A ˙++ (cid:17) ρ ˙ Bβa = (cid:15) ˙ − ˙ B ∂ − β q Aa + ˙ ζ(cid:15) ˙+ ˙ B ∂ + β q Aa (E.23)– 94 –t the boundary, they simplify to (cid:16) δ A ˙ −− + ˙ ζδ A ˙++ (cid:17) ρ ˙++ a = − ∂ + − q Aa (cid:16) δ A ˙ −− + ˙ ζδ A ˙++ (cid:17) ρ ˙ − + a = 0 (E.24)Thus ρ ˙ − + a are supersymmetric at the boundary. This fact allows them to have non-trivial holomorphic OPE. F Boundary conditions for gauge theory N = 4 super Yang-Mills admits many half-BPS boundary conditions and interfaces[10]. These boundary conditions preserve a set of supercharges which form a 3d N = 4superalgebra. These 3d N = 4 sub-algebras may be embedded in a variety of differ-ent ways in the four-dimensional super-algebra, depending on the choice of boundarycondition.In particular, if we look at boundary conditions and interfaces which descend fromIIB string theory configurations associated to ( p, q ) fivebranes aligned along three spe-cific directions in spacetime (say 456, rotated by SU (2) H ) we will find a correspondingtwo-parameter family of 3d N = 4 sub-algebras A d ( p,q ) which are permuted by S-dualitytransformations [8].The corner configurations considered in this paper preserve four chiral superchargesin 2d, which are analogous to the supersymmetries preserved by the boundary condi-tions in Appendix E. This 2d (0 ,
4) subalgebra will be a common sub-algebra to all the3d N = 4 sub-algebras associated to ( p, q ) boundary conditions of appropriate slope inthe plane of the junction.We can denote the (0 ,
4) supercharges as Q A ˙ A − . The 3d subalgebras will consistof Q A ˙ A − together with appropriate linear combinations of two sets of anti-chiral super-charges Q A ˙ A + and ˜ Q A ˙ A + . These have the properties that they anti-commute with Q A ˙ A − to translations in the plane of the junction: { Q A ˙ A − , Q B ˙ B + } = (cid:15) AB (cid:15) ˙ A ˙ B P { Q A ˙ A − , ˜ Q B ˙ B + } = (cid:15) AB (cid:15) ˙ A ˙ B P (F.1)We can organize the remaining supercharges into a set ˜ Q A ˙ A − with { ˜ Q A ˙ A − , Q B ˙ B + } = (cid:15) AB (cid:15) ˙ A ˙ B P { ˜ Q A ˙ A − , ˜ Q B ˙ B + } = − (cid:15) AB (cid:15) ˙ A ˙ B P (F.2)Consider now an SU (2) H twist of the Lorentz generator in the 01 plane, as inAppendix E. We have the scalar supercharges Q − ˙ A − and we may look for a way to– 95 –eform them. The common deformed subalgebra should contain, in particular, theGL-twisted supercharge with parameter Ψ.It is clear that we cannot find a deformation analogous to the one Appendix E whichbelongs to all the 3d subalgebras: each 3d sub-algebra would require us to deform Q − ˙ A − by different linear combinations of Q + ˙ A + and ˜ Q + ˙ A + . Instead, we will need to deformsimultaneously both the (twisted Lorentz scalar part of the) 3d subalgebras and thecommon 2d subalgebra.The deformation of Neumann b.c. compatible with a GL twist is well understood[5]. It is proportional to Ψ and thus vanishes when Ψ = 0. In a similar manner,the B ( p,q ) boundary conditions will be undeformed when Ψ is the appropriate rationalnumber. Thus for Ψ = 0 we expect to be able to reach the GL twist by deforming Q − ˙ A − by Q + ˙ A + , for Ψ = ∞ we expect to be able to reach the GL twist by deforming Q − ˙ A − by˜ Q + ˙ A + , etc. In general, the deformed supercharge should look roughly as Q − ˙ A − + ζ (cid:16) Q + ˙ A + + Ψ ˜ Q + ˙ A + (cid:17) (F.3)From the point of view of the 3d sub-algebra ( Q − ˙ A − , Q + ˙ A + ) preserved by Neumannboundary conditions, we are first deforming the 3d superalgebra to something like( Q − ˙ A − + ζ Ψ ˜ Q + ˙ A + , Q + ˙ A + ) and then the 2d subalgebra as in Appendix E. Similar consider-ations apply to the other 3d subalgebras.Again, the deformability of 3d boundary conditions is encoded in the requirementthat for every supercharge Q we want to deform by another supercharge Q (cid:48) , the bound-ary value of the supercurrent for Q (cid:48) should be the Q variation of a local operator weare deforming the boundary condition by.We expect all boundary conditions and interfaces which arise from configurationsof ( p, q ) fivebranes with appropriate slope to admit deformations of this type. Wealso expect deformability of 3d half-BPS boundary conditions to be rare. Analogousconstraints were found in [9]: Neumann boundary conditions coupled to generic 3d N = 4 matter are compatible with a bulk θ angle only if the matter theory momentmaps satisfy certain quadratic identities. Turning on non-zero Ψ is analogous to turningon a bulk θ angle. References [1] A. Kapustin and E. Witten,
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