JHEP05(2019)159 Springer May 1, 2019 April 5, 2019 May 24, 2019 : : : October 19, 2018 : Revised Accepted Published Received . Secondly, we propose a simple 3 ,N 2 ,N Published for SISSA by 1 https://doi.org/10.1007/JHEP05(2019)159 N Y b [email protected] , = 4 SYM. In most of the paper, we concentrate on truncations N to a general truncation . 1 3 N and Miroslav Rapˇc´ak W a 1808.08837 The Authors. Conformal and W Symmetry, Conformal Field Theory, Supersymmetric c

We study the structure of modules of corner vertex operator algebras arrising at , associated to the simplest trivalent junction. First, we generalize the Miura [email protected] ∞ 1+ -algebra modules, free fields, and Gukov-Witten W Ludwig Maximilian University of Munich,Perimeter Theresienstr. Institute for 37, Theoretical D-80333 Physics, 31 M¨unchen,Germany Caroline St N, Waterloo, ON N2LE-mail: 2Y5, Canada Arnold Sommerfeld Center for Theoretical Physics, b a Open Access Article funded by SCOAP ArXiv ePrint: their modules have a simple gauge theoretical interpretation. Keywords: Gauge Theory of highest weight charges.exponents of Parameters of vertex the operatorsGukov-Witten generating defects in function the in can free thedegenerate be modules. field gauge identified In realization theory the with and lastmodules picture. section, parameters of we more associated Finally, sketch how complicated to we to algebras. glue discuss generic Many some modules properties to aspect produce of of vertex operator algebras and Abstract: junctions of interfaces in of transformation for parametrization of their generic modules, generalizing the Yangian generating function Tom´aˇsProch´azka W defects JHEP05(2019)159 21 22 23 35 39 13 13 40 28 42 3 ,N 2 ,N 1 N Y 36 28 26 3 – i – 9 41 37 1 , 19 3 ,N 1 2 , 41 ,N 1 3 17 2 10 ,N 31 Y 1 ,N ,N 2 N 1 Y N ,N 1 Y 10 13 38 N 37 Y 5 7 ,N 0 38 , 38 0 , 0 1 , 36 Y 4 1 40 Y 5 18 — singlet algebra of symplectic— fermion Parafermions algebras 0 2 4 , , 1 1 (1) example N , , 1 1 0 b gl W Y Y = 1 , 0 , 0 5.2.6 Summary 5.2.7 Free field representation of degenerate primaries 5.3.1 Identity, box5.3.2 and anti-box Analysis at the level of generating functions 5.2.1 Singular5.2.2 vectors at level 1 Higher5.2.3 levels Virasoro5.2.4 algebra 5.2.5 Algebra 4.6.1 4.6.2 3.2.3 Miura versus screening 3.2.1 Review3.2.2 of The general case Y 5.3 Maximally degenerate modules 4.7 A comment on null states of 5.1 Surface defects5.2 preserving Levi subgroups Minimal degenerations 4.3 Zero mode4.4 algebra of Generating function4.5 for Relation to4.6 the free boson Examples modules of varieties of highest weights 4.1 Zero modes4.2 and generic modules Generating function of highest weights for 3.1 Four definitions3.2 of Y-algebras Miura transformation for 2.1 The corner 2.2 Line operators2.3 Gukov-Witten defects 2.4 2.5 Reparametrization of GW defects 5 Degenerate modules 4 Generic modules 3 Y-algebras and free fields Contents 1 Introduction 2 Gauge theory setup JHEP05(2019)159 ) N 63 64 W p, q ] and of 3 , . These 3 2 ]. 1 ,N 2 ,N → ∞ 1 N 1) interfaces be- N Y , 58 61 ] and its equivalence to Yan- 45 29 ]. Local operators of the twisted – -algebras 13 26 W – ]. Configurations of interest are ( 9 11 – 6 ] based on the previous work of [ ] that the algebras can be also viewed as 1 5 – 1 – ) and their ]. This makes it possible to use the techniques of N ( algebra. The study of this algebra has a very long 32 b gl – 53 ]. Recently, this algebra appeared in connection with 51 ∞ 29 25 , 1+ – 51 W 27 ]. The new perspective is based on a realization of VOAs as 18 46 47 49 5 , ) gauge theories leads to the VOA labeled as 53 4 2 3 , 55 1 N , 0 (1)’s example example example Y b gl = 4 super Yang-Mills theory [ 2 0 0 ]. Later, it was argued in [ ), U( , , , 3 0 1 1 2 1 , , , 0 1 2 N N W Y Y Y (1) was found [ ]. Later, it was gradually realized that there exists in fact a two-parametric gl 17 60 ), U( – 1 (2) from gluing from 14 N b gl 5.3.4 5.3.5 5.3.6 5.3.3 Free field realization and OPE of modules ] of supersymmetric interfaces studied in [ 0 , 1 10 , 1 The simplest configuration of the triple junction of D5, NS5 and (1 Y 6.2 Gluing and6.3 generic modules Gluing two 6.4 6.5 Gelfand-Tsetlin modules for 6.1 Gluing using free fields algebras [ family of non-linear algebrasequivariant [ cohomology of instantongian moduli of affine spaces [ integrability to study the properties of vertex operator algebras. tween U( corner algebras were originallysuper-algebras in identified [ in termstruncations (quotients) of of a the history. BRST reduction Originally, of a Kac-Moody linear version of the algebra was constructed as it was further explored inalgebras [ of local operatorsfour-dimensional within a topologicalwebs twist [ of a particulartheory configuration turn in out to the live at the two-dimensional junction and give rise to VOAs [ 1 Introduction The theory of vertexhistory. operator algebras Recently, a (VOA) is newand an way mathematical to enormously applications was study rich initiated subject VOAs in and with [ to a connect long them with various physical C More details of 7 Outlook A Transformation between primary and quadratic bases B 6 Gluing and generic modules JHEP05(2019)159 for -th (1.1) N (3) i . This R (3) i R generators. dimensional 3 3 ,N and by a three 2 N algebras is rela- operators of the 1 ,N , 1 + 1 1 -dimensional torus -dimensional torus , ,N N N ]. Generic modules 2 0 0 complex parameters , N N Y Y 0 N 1 38 Y N + for 1 complex numbers modulo 3 N with algebras defined previ- C N ] and [ (1) = by introducing two classes of 3 b gl 37 3 ,N 2 × ,N ]. It is well known that the kernel 2 ,N 1 . N ,N 1 N 35 1 , has an explicit construction in terms W − Y N ]. Gukov-Witten defects for the corre- 34 Y +1) ,N 30 3 0 , N 0 Y and taking the product of +1)( 2 – 2 – from the point of view of (non-freely generated) . In general, we argue that generic highest weight N seems to be more complicated at first sight. The of the first order differential operators (2) i 2 ] supported in the bulk survive the twisting pro- , 2 3 3 (1) = , 1 R , 1 , b 0 +1)( N gl ,N 33 0 1 )) supported at the two boundaries. Note that line 2 , Y Y N 9 × ,N ( N and 1 C for ]. Generators of the algebra are coefficients of an N N 5 (1) i Y and 36 W C R 1 , 1 , In this work, we identify 0 algebra should be parametrized by tori with the modular parameter being the canonical parameter Y 3 The representation theory of ,N N 2 ,N 1 N Y . We call the corresponding parameters lifted GW parameters. of the second type and ] and surface defects [ . We generalize this construction to N 2 9 C N , 8 , 6 ,...,N The situation of a general Apart from local operators living at the corner, line operators supported at inter- = 1 subvariety (the number of GW parameters in the setup) inside The gauge theory setup suggests thatpler. the parametrization Indeed, of we representations find should that be generic sim- modules can be parametrized by extensions of the Virasoroerators algebra of by spin generators 1,3,4,5 ofcan spin in be 1,3 the parametrized in second bydimensional the case a subvariety inside first was two-dimensional case subvariety studied and inside inmodules by [ of gen- the the boundary condition imposed onabove. the Such GW a defect that fusion has (orto been the the implicit choice full in of the a discussion boundary condition) lifts the representation theory of which is a productof of the Kapustin-Witten twist Ψ.imally These degenerate lead modules to correspond genericdimensional modules. to representations a On of the pair U( otheroperators of hand, can line max- be operators fused (parametrized with by the finite end-line of the Gukov-Witten defect. This fusion changes Generic modules. tively well-understood. Generic modulesthe are parametrized action by of Weylto group. be On parametrized the by othersponding a hand, gauge pair maximally theory of configuration degenerate Young are modules tableau parametrized are [ by known a complex order differential operator whichi is a product ofpseudo-differential first operators order differential operators first type, provides us with a simple way to determine the free field realization of dimensional operators together with free field realizationFree of field the realization. algebras. ously in terms of aof kernel screening of charges screening charges realizing of by [ the Miura transformation [ faces [ cedure. If welocal let insertions line at operators theGukov-Witten junction to (GW) generates end surface a at defects module theules. ending for at junction, the The interfaces corresponding the main also VOA. fusion objective Similarly, play of of the this the role paper of endpoint is VOA with mod- the study of modules associated to such higher JHEP05(2019)159 is by for (1.2) ) the ∞ ) 3 (1.3) (1.4) 3 κ ( i ,N 1+ N 2 x W ,N 1 charges of N i Y ψ (1) generated by and similarly for gl m n 2 -algebra generators) . h ) κ W + ( i corner. x n 1 2 . h Ψ 1 √ = − (3) i can be identified (up to constant Ψ x ) √ κ n ( i − 1 = x h 3 (3) j + x 3 . As discussed above, the algebra 1 h 1) interfaces is the second and the corner between ]. , , h on the highest weight state. Such an action − (3) i 38 Ψ i x or = ψ √ – 3 – charges whose poles are parametrized by − . The modules of interest can be defined in terms (2) j that parametrize the line operators (finite dimen- i i = x ψ ] and [ , e 2 i m − 37 2 , f n, m i h (3) i ψ parameters x + , h with the three families of lifted GW parameters discussed 3 Ψ n ) 1 1 N κ charges (eigenvalues of zero modes of √ ( i h x 0 A generic GW defect breaks the gauge group at the defect to = ) . A degeneration of a generic module appears when we specialize = i i 1 and h N W (3) j (2) i x are related to the canonical parameter Ψ of the Kapustin-Witten twist by x − κ h . This configuration can be further dressed by turning on a Wilson or ’t -algebra generators on the highest weight vector of a tensor product of (3) i 2 free-boson Fock modules. Parameters x − i W and positive integers 3 N N i, j 3. The change of basis of algebra generators allows us to translate + parameters , U(1) 2 2 2 , N × N When more parameters are specialized, one gets further degenerations associated to Further degenerations appear when the following specialization Moreover, we identify , We call the corner between D5 and NS5 interfaces with the gauge theory of the gauge group U( The parameters + 2 1 1) and NS5 interfaces is the first. = 1 1 , (1) i third corner. Similarly, the(1 corner between D5 and (1 more complicated Levigets subgroups. maximally degenerate If modulesoperators a that with maximal trivial can surface number be defects. identified of with them the are configuration specialized, of one line happens between GW parametersthe in other different pairs corners of for parameters. any integer for some sional SU(2) representations) at the two boundaries of the third two of the parameters specifyingare singularity of equal, the the complexified gauge preservedU(2) field gauge at group the interface isHooft enhanced line to of the the next-to-minimalappears preserved when Levi SU(2) the subgroup factor. lifted GW The parameters corresponding satisfy degeneration of the module shifts) with such highest weights, giving the thirdDegenerate perspective modules. on the maximal torus U(1) GW parameters such that a Levi subgroup is preserved at the defect. In particular, when the affine Yangian toand ( recover the Zhu varieties from [ above. In termsan of action the of freeN field realization, one can construct modules of isomorphic as an associative algebraan to infinite the well number known of affine generators of Yangian of an action ofis the encoded commuting in generators a generatingκ function of x JHEP05(2019)159 - ). 1 W p, q ]. -algebras. 43 , W 42 ] and their = 4 super Yang- 41 N ]. 5 ) from [ ] and comment on the main N 1 ( 0). gl , ) gauge theories. The corresponding i ] that local operators in the cohomology N ) associated to co-prime numbers ( D3-branes ending from the left and from 1 ) gauge theories as shown in the figure 2 (1) Kac-Moody algebra that serves as a -algebras associated to trivalent juncions 3 ] of the configuration with the canonical 2 b gl N 6 Y N N = 1 – 4 – , and 0 , 1 0 Y ) and U( N ) and U( 1 ], we proposed a construction that associates a VOA ] by finding a way to possibly determine all OPEs of 2 5 5 ) Kac-Moody algebra and corresponding N 1) and the D5-brane with (1 )-web configurations studied in [ N , N ( In [ p, q b gl ), U( 1 The simplest quarter-BPS trivalent junction of NS5, D5 and N 3 and deformed boundary conditions in such a way that the Kapustin- ) web of interfaces between U( )-brane. 1 ]. These triple junctions serve as building blocks of more complicated 1 p, q CP p, q , one can realize the fundamental and the anti-fundamental representation ∈ 5.3 ]. 40 , Let us restrict to Kapustin-Witten twist [ The gauge theory picture suggests that generic modules of glued algebras can be ob- One identifies the NS5-brane with (0 39 1) interfaces between U( 3 , junctions coming form various ( parameter Ψ Witten supercharge is preserved. It was argued in [ Mills theories with gauge groupsThe gauge U( theory setup descendsthe from right on a(1 ( was analyzed in [ we discuss the simpleprototype example for of the general the discussion in later2.1 sections. The corner There exists a class of half-BPS domain walls between four-dimensional algebra analogues. Generally, one obtains wild classes of irregular2 modules [ Gauge theory setup In this section, we brieflyplayers review the (line gauge operators theory setup and from Gukov-Witten [ defects) in the discussion of modules. Finally, identification of GW parameters is neededbi-modules for added generic in modules the to gluingture have trivial procedure. of braiding As generic with a modulesWe non-trivial of conjecture example, the that we a discuss subclass themodules struc- of induced modules from coming generic from GW Gelfand-Tsetlin defects modules can of be identified with of [ tained as a tensorcorrectly product identified. of The corresponding total modules numbera of of continuous sum each parameters of vertex of a with all generic GW the module parameters is numbers thus of D3-branes at each face. From the VOA point of view, this gluing fields by which wethe extend section the product ofassociated to Y-algebras. each interface In as particular, aare as Fock actually descendant discussed many of in possible a vertex choicestest operator for of in a free given examples) bosons. free thatrepresentant There field as realization the and long result we as conjecture (if we (and non-vanishing) include contour is integrals independent of of screening the currents choice along the of lines the to an arbitrary ( VOA is an extensionof of the the diagram tensor by a productedges fusion of of of bi-modules the associated diagram.pletion to line of The operators the free supported gluing field at proposal realization internal from discussed [ above points towards the com- Gluing of generic modules. JHEP05(2019)159 ] t ) 2 33 N and ) as 3 β . Since [Ψ]. ) and N ∨ 3 | ), U( T 1 N 1 ,N 2 N N ,N 1 ] by four real N Y 33 , and attaching it to 9 , . 8 C 3 factor corresponds to the ∈ ,N 2 z C ,N 1 C N Y has a natural grading by spin × 1) interfaces between U( 3 , ). These modules play the role of ,N 1 2 N of the Lie super-group U( | ,N 2 1 µ N N Y is the Cartan of the gauge group U( [Ψ]. For each choice of ranks of gauge groups ). In the GL-twisted theory, parameters determines the insertion of the corresponding 3 T D5 = (1,0) – 5 – N z ,N ( 2 u ,N 3 1 N N 2 plane of the gauge theory with Y 2 ), where N . The algebra , x 3 ,T 1 1 t x ,N , 2 t N NS5 = (1,0) ) gauge theory are labeled according to [ ,N lives in the Cartan subalgebra of the Langlands dual gauge group T, 1 N . The endpoint ( (1,1) η N C Y ∈ ) ∈ ) representations and line operators at the (1,1)-interface are Wilson z 2 N | 3 ]. Similarly, line operators at the D5-interface are ’t Hooft operators as- N 9 α, β, γ, η ( 4 The simplest trivalent junction of NS5, D5 and (1 , one obtains a one parameter family of VOAs parametrized by the canonical 3 ) gauge theories with a two-dimensional corner. The junction on the left shows the 3 N ,N directions. 2 GW defects in the U( Line operators supported at the NS5-interface can be identified with the Wilson lines 4 ) is left invariant under the Langlands duality, we do not distinguish them in this work. In general, the parameter ,N 4 N , x 1 3 parameters the Cartan subalgebra of the Lie algebra U( 2.3 Gukov-Witten defects Apart from the linealso operators survive the discussed GL above, twist.one Gukov-Witten Inserting of (GW) such the a surface corners GW of defectsfor defect the [ at the Y-shaped a junction, corner point one VOA. gets a new (continuous) family of modules line operators associated todegenerate representations modules of U( of and degenerate modules are characterizedgraded by component the compared fact to that a they contain generic less module. states in some vertex operator from theat CFT the point corner of with view. the line The endpoint process generates of aassociated fusing module local to for operators a living discussed finite-dimensional in representation [ sociated to U( Apart from the localported operators at each living of at theoperators the three supported two-dimensional interfaces at corner, are one part linethe of of corner operators the the at sup- twisted three point theory interfaces, as going well. from Consider the line infinity and ending at of the Kapustin-Witten supercharge are supportedrise at to the the junction vertex of domain operatorN walls algebra and give parameter Ψ. In the following, we often suppress2.2 the dependence on Line Ψ operators in Figure 1. and U( configuration of interfaces in the x JHEP05(2019)159 . In (2.2) (2.4) (2.5) (2.1) η ]. The around 33 and ωφ α + ), we see that A N = A . ) λ. α − → λ : α, η ( ) is related to Ψ in such a way that → 1 complex tori with modular parameters ν (2.3) N ) ) 3 ω M of the gauge group U( N η, T β, γ α, η N ( 1 η Ψ |
:( and z , . . . , λ Ψ and exchanges the role of 1 τ (3) i − 1 − 2 / S λ λ (2) i α 2 α µ → | ,N λ – 6 – 1 ∈ ) = M N 0 = Ψ diag( ,T λ ) in λ β, γ α, η (1) i α 3 ∼ λ → λ :( ) − (3) i z M η S ( η, , λ ( − A = 0. The parameter (2) i α → z labeled by finite representations of the gauge groups are supported , λ ) i of the twisted theory transforms as µ (1) i = Ψ λ M λ α, η λ complex tori of modular parameter Ψ. : :( N S S ] to deform the integration contour of the complexified Chern-Simons ) relevant to us transforms as 9 α, η is invariant under the T-transformation and the S-transformation simply λ . 3 Line operators ) transforms as ,N live in the Cartan subgroup 2 η ,N β, γ 1 Let us discuss S-duality transformation of the GW parameters identified in [ N is a closed combination at the interface (modulo a gauge transformation). Since both Y and were argued in [ We see that multiplies the Gukov-Witten parameterlater by sections, we 1 will seeof that this transformation is consistent with the triality covariance The complex parameter pair ( under the S-transformation andhand, it the pair is ( unaffected by the T-transformation. On the other A α the corresponding monodromies (andlabeled Gukov-Witen by defects points in in the GL-twisted theory) are parametrizes the monodromy ofthe the defect, complexified i.e. gauge connection near the defect at the origin γ theory. On the other hand, the combination Figure 2. at interfaces and givethe rise diagram to are degenerate labeled modules by Ψ of and give VOA. rise GW to defects generic attached modules. to the corners of JHEP05(2019)159 2 − by i N (2.6) C U(1) . Similarly × α and conformal dimension Z Ψ at the corner can be identified ∈ lifting the parameter A m Z . and the line defect supported at the 2 Ψ ) η w ∈ Ψ − m . Line operators at the D5-boundary were z 2 ( n 1 ∼ 2Ψ – 7 – complex-dimensional torus of in each corner to ) w κ ( N J ) z ( J that lifts the parameter Z normalized as ∈ J 3 living in the , configuration, line operators supported at the NS5 interface produce n 2 1 , , 0 , (1) example 0 and conformal dimension b Y = 1 gl ) since modules related by the Weyl transformations are gauge equivalent. i Z κ N = ∈ . The boundary line of the surface operator can be fused with line operators (1) current 1 (1). This example is extremely important since all the other algebras can be , κ n for b gl 0 b gl N , ) 0 C κ = ( Y . On the other hand, GW defects are parametrized by a complex torus with the λ ∈ ], line operators supported at the NS5-boundary were identified with electric modules The insertion of the complexified gauge connection Finally, let us note that throughout the discussion above, one needs to mod out Weyl For generic values of GW-parameters, the defect breaks the gauge group to the maximal When a GW defect ends at an interface, one needs to further specify a boundary 1 2 , 1 ) 0 , κ m ( 0 2 Ψ ˜ In [ of charge identified with magnetic operators with monodromy Ψ obtained from a fusion (coproduct)ple algebra. combined with the triality transformation of this sim- with the groups of U( 2.4 Let us illustrate howY above gauge theory elements fit nicely with the simplest example larger Levi subgroups. In thea case trivial that interface GW (there parameters are aremodules maximally no specialized, to singularities we in be have therepresentations maximally bulk) of degenerate. and gauge we groups expect The (labeling the corresponding line corresponding operators modules at are the interfaces). labeled by finite of the preserved SU(2) gaugeone group. gets In a the discrete parametereach set space pair of of codimension of the one lifted Cartanchamber-like GW walls structure elements. parameters, corresponding with to the The degenerate modulesmore full modules degenerating walls, parameter at for we the expect space walls. further of At degeneration generic the to modules intersection appear. of thus These has intersections a correspond to for the corner VOA. Foris special preserved values and of we parameters, expecti.e. a the Levi the corresponding subgroup representations associated of to Vermamonodromy the be parameters module gauge (partially) are group specialized, contains degenerate, the some next-to-minimalis Levi null preserved. subgroup U(2) states. One can decorate such For a example, configuration by if line operators two in some of representation the a defect with charge other interface creates a vortex ofin monodromy the Ψ other twomodules corners, coming the from line fundamental operators domain at of the the corresponding torustorus two boundaries. is at the lifted defect. to Corresponding the modules full are going to be associated to generic modules condition for thelifts defect. Weλ will see laterdiscussed that above. the Such choiceexample a of in fusion the the changes the boundary boundary condition condition for the GW defect. For JHEP05(2019)159 Ψ as / and C (2.8) (2.7) in the n λ + 1, whereas ˜ λ → + 2 + Ψ and interchange. ˜ λ λ η = ˜ λ + 2 + Ψ and that can be identified with λ α C + Ψ. The module coming from . ∈ . (1) Kac-Moody algebra normal- ˜ λ  2 ˜ λ b Ψ. This is consistent both with ˜ gl λ 1 Ψ → . 1 ˜ λ/ 2Ψ 2 − ˜ λ ) ˜ Ψ w − − . The roles of (1) algebra. GW-defects are labeled by a point . The charge of the corresponding module 1 ˜ z Ψ = 1 J b gl (  (2) thus lifts to Ψ 1 0 ˜ λ , √ ∼ λ 1 – 8 – , ) ˜ 0 Ψ] leads to the = [ Y w 0 λ ˜ ( , J 1 J , ) 0 z Y 1 ( [Ψ] = J 1 , 0 , Ψ 0 and conformal dimesion Y ˜ λ . The GW parameter at the corner of 3 A The lattice structure of modules of the eigenvalue. The fusion with an electric module shifts it by one 0 Let us show that transformations of parameters are also consistent with the triality Note that the S-duality transformation exchanges NS5-brane and D5-brane and the J lift the torus along the full complex plane parametrizing generic module of the algebra. Lattice n ized as Consider a GW defect with the parameter relation The insertion of orientation of theand diagram the gets transformed reversed. liftedthe The GW transformation transformed parameter of level becomesconformal degenerate dimension of modules of the the and generic algebraand the module so is is unlifted is invariant 1 under the GW the charge parameter. S-duality if transformation we renormalize Note that shown in the figure the the fusion with a magneticthe module GW shifts defect it by has Ψ, charge i.e. modular parameter Ψ parametrizing the monodromyin for the the bulk. complexified gauge If connection defect the with GW line operators defect supported ends atby at the 1 the boundary. and NS5 Such lifts a boundary, the linewith one operator torus modules can shifts of the fuse supported the charge the at Gukov-Witten end defect the line in D5-boundary of the the lifts real direction. it in Similarly, fusing the Ψ direction tessellating points correspond tocorresponds dyon to modules the GW-parameter. offundamental For the example, domain algebra the are modules and related ofmodule the by charge of fusion position charge with 1. in the the electric fundamental module domain of charge 2 and the magnetic Figure 3. in the torus of modularΨ parameter Ψ. Fusion with electric and magnetic modules of charges JHEP05(2019)159 Ψ / (2.9) (2) (2.11) (2.12) (2.13) (2.14) (2.15) (2.10) and Ψ ˜ λ ) group i Z h = , Ψ normal- √ (3) ˜ λ [Ψ] that equals J/ 1 , 0 = , 0 Y (3) . ∂φ Ψ 1 = √ with the coefficient in the by 3 − (3) . . (3) . J , h 2 x subgroup of the SL(2 Ψ (2) ) 2 (3) ˜ λ 3 √ η w = 0 , h 2 S . Ψ 1 1 = h 3 − √ h (2) (1) h 3 z + η η ( = 1 3 + i 2 h h ) 2 (3) 1 h 1 equals Ψ 1 h α 1 w + + 1 ( − h , h + h φ ˜ 1 + (2) (1) 1 Ψ Ψ (2) = (3) α α ˜ λ – 9 – √ − 3 2 ∼ − p x h h h 1 ) − parametrizing defects in the third corner, one can (3) J/ ˜ w λ q = = = ( (3) , h exp Ψ 2 ˜ 1 2 1 λ = (3) (2) (1) √ h h J x x ˜ ) Ψ − = z 5 ( are determined up to the overall rescaling. The VOA is − (2) , h (3) 1 ˜ i λ (3) Ψ = x h Ψ J 1 p √ = 1 h example, we can identify the parameter 1 , 0 , 0 Y Ψ, we see that the two GW parameters must be indeed related by √ In the / In the following we will drop the normal ordering symbols and we assume all the exponential vertex 5 (3) ˜ ized as operators are normal ordered. exponent of the vertex operator in the free field realization of the module with the current and similar combinations in the other two corners Instead of the lifted GWconsider parameter the combination independent on such a rescaling.for example Up as to the rescaling, one can relate parameters algebra and its modules, let us introduce parameters Note that the parameters consistently with the above discussion. 2.5 Reparametrization of GWThe trivalent defects junction of interestof is S-duality invariant under transformations. the To get manifestly triality invariant parametrization of the Comparing it with the charge withλ respect to the normalized current of with respect to the normalized current JHEP05(2019)159 k ), ≡ ∞ + 3 3 = 1 1+ ,N (3.1) (3.2) (3.3) ,N 0 (2.16) , vertex k,N 2 W 0 (3) + 3 Y together 2 . Jacobi α ,N N κ 1 h k,N + N + ,... 2 1 4 , N N 3 Y , + normalized as 1 . ) κ 1 2 N = 1 and ( λ λ i 3 ∂φ − ,N 2 = ) ,N , 1 κ ( 2 N ) of spin J Y ]. w i 1 25 − W z for Ψ = . in terms of truncations of the . ( 3 3 of unit charge corresponds to = 1 h 6 ,N = 0 generated by a singular vector at level 2 κ 2 ] of the algebra up to three parameters is an extension of the vertex operator al- 3 2 3 3 h h 3 3 1 1 λ ,N N ,N 25 λ M 1 ∞ h 2 ,N – with currents 2 N + + ,N 1+ 0 Y 23 1 , ,N 2 – 10 – 2 2 1 1 N , 1 W ∼ − λ λ 0 N N I satisfy ) I Y / i + w + = 1. λ ∞ by primary fields ( 1 ) 1 1 1 . ) is invariant under a constant shift of all integers κ , the triality tranformation simply permutes λ 1+ generators of algebras and (3) ( λ 3 T i N . η 3 J h 0 ,N W 3.3 , ) N 2 0 ,N , z 2 1 ( ,N W ) 1 Y ] that for each triple of non-negative integers ( ,N The algebra κ 1 ( N . The invariance of the charge of the current normalized to [Ψ] = Y N J ) 3 . 30 can be identified with shifts of exponents of Y κ , ( ,N ) 5 ∞ 2 κ , η ( i , one can define the quotient + 1), whenever ) i ,N 1+ x contains an ideal κ 3 1 λ ( N N α W ∞ Y are again exponents of the corresponding vertex operator. We will later ) 1+ κ ( + 1)( W x ). The first few 2 3 ) subject to the constraint N 3 (1) to arbitrary ,N b gl 2 , λ 2 × Note that the constraint ( It was argued in [ In the parametrization using In the other two frames + 1)( ,N Some truncations of this sort have been recently discussed in [ , λ 1 1 6 N 1 N N λ ( will thus have identicaldiffer operator by higher-spin product null expansions. generators forming an We ideal expect inside that the bigger the algebra. two One algebras might For these values of the algebra ( gebra of the stress-energyidentities tensor fix all( the structure constants [ alize the Miura transformation constructionW of the kernel of screening charges for 3.1 Four definitions ofTruncations Y-algebras of 3 Y-algebras and freeIn fields this section, we reviewalgebra, the the definition kernel of of screening charges and the BRST reduction. Furthermore, we gener- parameters see that parameters operators also for general with parameters identity is manifest. whereas the magnetic module to In this parametrization, the electric module JHEP05(2019)159 ] ] , ]. ∞ 46 re- 24 26 and 45 W k (3.4) , ], the + 5 N 3 -bases) 24 U k,N + 2 k,N + 1 N Y . Finally, even though 3 (1) as discussed in [ -algebra, the additional gl ,N Y 2 and the parameters ,N ]. Secondly, our generalized j 1 λ N 44 Y . is isomorphic as an associative 2 2 3 λ ∞ 1 λ λ 1+ − W = as described in the section 4.4 of [ 2 0 ]. 1 , algebra that does not contain any null fields 0 24 , 0 Ψ Y – 11 – (2) Because of the presence of the composite primary b gl and 7 1 N, α , 1 , can be the determined (with the normalization ambiguity = 0 i = 4 SYM setup when one considers non-trivial line and Y 3 and not its simple quotient λ W k N + 3 with 1 , . k,N 1 , + 2 2 A Y The vertex operator algebra k,N does. -basis or the quadratic basis) related to the Miura transformation [ + contains null states, when glued with another 1 1 , U 1 N k , 2 Y + field. Generators 3 Y k,N ... + parameters permuted. 2 )+ i 3 λ k,N Identification between the triality-covariant parameters The transformation between the primary and the quadratic basis is not known in Apart from the primary basis mentioned above, there exists another useful basis (some- W + It is interesting that in the semiclassical limit, i.e. when the VOA simplifies to a Poisson vertex algebra, used in Miura transformation and the structure constants in the quadratic basis is [ 3 1 7 0 N W Affine Yangian. algebra (after a suitable completion) with the Yangian ofthere affine is a closed-formwhich determinantal has formula very for similar transformation form between to primary the and formula quadratic for basis Virasoro [ singular vectors. Note that there arewith indeed three possible identifications (and the corresponding still undetermined) byfunction) a with further the requirement compositeare of primaries. given the in First appendix orthogonality few (vanishing primary two-point fields determinedα in this way fields, the primaryif basis we is decouple notsubalgebra, the uniquely spin one determined one is bycondition field the still of from primarity not being condition. the able primary.( rest to Starting Even at of uniquely spin the fix 6, algebra the there appears and primary the work generators first with using primary the composite only the operator product expansions into this guess basis a have closed only form-formula quadratic for non-linearity. all This OPEs allowed [ general but can be calculatedthe spin generators by in spin, the i.e. quadratic we can basis. construct primary fields in terms of times called the Generating fields of this basisof are the not quasi-primary triality and frame alsochanged (so explicitly by there depend the are on action in a choice of fact the three different triality). bases On of the this other kind hand, that are their inter- main advantage is that alizes exactly Y gluing fields discussed in later sectionsFor makes example, them gluing not null withresulting respect algebra to is the full expected algebra. toeven be though the ters vanishing (algebras with allto null consider fields removed, the the generalpears simple case quotient). naturally for It both is the insurface convenient following operators the three and reasons. inversion the of First, Miura dual transformation the geometric realizing general picture a case from free field [ ap- representation of be tempted to use these shifts and restrict only algebras with one of the integral parame- JHEP05(2019)159 . 3 3 3 ,N ,N ,N 2 2 ] for 2 (3.5) (3.6) 5 ,N ,N ladder ,N 1 1 1 , where i N N ) N e Y Y κ Y ( i φ ] and [ ] is in terms 1 bosons of type 35 κ , N 34 -th order differential ]. They were defined is ] in terms of a BRST and lowering N 1 13 with free bosons i ∞ – 1 screening charges. The f 3 3 1+ 11 N − N 3 W + + studied in [ N 2 2 3 N + N ,N 2 ] is generated by an infinite set of + 2 + N 1 and realizing that both constructions . ,N 29 1 = 4 SYM [ 3 1 0 + , h N N (1) are isomorphic to VOAs 1 1 2 , N = 0 Y 1 j h gl , one factorizes an ] and reviewed later. The explicit form of N 3 1 Y ] in general, based on the previous work h ,...,N h h ,N -parameters of 0 35 0 44 , λ , + 0 ψ and = 1 2 Y 34 – 12 – − i 2 h , constrained by 0 , = + 0 3 . To each neighbouring pair of free bosons, we can j ) 1 Y h κ λ h ]. For ( i φ and 48 Consider a set of , 2 36 8 , h 1 truncations and [ Another definition of h together with an infinite set of raising (1) [ Y-algebras were originally introduced in [ N i b gl ψ W × N ]. W ] for 47 , = 26 ]. The affine Yangian in the basis of [ 30 ,N , 0 32 , , is the first (central) element of the commutative Cartan subalgebra of the Yan- 0 29 3 labels a type of the boson and Y 0 , , 30 ψ 2 , , 26 , Below, we give an alternative way to construct the free field realization of We are grateful to Mikhail Bershtein and Alexey Litvinov for pointing out this relation. 5 29 8 , = 1 . Let us pick a fixed ordering of reduction translating the boundaryas conditions in a combination ofpair the of Drinfeld-Sokolov Kac-Moody reduction super-algebras.a and summary. the We BRST refer coset reader reduction to of the a original work [ free field realization. Onealization can by check comparing that the the results two for are constructions essentially give local, the i.e. same involve free only field operations re- on theBRST pairs construction. of neighbouring bosons. The construction is based ontion a for generalization of theoperator standard as quantum a Miura product transforma- of firsttor order by operators. certain pseudo-differential Replacing operators these which elementary we first order describe opera- below, we obtain the desired associate a screening charge accordingthe screening to charge [ depends on thecan type be of then the defined neighbouring pairresulting as of a algebra free commutant is bosons. of independent allthe of such corresponding the Fock space choice depends of on ordering the but ordering. the way it is embedded in Free field realization. of subalgebras of freeκ bosons. κ where gian. Specializationsas of proved the in affineof [ Yangian [ of The map between these parameters and the commuting generators operators. As wethis will Yangian point see of below, view.locality the and The representation triality conformal theory symmetry fieldthree is simplifies theory complex manifest, significantly interpretation. parameters but using one The loses structure the of manifest Yangian depends on 27 JHEP05(2019)159 , k N U (1) 36 (3.7) (3.8) gl (3.10) (3.11) of [ = 1 or N 2 ,N λ 0 , 0 , Y k − = 1, N ]. ) ). Operators 1 ∂ λ 35 0 , 3.4 α 34 )( z by ( ( k . Consider a set of U ∞ ,N 1+ 0 =0 N , k X 0 W Y ) (3.9) ) = 2 z ) z ( corresponding to j j ( w − − jk (3) ∞ λ (3) j N δ − ) J ) R 1+ z ∂ ∂ ( + 0 ], where differential operators of degree 0 W =1 N α α ∂ Y 3 j ∼ 0 50 )( )( , ) ,N α z ≡ z 2 (not necessarily positive integer). This is what ( ( – 13 – w 49 j j ( ≡ )) which conventionally appears in the Miura trans- ,N λ k U z U ) 1 ( 3 J z ) N λ ( =0 N ∞ =0 N z j X Y j X J ( (3) = j J + R = 1. We will see that the usual procedure works even in N ∂ 0 3 λ α ( ) via is related to the parameters of ··· z ,N = 1 but we have to replace the elementary factor and relate it to the free field realization of [ ( 0 0 )) , 2 k 3 α ]. 0 z λ U ( ,N Y 1 2 45 J ,N 1 + N ∂ Y = 1 or 0 ) with OPEs 1 α z λ ( ( j J integer can be naturally extended to pseudo-differential operators of the form N = 1 (as well as their conjugates). The usual Miura transformation in our conventions ] to general 3 which now make sensewe for need all because values the of parameter formation takes a non-integral valuesecond for asymptotic the direction. bosonic representations associated to first and for by a pseudo-differential operatorfields. with This generalization an is infinite commonequations in number (e.g. the of KdV context or coefficients ofof KP integrable which the hierarchies hierarchies), are of form [ differential local λ has all nodes ofthe type case 3 of with 3.2.2 The generalOne case can extend thetypes. Miura For that transformation it to is importantto the to remember three case that we different where have free three there types field are of nodes representations nodes corresponding of of different where the parameter and their normal orderedproduct products expansion and [ derivatives form a closed algebra under operator currents and define operators Let us now give a48 generalization of the well-known Miura transformation for 3.2.1 Review of Firstly, we review the standard Miura transformation for 3.2 Miura transformation for JHEP05(2019)159 . ) 1 z ( − has (1) j (1) U j (3.15) (3.16) (3.12) (3.13) (3.14) (1) j U ]. They (1) 3 U ! U ! 24 (1) (1) J 3 J 24 ∂ 6 2 3 ∂ h 2 ] h + 30 + + ) associated to a single ) (1) ))) generators, all expressed ∞ (1) J j (1) J 1+ (1) U . J 6 symmetry. Fixing a positive 2 is the order of the pseudo- 3 (1) W J 2 (1) h = 1. In this situation, we know 1 2 (1) ∂ J ∂J . ( (1) ( h h ( 24 1 J 2 2 1 h λ ) N h (1) h w J + (1) − + 0 ( ↔ − − α ) N )) = (1) z (1) ( = J (1) representation. In the following, it will , 1 (1) ( J N J h 2 ! ∼ − (1) 1 ∂J

h (1) = λ ) 2 − (1) 6 1 h 8 φ (1)  3 (1) h w J 1 2 h 2 ( h − λ ( – 14 – 3 − h 1 = ∂J (1) ( (1) h∂J -basis (which is itself conventionally associated to = = = = = 2 − J J U 1 2 3 0 ) h ( + 4 h h h z ψ (1) 3 ) (

+ λ    (1) (1) )) = 2 2 fields are uniquely determined from the OPE of J J 2 2 2 h h (1) (1) (1) (1) j J − − J N U ( 3 3 (1)

4 J ( ,      , requiring just the commutation relations spelled out in [ 2 2 2 (1) (1) 1 1 1 h h h (1) N ∂J J ( − − − h 2 2 2 (1) . Choosing for convenience the parametrization as in [ , but in the quadratic  + J   (1) to be (1) = = = = φ φ (1) (1) (1) (1) (1) 3 4 1 2 ∂φ U U U U ≡ Let us first consider what happens in the case that (1) The expressions for higher But even the general pattern is not very difficult to understand: first of all, each sign, the expressions for the first few fields are Having fixed all thefields parameters in of terms of algebra,are we uniquely can determined up now to find the the conjugation expressions for From the Miuradifferential transformation operator point corresponding ofbe to useful view, the to this chooseJ the normalization coefficient of the two-point function of the current we need to require to get a representation on one free boson, i.e. from triality that therefree exists boson a freethe field third representation direction), of there isin terms an of infinite number of non-trivial JHEP05(2019)159 by 1 3 h h node ) (3.20) (3.17) (3.18) (3.19) (3.21) ∂ 0 (1) to be the α φ j k U 9 m OPE is equal  (2) (1) J J 4 1 − (2) − 4 a J k 3 ∂ operators are uniquely h . . j 1 1)! + (2)  − j − 1 3 3 k − U . h h − (1) h k ) 3 k a ( ∂ h N = 1 and so in this case 1 3 0 2 2 −  ]. h h α − a (2) 2 1 k 2 51 )( 1 λ h m h z  k ( 1 − ! 1 + k , all other (1) =1 − j = 3 2 Y j k m U − 2 − (2) 1 1 3 8 a a = j U =1 2 2 =1 ∞ − Y 1 to be the power sum symmetric polynomials. k j X h h 2 j j  h = + a the calculation is entirely analogous and in fact – 15 – 2 + + j 1 . We require h 1 3 2 2 = 1 h h h jm (2) − − operators that we can construct out of a free boson. ) − + a a φ ∂ j (2) 3 2 1 1 1 ... 2 4 0 λ 2 − − − + α ↔ −  X 2 2 ( = k h ha ha ha m ≡ +2 + + + (2) ) 1 1 + z 1 1 1 1 N  ( m ) is thus given by the pseudo-differential operator 2 6 4 − 2 − 3 − − 1 24 a a a  4 a (1) =1 − Y j k R (1) k → → → → . Choosing the sign of N (1) (1) (1) (1) (2) 4 1 2 3 − U U U U N 1 in each term counts the number of derivatives appearing in the operator and  1 h =1 − Y j k = (1) j U As observed by A. Litvinov, this Miura factor can be actually interpreted as a dressing of ( 9 The current is again normalizedto such this that the value quadratic of pole of the a bosonic exponential vertex operator. This is also discussed in [ we can just make a replacement In the case of representation of type where everything is normal ordered.of the The diagram total (see Miura figure operator representing the These are exactly the coefficients appearinghomogeneous in symmetric Newton’s identities polynomials if and we thinkOne of can thus also write a closed-form formula The power of the combinatorial factors can be most easily seen using the operator-state correspondence: Next, there is a sum of all dimension an overall multiplicative factor JHEP05(2019)159 ) z ( (3) 8 (3.22) (3.23) (3.24) (3.25) R ) z k ( m (1) 7 as shown in  ! ! ) is j (2) 3 ( j (2) 2 (2) ...R h J φ − J (2) 1 ) h J 3 3 z h 24 6 φ 2 − ( ∂ 3 k ∂ N ∂ (3) 8 (1) 2 − + − 1 + φ 2 R ) h − 3 ) ) 1 2 k h . z ))) j (13) N − ( 78 h (2) (2) k + − S J (2) 1 2 J (1) 1 2 3 (1) 7 1)! h 1) h h h J h 1 R φ (2) . The algebra can be found by 2 ) 6 (2) − N 4 − , J ( ∂ (2) bosons of type ) 1 2 0 , k (31) 67 J ∂ ∂J j 3 ( ∂ ( α ( 24 0 S ( Y N (3) 6  α )( (2) − φ z + k )( J ( z ( ) m )) ( (3;1) k 56 algebra and multiply the corresponding (2) j j 1 ! (2) (2) (2) S k 4 U U , (3) 5 J J ! 2 1 ( m , ∂J φ h 3 =1 =1 associated to the lines of the chain of free bosons. ∞ ∞ (2)

j 6 j 8 X X (2) Y h (2) ) j =1 J (3;1) 13 45 2 78 2 Y k ( + + – 16 – ∂J h S j ∂J (3) 4 2 3 3 3 ( = (2) h h h j φ − 3 J ) − ,...S + ( N ) ∂ jm + 0 (23) 34

2 + )) (2) (12) 23 ) )(4 α h 3 S ... ( 2 2 2 J h (2) 3 2 ), their normal ordered products and derivatives form a (2) + ,S N h h X φ z 2 ≡ + J 2 2 (2) ( 1 m j ) J h (1;1) − − (12) (2) 12 ( 23 z 1 h U +2 ( S N 4 J S 1

( (1) 2 ) )(3 )(3 ) m (2) ∂ φ 2 2 2 (2) 0 R  h h h α . Commuting all the derivatives to the right (recall that even for (1;1) 12 ) is now ∂J − − − j (2) ( k S (2) κ (1) 1 ( j N − J (2) j φ ) = ( R algebra: pick an arbitrary ordering of U z − = (2 = (2 = = (2 ( 3 1 R ,N  (2) (2) (2) (2) 2 3 4 1 2 for a particular ordering of the An example of the ordering of free bosons for 1 U U U U ,N =1 − are certain normal ordered differential polynomials in the free boson fields. The 1 4 Y j k N j Y = U We can use these newly constructed building blocks to find a free field representation (2) j U where statement is that the fields of any the figure Miura operators non-integer powers of derivative thethe generalization end of a Leibniz pseudo-differential rule operator still of applies), the we form find in and the Miura pseudo-differential operator representing a node of type The formula for of kernels of screening charges determined and we find Figure 4. multiplying the Miura pseudo-differentialas operators shown in in the the figure. order Alternatively, one can construct the free field realization as an intersection JHEP05(2019)159 . = in +1 ) 3 κ i,i ( i ,N (3.27) (3.28) (3.29) (3.30) (3.26) S 2 J ,N 1 N Y . One associates a 4 . On the other hand, 3 ,N 2 ,N . , 1 i i 2 i N . ) 2 +1 +1 Y +1 i (3) (2) i w ,κ (3) i +1 1 = 3, the corresponding screening φ φ φ i,i κ − 1 3 10 2 δ S h h z h +1 fields are still those of the quadratic ( i + + κ 3 + j ker h U with the corresponding currents 1 = 2 κ i (3) i (3) (3) i free bosons which represents − h i h φ φ κ 3 φ 1 1 2 3 κ 2 N h h h h N + to a module with the highest weight vector =1 2 − − ]. Examples will be discussed in later sections. \ − i ) + h h N h κ 2 ∼ − – 17 – 24 ( i + 1 of type ) N exp exp N ∂φ ) exp w + κ ( ( i ) = 1 dz dz = dz 2 φ 3 κ N 2 . One gets similar expressions for the other three types ) ( i I I 2 I κ ,N maps the vacuum representation of the product of the , J ( i 2 0 ) , = = J = 0 z ,N +1 ( 1 Y ) +1 +1 N i,i 1 +1 κ Y (32) (3;1) 1 S i,i i,i ( (3;2) i i,i S S J S is the screening current associated to the screening charge free bosons can be defined as an intersection of kernels of screening charges κ ] construct a free field realization of the same algebra as a kernel of +1 3 N i,i ,N j 35 2 , 1 screening charges acting on the tensor product of the current algebras ,N 34 1 ]. − N 3 Y 35 parameters permuted. To a pair of free bosons of different type (say ordering N ), one associates instead the screening charge , where i i h + 0 +1 | (2) i 2 φ normalized as N (0) ) The screening charge Consider a fixed ordering of free bosons such as the one in the figure × The commutation with the spin one and the spin two field gives two possible solutions as in the case κ + i ( +1 10 1 -basis with structure constants given in [ (3) i i,i of the Virasoro algebragenerator. but only one is preserved by the requirement of commutativity with the spin three and similarly for the other five combinations. current algebras generated by j The algebra with the φ These two can be determinedvertex from operator the commutes requirement with thatfields the the in free zero the mode field Virasoro of realization algebra the of exponential the spin one and the spin two If the two free bosonscurrent are can of be the same chosen type, to say be either or N above. Let us definethe screening form of charges [ for each such ordering andscreening check charge that to they each neighboring are free of bosons (lines connecting two nodes of the chain). ∂φ we have the associatedthe free authors field of realization [ of the algebra terms of free bosons.U Furthermore, OPEs of these 3.2.3 Miura versusTo screening each ordering of closed subalgebra of the algebra of JHEP05(2019)159 gives (3.33) (3.34) (3.35) (3.32) (3.31) (3) 3 φ × (2) 2 φ × , one can pick either (3) 1 φ (3) 3 φ , × associated to Gukov-Witten ! ) or one of the first type and ]. (3) 2 2 1 3 φ h h 2 35 ,N , one has again a choice between 2 − 3.28 × ! 2 1 ,N 2 1 1 (2) 3 from the Miura transformation and h h 1 h h (3) 1 φ 2 1 N , φ interchanged. These two matrices are 2 1 − 1 Y ) or ( h h , × jk 2 1 2 0

δ 2 h h h Y 3 1 2 (3) 2 − h 3.27 h h . j φ ! ↔ 2

κ 3 1 h − ! 1 1 h × and h h 1 h 2 h , 3 1 (3) 1 1 1 1 1 1 – 18 – 0 h h , − , φ 0

,

Y ) leading to the following two overlap matrices ! = ]. ! 1 1 jk 1 2 35 g ]. The last, symmetric ordering 3.28 − h h 1 1 2 1 2 35 2 − −

) and ( 1 2 h h ) lead to different matrices from [ − 3.27 3.28 ) and ( 3.27 If one of the three free bosons is of a different type than the other two, say 332, one has If all the three free bosons are of the same type Consider now a triple of free bosons neighbouring in the chain and let us compute the 4 Generic modules Let us turn todefects. the discussion of We generic start modules with of a review of the algebra of zero modes and how to parametrize Comparing the free field realizationsfrom of the kernel ofY-algebra screening labels, one charges can together see with that the the two triality free symmetry field permuting realizations the are the same. which is of the formof 5. overlaps Finally, if all the bosons are of a different type, one gets the matrix that are of thean form overlap 4 matrix and of 3 the of form [ two possible orderings. Inthe the screening first currents case, ( together with matrices with theof parameters the form 1 and 2 from (2.24) of [ currents ( both screening charges to bethe of second the one same of type thetwo matrices ( second type. In these four cases, one gets respectively the following matrix of inner producs ofrespect the to corresponding the two metric exponents given of by the the screening normalization currents with of two-point function We will see that the different choices of ordering and different choices of the screening JHEP05(2019)159 ) + ]. act. , we j 52 X,Y W 4.2 ] = ( Y acting on the [ ? ] ∞ X 1+ W from the mode expansion of ... ] + 2 (4.1) − ) z ,Y ( 2 ]. If the Zhu algebra is commutative Y X in the primary basis. The next two 1 53 [ − 3 ] + [ ) 1 11 ,N X − ( 2 h ,Y ,N 1 1 leads in general to irregular modules of the Y N dzz 3 Y – 19 – ]. We can thus define a one-dimensional module I ,N 2 25 πi 1 ,N 1 2 N ]) the variety of highest weights is the spectrum of the Y = 0 25 X describes a general structure of the variety of highest weights case [ ). Starting with a highest-weight vector anihilated by all positive representations can be induced from representations of the subal- 3 4.3 3 ,N X 2 ( ,N h 2 ,N 1 ,N N 1 Y N Y . (2) Kac-Moody algebra. 4.6 b gl ). The section u ( a field of spin modes on the space offorms highest an weights, infinite-dimensional the representation space of of the highest zero weights mode itself algebra. generically with trivial action ofhighest the weight positive modules modesglued of on algebra. the We highest willthe weight later vectors. illustrate this Gluing phenomenon of on the simplest example of appears under the name of(as the in Zhu algebra the Zhu algebra. ψ Even in the case when the module of the glued algebra has a trivial action of positive Not all the modules produced by gluing are induced from the algebra of zero modes In the math literature, the algebra of zero modes acting on the highest weight state Let us add few comments: The Zhu commutative product is defined as a modified normal ordered product [ X 3. 2. 1. 11 corrections. The correctionsthe are normal the ordered commutators product [ acting on the highest weight state. For a more precise comparison see [ on the highest weightweights state. leading to The a existence variety of of highest null weights. fields thus constrains possible highest modes, one can show thathighest the weight algebra vector of is zero commutativefor modes [ the of zero-mode truncations of algebra byIf prescribing there how are zero relations modesVerma in of module), the the the space strong zero of generators fields mode (which of show the as corresponding singular null vectors fields of must the vanish vacuum when acting gebra of zero modes for field realization of modules.the Finally, section we give two examples of variety of highest weights4.1 in Zero modes andA generic rich modules class of review a compact way totions parametrize highest weights in termsparametrizing of Yangian generic generating func- representations of sections state the generatingwith function Gukov-Witten for parameters such and representations parameters and of Fock related modules its in parameters the corresponding free modules of a VOA induced from modules of the zero mode algebra. In the section JHEP05(2019)159 = 0 = 0. (4.6) (4.2) (4.3) (4.4) , 0 h h . ∂χ ) 3 − ) w 16, and ) / − w z ( 2 representation of = 1 (( / T. ∂χ (4.5) 4 h O ) of dimension 1) ∂ z = 1 ( . + = 0 it requires 9 2, Virasoro primaries. We 5 = 0 ) c − χ ) / ) h w 2 i − w , w h w ( h 1 ) ( | − h = 1 − χ , z 1) 4 ∂T 2 z TT ) h ) ( ∂ 2 w 1)(16 − ) (4.8) w w + = 0 it does not give us anything ∂ ) (4.7) and ( z h − − z ( 2 to vanish. The most singular (sixth − h ( 1 ) ) , z + 7) h ∂χ 2 16 z 2 χ / / w w 264( ( h h 1 1 ( + 16, while for 1)(16 χ χ − − T 12(2 2 2 / 2 2 and ) z ) ) − ∂ ∂ ( 6(64 6 + 3 4 w w h φ ( = 1 ) + + acting on the highest weight discussed above − − − (2 h w 4 ) ) ) h hχ 6 z ∂T ∂T ( ) z z ( – 20 – w φ χ w 4 )( )( )( / = 4 2 ∼ 1) − 1 / 16 i ) 1 / 2 and 6 z − 4 T χ h 1 ) ( ) w | / ( ) h T χ w w z )) + 186( χ T χ ∼ ( 17)( ) 4( = 1 − 6 − ) z 3( As an illustration of possible restrictions that arise in the ( φ − TT z z w h 1)(16 5 ( ( ( ( T h T T − ) (8 dzz h z 16 h ( / (2 I 48 T h 4 = 128( + 2 it requires πi 1 = 1 ] that the vacuum representation contains a singular vector at level 6 / 6 2 vanishes in all the correlation functions (which we would need to do for h ∼ φ 54 6 ) = 1 φ w h ( χ ) z ( 6 φ From this simple example we see that the singular vectors of the vacuum representation Let us look at the quartic pole more closely. For It is interesting to look also at the conditions following from the vanishing of the lower carry interesting information thatwe constrains impose the that spectrumexample of in primaries a of unitary the theory),we we theory. also find find If that their there singular are descendants. only three possible primary fields and to be zero and for to be zero.could These proceed are further just and find the other singular relations vectors coming of from the lower order poles. which is theinvariance usual of singular the vacuum). vector of the vacuum representationnew while at for level 1 (translation The quintic pole vanishes for and the variety of highestThese are weights consists the allowed of primary three fields points of the Isingorder model. poles in the OPE and require the operator product expansionorder) of term is precisely the zero mode of with the corresponding primary field The requirement of vanishing of thefor null the field VOA. in In any our correlator case, constrains let possible modules us consider a generic primary field presence of null states thatthe are Virasoro quotiened algebra out, let us consider the It is well-known [ Example — Ising model. JHEP05(2019)159 is ∞ (4.9) (4.10) (4.11) (4.12) (4.13) 1+ 1. The W ≡ 0 ) is given by u u ( ψ and ,N 0 , ) and 0 ) u Y 0 ( 3 α U -plane, so we may factorize 1) ,N u 2 N. highest weights, i.e. eigenval- − . ,N -basis is particulary usefull for k 1 = 0 . N U ) 0 N α 0 j 0 + ( +1 . Y 1) α j ) ψ α u ∞ u u − − ( -generators of 1) − 1+ j ( j U u k =0 ∞ ( − charges of the highest weight state and , ψ ( U j X u W 0 j 3 U ··· j − ( α h ) We will not be able to write down explicitly 0 ψ of j 2 0 − = Λ h α 12 1 3 u Nα u − h + – 21 – − u u . To label a generic highest weight representation u − =1 N . Analogously, a generic representation of -th order polynomials in Y j )( , h ,... algebra are labeled by N u 1 3 ) = ) = 1 + ]. We will specify the module by the eigenvalues of the , − − u ) = u ,N 2 ( ( ( 0 u , ,...N , 30 = ψ ( ψ , 0 3 =0 N 2 , Y U k X 29 2 h 1 , ≡ h ) = N u ( from [ charges. U i with the generating function given by algebra in the U-basis contain only quadratic non-linearities with all the structure × W j , e u ∞ i ]. ,N (1) 0 , 1+ 24 , f 0 b ], the transformation between generating function i gl generators on the highest weight state encoded in the generating function W Y ψ -charges of the quadratic basis i 30 ψ U zero modes for 1 and its truncations, it is convenient to introduce a generating function of the are the eigenvalues of zero modes of the i j ∞ W u 1+ Another possibility to encode the highest weight charges is in terms of the generating OPEs of the W 12 These relations allow usthe to corresponding translate between constants fixed in [ if we identify the parameters as it and write As shown in [ where generating function is a ratio of two function of description of will see that the modules canof be generators easily parametrized using thecommuting Yangian description in temrs independent generator of spinof 1 highest weight charges. Generating function of highest weights. the generating function of highest weights in the primary basis of the algebras. Instead, we Generic highest weight modulesparametrized of by a the VOA with actionmodules a of of commutative such the algebra zero of modesues zero on modes of the are highest weightspecified state. by For an example, infinite set of higher spin charges of the highest weight state, one for each 4.2 Generating function of highest weights for JHEP05(2019)159 ) in ,N 2 , 0 , 1 , one , 0 (4.17) (4.16) (4.14) (4.15) i 0 Y ,Y N 0 are chosen appears at ,N, + 1 and its j and 0 3 n 1 , ,N ,Y 1 from OPEs. 2 , 0 , 0 0 . ,N Y 1 ) ) are removed by the N, 0 N 0 Y , we have a manifest Y i > n α j +2 Nα n x 1) for − − ,W i ) N N W ) in the spectral parameter ( Λ +1 does not vanish, one can use . u n 1 ( − − ψ +1 − , there will be less states at this u N i n ( , J, W Λ 3 N ( W 3 h the problem outlined above becomes , ··· − + 1) h j ) 3 − x +1 u 3 0 1) ( n j α N ,N − x 2 − 2 + 2. For large enough values of ··· u j − − ) . Using the variables , ∂W n ,N 0 = 0. ) 2 3 u 1 + 1)( n α + ( h j Λ , we find 2 N j – 22 – =1 N − 3 U + Y j − N 2 j ,N u T,W x Λ = Λ 2 ( field from the OPEs. At the next level, three more )( ) = , j 0 − ,N u + 1)( x )) 1 ( α +1 . One can continue this procedure and (assuming that 1 u n n but one also gets one extra condition. This condition can ψ N − )( N +2 1 Y W 1 1 n , where Λ W J, W Λ = ( W n ( − − n J, u ( u -generators and need to be removed as well. The first con- ( , ( specify the positions of poles of are finitely (generically non-freely) generated vertex operator alge- ) W n 3 j ,...,W x 2 ) = ,N 2 u ( ,W ,N J, ∂W 1 ψ 1 ( that needs to be removed in order to get the algebra N , W ) Y ∞ ∂J 1+ n W W + 1. Assuming that the coefficient in front of ( , n n One can see that for generic values of In this way, one can solve many null state conditions by restricting to a finite Plugging in the product formula for W 2 level as can beWe will easily also seen see from thatactually the some non-trivial. constraints box-counting will and be as trivially we satisfied and will only see somerather in of complex. them examples The are null below. states have been fully identified only in the case the restricted setstraint of appears genericallycan already see at level from∂ the box-counting thatabove there argument. One are has begebra still 4 of 12 constrains zero null that modes. lead states to Note at a that non-trivial this for conditions small level on values the but of al- only of be used to removethere the are field enough conditions at each level) one cannumber remove all of W-generatorssome but null generically states (apart remain. from the These are case going of to be composite primary fields formed by state of level this null field tonull eliminate fields the appear. Twonormal ordered of product them with are the derivative of the null field at level The algebras bras by fields The finite generation can be seen from the structure of null states of the algebra. The first plane while the zerospermutation are symmetry at of positions the generating function,such while that the the shifted vacuum variables representation Λ has Λ 4.3 Zero mode algebra of we can rewrite this as i.e. the parameters Defining JHEP05(2019)159 ]. ], n 13 58 – 59 , (4.18) (4.20) 55 , 17 charges [ where i 25 n ψ ∞ dimensional 1+ 3 W N + ,...,W and we conjecture 2 1 has a finite number N W i null + ∞ 1 X N 1+ . We will now generalize . W 3 ,N 0 , N 0 + (4.19) Y 2 ) equals a ratio of two Drinfeld N u ( + = 0 1 ψ 1 i N . . Although we will not be able to i ) i n i n w | q C w ) | i i − ) that parametrize generic representations w w ( u i ( (1 -tuples of partitions, i.e. = f are finitely generated by 2 ψ 3 i 3 =1 i N ∞ = – 23 – Y ,N n ,N w 2 i | + 2 i 0 2 ,N have been considered rigorously in math literature [ ) w ,N | i ) = 1 1 0 N 0 , where modules of each of the factors are parametrized , q of the modes at each level are independent, giving rise N N 2 ) 3 ( W 1 2 3 , − ( + Y Y N can be simply parametrized. 0 N , 1 N ⊗ i null 0 3 + N,N N Y 2 + X Y ,N ( N 2 2 ⊗ 2 + 0 N ,N 1 , N and 1 1 N , + ⊗ 0 N 0 , χ can be also guessed from the free field realization of the algebra 1 1 Y Y − N 3 ⊗ ] and lead to nontrivial constraints on the allowed highest weights. 1 ] that a highest weight representation of N 0 N,N , N on the highest weight state subject to the null state conditions. As in the dimensional subvariety inside from the literature. 0 Y , 30 parameters respectively. The dimensionality indeed matches in examples ⊗ + i 38 , 1 ) to a generating function , 2 3 0 ). In particular, generic representations have a finite number of states at , 3 2 . In particular, we conjecture that the complicated variety parametrizing Y W 1 , 3 N 37 N continuous parameters of surface defects available in the configuration. The N 0 4.16 ,N N,N, Y 3 4.17 + 2 Y algebra. The conjecture for the dimensionality comes from the existence of + N 2 ,N 1 and inside 1 N N 2 + N and 3 N 2 W Y 1 ,N + , ,N 2 N 1 1 , 1 Note that the above discussion also implies that the character of the module with 0 ,N The cases + N 1 N Y 13 1 N polynomials of the samethe degree. formula ( Thisfor is all indeed true for modules of the algebra As we have just seen,is truncations given by ( level one. Following the usual notionit of was quasi-finite representations argued of in linear of [ states at level 1 if and only if generating function subvariety, one can use negative modesat higher of levels. the null Only conditionsto to the remove above appropriate character. states 4.4 Generating function for A general state ofnegative modes a generic modulecase of of zero modes, the where the algebra null states can were used be to carve constructed out an by an action of Y by of generic highest weights counts explicitly construct the nullexplicit states parametrization of in the general variety byof in generalizing the the terms generating of function of primaryN fields, we willnumber give an The highest weights are constrainedthat by the the existence resulting of variety null of states highest weights of the algebra of zero modes is From the discussion above,structure of one the can variety of stillare highest given draw by weights. the a As subvariety conclusion argued inside above, what the the space will possible of be the highest highest weights the weights general of zero modes the literature [ JHEP05(2019)159 ) 3 3 2 + N ,N (1) 3 1 , h 2 + 1 2 N , λ h ,N (4.21) N 1 factors factors + N (1) 2 i 1 N Y N N generators generators , λ W C W j i (1) 1 -parameters. ψ . λ W λ 3 1 , N 0 remains the same , ⊗ and the definition 0 0 Y 3 α ,N × 2 2 0 ,N , N 1 1 , ⊗ N 0 Y Y modes and Yangian genera- is additive under the fusion. × 0 charges for arbitrary ∞ 1 ψ 0 i , N 1+ 0 ψ , ⊗ ) (4.22) 1 W u Y to the zero modes of ( i ⊂ (2) ψ 3 This coproduct of the affine Yangian also ψ ) ,N ), we see that this requires ( 0 u , 14 ( 0 3.4 Y (1) . Note that the parameter 3 should be simply a product of three – 24 – × ψ 0 3 ,N The gauge theory setup suggests that the modules , 0 2 , ,N 0 ) = 2 ,N Y u 0 ( ,N Y 1 ψ N × and Y 0 0 , , 0 2 , Using the map between . 1 ,N N Both the Miura transformation for 0 A Y Y , ⊂ 0 , 2 0 , 1 ,N ). The full free field realization therefore suggests that the generating 2 N ) to be proportional and the fusion is simply additive in in the free field realization above can be identified with one multiplica- Y ,N 1 (2) 3 1 , 0 4.16 N , , we can precompose this with the coproduct if needed to obtain slightly more general , λ 0 Y -parameters this is the same condition as found above. u Y λ (2) 2 3 is no longer a finite linear combination of other generators and their products, dimensional tori (modulo Weyl group) that we expect to be lifted to , λ parametrizing the algebra are the same while the 3 ], we can translate the fusion to Yangian variables. The coproduct of ≥ (2) 1 . A parametrization of the variety of highest weights can be recovered after changing 3 N λ j 30 h Such a parametrization of the variety of highest weights is natural the from point of Since the Yangian has a non-trivial automorphisms, like the spectral shift automorphism translating + 4.23 14 2 this picture suggests that genericallyshould the be contribution independent. from GW-parameters in each corner the parameter coproducts. This is actuallya what is vacuum needed representation. if we want the fusion of two vacuum representations to produce Gauge theory and braneshould picture. be parametrized linearly.N The GW parameters that labelby modules boundary live conditions in imposed the on the GW defect ending at the interfaces. Moreover, analogous to the usual onessuggests in a finite simple Yangians. form of theassociated generating to function each in corner. terms of Theand a compatibility product of of parameters three inIn this case terms requires of that tween VOA description and the Yangian description.weight state Fortunately, when (corresponding acting to on a a primaryadditional highest field terms via drop the out operator-state and correspondence) we these obtain a simple formula and ( Yangian point oftors view. [ with but involves an infinite sum. This is related to the non-local terms that enter the map be- Each factor tive factor in ( function of a generic modulecorresponding to of in the fusion procedure. From the formula ( according to the appendix Free field realization. in terms of a kernel of screening charges give an embedding of the algebras of the form view of the coproductviewpoint structure and of the the affine gaugedown Yangian, an theory but explicit perspective. also formula for fromin the After free generating field stating function realization these of the motivations, variables we from write the affine Yangian generators JHEP05(2019)159 for m (4.24) (4.25) (4.23) 2 h + and the fact n 1 1 , . h 0 , 3 ]. [Ψ]. The process 0 h 3 Y , 30 − ,N , 2 (3) j 2 h x 29 1 ,N (3) j 1 − h x 0 N u ψ − Y , see [ i acting on the highest weight − u . h 3 3 3 = h =1 ,N 3 N 3 3 Y 2 and j corresponding to the trivial GW N 0 ,N 2 1 ψ m h (4.26) − N 2 2 algebras from each corner Y − h (2) j h , λ = 1 x 2 3 N + (2) j 15 h 3 N 3 − x n W 1 λ 1 N u h − − h 0 1 + ψ u h = ). 2 1 2 − – 25 – i by a lattice vector. 2 =1 λ charges for N N N 2 N (3) i Y = i j − x (3) + 2 ψ 1 x h 1 = 1 . λ 3 0 N − (1) j ψ ,N x 3 2 (1) j , λ h − 3 ,N x 2 1 h h u 2 N − 1 The discussion above motivates us to write down an explicit h Y h u 0 ψ 1 =1 N should have a natural generalization for − Y j can be identified with the lifted Gukov-Witten parameters in the third N = ) κ 1 ) = W ( i λ u x ( amounts to a shift of ψ (3) x . We will later see that that Z ∈ Note also that the generating function is manifestly invariant under the Weyl group Parameters The coproduct from the point of view of the gauge theory corresponds to increasing The algebra is invariant under the simultaneous rescaling of from the expression above, one gets 15 0 will also see thatparameter the fusion of a degenerate module withassociated a to generic the three module gauge labeled groups by U( a corner. This canthat be each seen multiplicative factor from corresponds theparameters to comparison one themselves of such can factor. then, The be U(1) m unlifted charge identified Gukov-Witten for bydefect modding (and out a by possibly non-trivial the line lattice operator) corresponds to a degenerate module. We satisfied by parameters of one gets the correct expression parameters and the truncationψ curves are reproduced correctly. In particular, extracting Identifying the scaling-independent combinations Note that the expression is manifestly triality invariant, depends on the correct number of other pictures discussed above. Generating function. formula for the generating functionstate of by simply multiplying contributions from the rank of gaugeinverse groups process in to the Higgsingreduces three the the corners gauge theory of group. that thethe This corresponds diagram. procedure coproduct to can of One be separationis can performed of independent in look D3-branes on each at the and corner it gauge suggesting coupling that as suggesting an that Ψ is constant in agreement with the JHEP05(2019)159 (4.29) (4.30) (4.31) (4.28) 16 and . point of / 002 = 1 + 1) + 2) Y u u h ) let us discuss factors in order 2 4)( 3)( h are the coefficients = 1 ( 4.23 1 − − = 0, i j 3 h q u u N = 3 and one zero-pole h h ( ( − 1 . h 2 ) ) = w u , . . . , q jk . ( 1 δ )) while − 2 i , we find q = 1 (4.27) | / u j 0 z 1 | ( j 3 charges that have been already ( q ψ ∂φ 3 i   ) is given by j h ψ j = 2 ( φ parameters to be integers, 1 i ≡ j κ h ) (4.32) j ψ j q 1 ) implies that it should be possible to h N , ψ z J h ( h j 2) 2) =1 N , h J j X / / 4 4.23 5 . We reintroduce the   − ∼ − ]. =1 N j , . . . , q ], we find them to be are the charges that appear in the exponents j X − + 1 1 J 2 30 = j ) q u u point of view). Up to an overall translation in | q 2 30 – 26 – w , k ) = jk q = exp 2)( 2)( z g − 210 29 / / ( i jk Y 1 1 3 z g ( N U − − 4 ( = ∼ u u i − , h ( ( ) whose zero mode is N w , . . . , q = ( 1 2) and can be thought of simultaneously as a truncation of ∞ = 3 k 2 , ) = q algebra. Choosing the | 1 J h 2 u 1+ ) h ( / ) holds, the formula ( z , . . . , q 1 W ( 16 210 1 j / − q 3.6 Y 1 | J , from the generating function of 0 3 j, / ) κ J ( i To illustrate the power of the simple generalization of ( x , ψ 2 minimal model of Virasoro algebra, lies on two intersection curves. Its 2 / 6 so that ( u − / is the metric extracted from the two-point functions of the currents, = 1 u ) as product of two zero-pole pairs separated by distance u c jk = 1 ( 2 which are exactly the conformal dimensions of the Ising model. g 0 ψ ) = / -space (spectral shift) there are only three possible solutions: algebra as well as u ψ The U(1) current of u ( 0 = 1 -parameters are (2 002 ψ under the scaling symmetry of the algebra [ Our conventions for charges areof such vertex that operators (and inof positions the of first zeros order and poles polesto of of make OPE the with currents expressions manifestly triality invariant and also of definite scaling dimension Acting on this state with the zero mode of current where representation with generic chargesthe can vertex be operator obtained by acting on the vacuum state with 4.5 Relation toThe the parameters free boson modules related to the Gukov-Witten parametersfor can be the also related vertex to operators exponents in in the expression free field realization. A highest weight vector in free field the Extracting the conformal dimensionsh [ and write view) or alternatively as two zero-polepair pairs separated separated by by distance distance one simple application and findbeing primaries a of the Isingλ model once again. TheY Ising model, Applications. JHEP05(2019)159 ) u ( ψ ) has (4.37) (4.38) (4.41) (4.33) (4.42) (4.39) (4.34) (4.35) z ( (2) 2 k R ) ∂J j z j ( κ ∂J h k (1) 1 κ R h jk X ) ) + z 1 2 z ( z ( . . R )(
− i . k 2 (3) 2 j ) (4.40) k k J h N ) R . j κ κ z 1 ) 3 ( ∂J J h h 3 − z ) h ( k h j ( 3 h 1 κ 2 3 N ≤ − h h N k kcentral charge w w w generator. = 1 (4.43) w 1 0 W ( ( , λ W − 1. Generically, starting from spin 4 we can − 1 3 2 − , T T − · 1 1 2 λ + 3 z . The OPEs of = truncation requires W z 2 = Y ∂ ( 2 · 0 3 + 1 – 28 – , 4( B ∞ 3 λ 1 − c , 1 λ + 1 1 6 λ + Y ) 2 times a free boson as further discussed at the level examples without the necessity of going through the 1 1 w 3 − λ 1 generator free for later convenience. We could absorb ,N − = 2 3 z . c to be parametrized by two continuous parameters, while ( ,N 1 W 1 0 λ ,

N 1 , Y ) in appendix 0 33 1 by rescaling the u Y C ( 0 33 ψ that would reduce the number of parameters by one. From the ∼ C algebra as truncation of ) 3 algebra at 0 , w 1 3 , W ( is the simplest truncation of 1 3 W Y 0 algebra have in general three highest weights. We thus need to find a , W 1 ) — singlet algebra of symplectic fermion , 3 1 z ( 0 Y , 3 1 , W 1 × W Y We are now interested in constraints on generic representations of Considering physical reasoning asgeneric well representation as of fromthe the U(1) free fieldsingular representations, vector we in would expect the We kept the normalizationthe of structure constant a quotient of the of generating functions algebra coupled to a spin 3 current which has OPE independently of the value of uum representation appear atuse level these 4 singular = vectors 2 toan eliminate algebra the that higher spin is generators generated of by spin fields 4 of spins 1, 2 and 3. Therefore we identify From these constraints, weformula, we learn find that as we will see, itof can the be central understood charge. as First (a simple of quotient all, of) the as well as the usual constraint tedious calculation of the nullconstraints constraints on in the the primary zero basis modes and of translating the them4.6.1 null to fields. the The algebra Finally, we are ready tovariety of show highest how the weightsfree in generating field the function realization, ( examples we canvariety construct of all highest modules weights). (there Thethe are knowledge variety no for of further all the restrictions the generating on other function the allows to determine 4.6 Examples of varieties of highest weights JHEP05(2019)159 × 16 0 , and 0 , 1 1 (4.53) (4.50) (4.51) (4.52) (4.47) (4.48) (4.49) J Y ... ⊂ ... 0 direction, + , 1 2 , 9 + 1 J q  8 Y + gives us a non- q T 1 4 e J ∂ 6 + 170 2 N 27 8 + 117 q 7 − q ) to be in . 3 + 106 TT ) is necessary and sufficient, + 67 110 2 7 compared to the situation in W 6 Y ∂ 3 q q ( 0 ∂ , 4.51 7 9 . 1 , . 2 − 1 2 / + 63 ) ) Y + 40 on the generic highest weight state 2 3 ) + 6 in terms of two U(1) currents + 1) w 5 ) q +1) o jk q k 0 (2) rotations in the space of free bosons j 6 h . , ( δ − q 1 and check that the generic modules can 2 j O , N (8 q z J 1 ∂T W 0 − 2 ( j , + 38 Y ∂T ∂T + 21 1 h ( + , 5 1) (1 4 1 ∼ 12( q 1 0 33 q 9 Y − 19 36 J ) =1 ( C − – 29 – ∞ k has w ) = ( =0 0 + 21 3 + Q k , ∞ j + 12 J )) + 1 4 2 3 J , 3 is q ) 1 P q w z Y 3 TT ( T ∂W ( j = + 6 J + 12 T 2 0 = ( 2 ) 3 × W q k 8 9 = 8( q q o  6 + 3 − + 6 0 j From the general fusion ideology we expect that 33 N q q 2 C (1 q =1 1 + j k ) + + 3 3 ' q Q W j 3 =0 + ∞ j s X 1 + W q 1 ' at the generic value of the central charge. The first null state is the even − = ( ) = 1 3 e q 6 ( =0 ∞ N Y j vac × W , i.e. that there exists a representation of χ 3 0 =1 , In this section we are temporarily using a different normalization of U(1) currents than in the rest of Y s with OPE 1 , 16 0 2 With this normalization, we areso still we free may to with make no loss of generality align the U(1) current of the paper. Free field realization. Y J primaries from three to twocould proceed which further is by in studying the accordancenew singular with vectors (independent) what at constraints. higher we levels expect. andwe In possibly will In discover order principle, construct to we aindeed show free by that field realized. ( realization of vanishes gives ustrivial identical constraint zero while the similar requirement for This is the constraint we were looking for. It reduces the dimension of the space of generic and the second one is the odd field Requiring that the action of the zero mode of We see that atU(1) level 6 therequadratic primary are composite two field null states in while the vacuum representation of singular primaries at level 6.vacuum We representation can of see U(1) this by looking at characters: the character of the general reasoning, we expect the first relation to appear at level 6. In fact, there are two JHEP05(2019)159 = ∞ c (4.57) (4.55) (4.56) (4.58) (4.59) (4.54) would and the − J ∞ T (2) 1 3 × /h 3 ∂U 2 . ) h /h ) z 1 1) (1) 1 ( h ∂ U(1) algebra such that ) U 0 − − ) ∂ J α 3 × 0 ( − 2 h q α ∂ i ) are realizable in terms of ( . 1 4 − i ) , ... 2 1 z 1)(2 J ) and with central charge 4.51 ( h ... + z ) + 9 which is indeed the case. This − ( − 2 z + − J + ( − − )( ∂J 2 ) , q ... current with itself and also that the − = 1 2 ) ( − ∂ with ( (2) 1 2 ) (2) 0 1 − 3 + J 0 ∂ U ∂J q 0 33 α , (2) 3 0 1 ( W ) + 1 − C ∂U , − − (1) α U z 2 = J 1 1 ( (2) 2 1 ∂ ( 2 Y U h 3 )( ) J ∂ 3 2 U (1) z ( 9. We can verify that there are no fields other − 2 ) + + ( 3 2 U − 2 + 1 ,J h – 30 – √ ) + 1 U (2) (2) 2 1 − + = z Let us see what free field representation we find by − − J U U 1 ) + 2 ≡ ( 1 0 33 − ∂ ))( 2 1 2 h ) (1) (1) ) commuting with 0 1 1 − , w C − ( − z ∂ J α 1 U U J ∂ ( 0 ( 1) ) h − α ) = ∞ + + z ( J (2) − z ( 1 T + ( ( 1 2 U − (2) (2) (2) (1) − 1 2 3 1 U ∞ q J (2) 2 U U U ( U + T + − 2 anyway. 1 + + + + q 1 ∂U h − 1 − 1 ) = ( (1) (1) (1) = h ∂ h × 1 2 3 z = ) is satisfied if and only if ( U U U + 3 are ∞ = . We find that the conformal dimension with respect to ∞ h 3 = = = c − ) = W 4.51 q z 1 2 3 W ( U U U R 2 and the requirement of existence of spin 3 primary constructible from charge is − − Consider now the highest weight representation of the U(1) J = 2 is ∞ By commuting the derivatives to the right we find applying the Miura transformation explained above.of two The basic total Miura Miura operator factors is associated a to product first and second asymptotic direction and the relation ( means that all thetwo generic free bosonic representations currents. of Free field realization from Miura. the spin 3 charge of The normalization coefficient is now than descendants of thedimension identity 6 in singular the primaries OPEc vanish of identically. Noteforce that us we to choose did not need to require We can also find one spin 3 primary field commuting with the unique stress-energy tensor − Denoting the normalized orthogonal combination JHEP05(2019)159 T (4.63) (4.61) . Boot- 2 )) , 1 2 , 2 2 J 0 J 2 Y 2 ∂J J ∂ ( (4.62) ) 2 operators with algebra. There 1 2 3 )) j J J − 2 5 2 h h ( 2 ) + 1 ) U (4.60) 2 2 2 5 was carried out J 12 2 2 ) W 2 2 , w h h 3. Finally using the 1 J ∂J 1 4 h J 3 , 1 ( − J 2 ( , − 2 1 1 2 /h , − + z 3) ∂J J 2 h ( ( 1 1 ∂J h − ( J 2 − = 1 2 2 ∂J h 2) + 1 ∂ j h )) 1 + 1)( ∂J 2 2 ∼ − 2 2 h 2 − J 2 2 h J h − 2 ) with unit normalization and h 2 h h 2 + 1 1 2)(4 2 2 ( ∂ w 2 − h J 2 ( h 12 ( J 3 1 h 2 − − ) + 2 1 h 1 J 2 2 J + 6 + 2 ) 1)(3 ∂J J )) − ( h 2 z 1 2 1 ( ) 1 h hJ ( ) + h 1 fields with − J J 2 2 2 2 2 1 2 ∂J J 2 j − − − J 16 h ( J h 2 2 h ) − 2 U 1 ( 6 2 2 J 1 h − J ) (2 3 ( ∂J 1 2 J 2 4 + 1 ( 2 h ( ∂J J − h 2 2 h 1 2 h – 31 – h J 2 J 11 2 ,J is the chiral algebra of parafermions )) + ( − ( − + 1 )) + + 1) 2 h 3 2 4 1 2 2 2 2 2 J + 2 h = ∞ h J ) h 2 2 h − 1 )) + 1 h J ( − w h 1+ 2 J J 2 2 ( ) + − 1 h J ( 2 J 2 h ) + 2 − 1 W 2 ∂ 2 J 5 J )) /h J was chosen to be J ( z J 4 vanishing (as they should). To compare to the previous 1 1 ( 1 ( ( 2) ) + 1 in terms of free bosons, we find J we find in the primary basis J 2 h J 2 3) J 1 ≥ ). There is an infinite number of non-zero 2) J ) + j − ( J 2 A j − U ( ∂J 2 − J 1 1 2 ) + ( h 1 ∼ − 2 3.12 J J ) + ( 1 h -algebras generated by primaries of spin 3 2 ( ( ) h 1 2 J + 1) h ∂J 4 J 6 1 J w ( 3 W 2 6 1)(3 1 ( 4 h J 1 h h 2)(2 1 ( + 1 + 1 J 6 h − 1)(3 ( ( 2 − J 1 3 2 2 2 2 1 − ) − 2 h currents, 2 − J h h h J 2 z 2 2 1 h 2 − h j ( 2 h 2 h − − + − h + 1 2 h 2 ( 2 (2 — Parafermions W J h 1 = = = 2 (2 + + + J 2 , 3 1 2 1 , = = = 0 W W W 3 1 2 were expressed in terms of the orthogonal combination, if we choose the orthogonal Y U U U 3 ], where two solutions were found. The first one is the standard W 4 but they can all be read off from OPE of 60 ≥ 4.6.2 Another interesting truncation of strap construction of in [ and combination to be the current we exactly reproduce the formulas of the previous section up to an overall normalization. with all other discussion, where the current and with conventions in ( j transformations of appendix in the normalization Plugging in expressions for JHEP05(2019)159 ] 3 62 W [ 3 (4.65) W OPEdefs of 17 0 ] 00 4 0 ] ] 3 5 W 3 W W .... ]. The algebra was further 3 3 . From the general dis- W [ 22 0 W W [ 012 [ ) + OPEPPole 0 00 z Y (34) 45 )( (35) (33) C 55 4 55 C C W two singular vectors at level 8 so 3 ] + 4 c ] + ] ] + 5 3 3 W ∂W 3 ( W W W 4 7 3 3 3 W [ W W W with a singular vectors starting at level 8 [ [ [ (4.64) ] ) + (34) 2 45 3 U(1) was discussed in [ z , / 1 (35) (33) (33) C 55 44 55 , )( W are indeterminate. The expressions for other . 0 which is freely generated by primaries of spin 4 3 C C C 0 + Y – 32 – 5 C W 5 5 + + [ ∂W ] + (35) W 55 4 4 4 W W 3 4 C 5 (33) 5 W W W 34 35 45 W W ( 4 4 4 4 C C C 33 44 55 7 3 remain undetermined and we express all the remaining C C C W which has for generic and + + + [ − 3 4 3 5 + + + 34 00 012 C (44) 1 W 1 W W 1 55 as a quotient SU(2) Y ) = (33) 55 0 3 0 4 3 0 C 33 34 44 35 45 55 z 2 , ( 1 C C C C C C + 0 , ] ,C and 0 0 4 Y → → → → → → -algebra generated by fields of spin 3, 4 and 5. We can also assume 4 33 W (34) 45 3 3 4 4 5 5 5 C W C , W W W W W W W [ 3 3 4 3 4 5 0 33 . With this spin content, there are two algebras that solve the Jacobi C W W W W W W c ]. ] denotes the primary operator which is the leading regular term of 3 61 W symmetry under which the spin 3 and 5 fields are odd. The ansatz for operator 3 2 W We are interested in the generic highest weight modules of Let us consider Z Coset representation of 4 and 5. The other one is 4 and 5, there are no additional redefinitions possible. We fix our conventions such that 17 , , structure constants are given in appendix cussion of the fusion procedure, we expect these to be parametrizedstudied by in 3 [ complex numbers. central charge identities. One of them3 is the algebra the Jacobi identities arethis, satisfied the only constants up to these singular vectors. As a consequence of of these coefficients can actuallynormalization be of fixed, the because generators, weprimary i.e. still composite we have appears have the at 3-parametric freedom3 spin to gauge 6 change freedom. the while Since ourthe the algebra coefficients first is generated bystructure primaries constants of fixed spins by Jacobi identities in terms of these three constants and the We can conveniently extract thesewhich primaries automatically using performs the function the20 primary free projection. coefficients Our appearing next in goal the would ansatz be for to OPE fix using the the Jacobi identities. Not all e.g. [ OPE after subtracting descendants ofAnalogously primaries appearing for in the the singularregular composites part terms, of involving the i.e. derivatives OPE. which we extract from subleading We only list thealways primary fixed fields by appearing the in Virasoro algebra. the OPE, We also the denote coefficients the of primary descendants composites are by brackets, that need to be factorized in order the bootstrapthe equations to be satisfied. product expansions of primary families is exists one more solution corresponding to JHEP05(2019)159 0 ] 4 4 4 w W 3 3 w (4.66) (4.69) (4.67) w 4 33 W 4 33 C ch C + 6) 00 6392 c ] 0 33 4 ] 2 3 − + 138) 5 C W w c h 3 W + 19 . 4) 2 3 5 c 2 gives identical zero if with spin 1 to 5. We W c − [ w 0 W ] [ . c + 12) 5 ] 4 34 5 + 33 + 117 2 012 4 0 4063 33 c 2 C C h Y W 2 c − C W 5 3 4 5 − 33 34 4 2 w − C c C h 388 W 3 5 + 9 W 2 3 [ 4 33 870) 2832) c c + 114) w − 2 4 w − h 4 C 0 c 33 2 w 5 h − − 34 3 + 2) + 3 3 ) w ch 2 C 117 c c c 0 C w 33 ) h w 4 33 5 34 3668 0 22 33 + 2 4 0 C − 33 33 4 33 C C C + 2 2 − C C − C 870) 2 0 + 114) 33 ch ( h ) h + 24)(7 (4.68) + 533 c + 218 C − ] ch 4 33 c + 1214 + 114) 2 5 56 c c 2 C c ch ( − c W 2)(17 1)(7 + 114)( 314 + 288) 4 + 114) + 114)( 31 c + 114) + 3376 c − c − − c W – 33 – − c + 10)( 2 [ + 533 c c ( (7 c 3 2 2 2 (7 5 34 ch 1)(7 (2 c + 22)(7 + 114) 2 c + 10)(29 + 114)( C − 0 33 + 464 c c − 18( 3 2 + 13)(45 + 22)( c c ) h 2 C c c c c 2 )(29 + 22)(7 8( + 22) ( + 22) 4 33 2 c 1)(5 + 22) c c C 1)(7 + c + 44264 1)(5 − (5 ] + 00 − 2 2 3 ] + 22)(7 (5 1)( c + 10) 1)(5 + 7)(2 + 380 1)(5 − 3 2 c h + 10) symmetry. The even spin 8 field is the linear combination c c 1) c 3 W 2 (2 c − − c ( c 3 1) − W 2 c 2 + 114)(56 (5 c c 3 − c 2 c Z 2( 54( c W − c (2 3 W 1) (2 c − c (7 2 2 W + 10)(2 (1288 + 10) + 118 − − = [ 2 + 10)(2 c c 4 c ) c e c c = [ + 10)(2 8 + 10) + 7)(2 + 10)(2 c o + 10) N ( 9 c + 4 c c c + 7)( + 7)( ( ( ( ( N ch c h + 7)( 2 2 + 7)(2 c 2 c (10 + c c ( ( c h c 12 4 36 c h ( ( c ch 2 − +3 − − c 3 36 7468 − +6 − − − +54 0 = 72 The first constraint appears at level 8. There are two singular vectors at this level, one 0 = gives us the second algebraic relation and its action onsecond the relation highest of weight state theagain is variety of one identically highest zero. odd weights and This wewe one means need act even that to with field. to look its find This zero at the time mode level the on 9. the even Here highest field we weight [ have state. On the other hand, the odd field on higher spin charges of the highest weight state. The odd spin 8 singular field is [ Its zero mode when acting on the highest weight state gives us a constraint thus need to findto two expected constraints 3-dimensional that subvariety. reduce the space of highesteven weight and representations one odd under Including the U(1) degree of freedom, we have 5 generators of JHEP05(2019)159 j ∞ in W 2 1+ , 1 (4.72) , W 0 Y 3 ∂J 2 ) 2 3 − h 3 h h 2 2 ) J 3 3 h ∂ J 0 J 3 3 ( ) α ∂ 2 3 (4.71) − 1 ) )( J 1) 2 3 1) 2 1 = 5 to 2 + 1 = 3 using − J − ... 6 − ( ∂J − ∂J 2 2 2 2 1 h 1 + )) 1 h currents are those of J 2 3 h 2 ( ∂ ( − − J h j − 0 2 h 3 2 2 2 U α J h ) + h h ( − ) + 3 2 2 (2) 2 1 ) J − J J 3 3 U − ( ) 2 algebras and their singular vectors. 3 2 3 ) -currents J + ∂J ∂J 3 3 h J ( ( 1 ∂J U 3 J 2 1 ,N J 1 J − 1 1) 2 ( 1) ( J 2 ))( ( 2 h 2 ,N − − h − z 1 2 1 h 2 1 ( 2 2 h N − 2 2 h h − h − Y (4.70) − J )) + )) ) 2 – 34 – 3 (the OPEs of h 2 3 2 + 3 z 2 2(2 h + J ( h J 2 2 2)(2 3 ≤ 3 ∂ J 0 ) + (2) 3 J j ∂J ( − − ( 3 ) + α 1 R 2 3 1 ) 2 J ) J 2 J 2 J h ( z h 3 ))( ( − 1 ( J J ( 2 2 generators of spin 6 and higher are identically zero when z we will actually check that there exists a three-parametric J 1 J 2 with We can construct a free field representation of ( h 3) ( J j h 2 (3) 2 1 , j ( − 2 ) + 1 J − − 1) R , 1 ) + ( U W 2 3 + 1 ) 0 3 2 2 J J + z − 1 Y − J h 2 2 ( − 3 1 2 ∂ − 2 h 2 J 0 6 J ∂J h h (3) 1 ∂J 1 h α )) + ( 2 2 h J 2)(2 + R 3 1 2(2 − − − 2 J − ) + ( 2 = ( J − h 2 ]). If we are interested in primary fields, we can use the formulas given 2 J = = ) = 2 currents are non-zero but they can be calculated in a straightforward way J ( h + has already quite a long expression which we don’t need to write explic- z 1 1 24 h ( 1 2 j ( 1 3 J J or apply directly the orthogonalization procedure and the result for U − J + W W R W A = = ( = ( 1 2 3 U U U The current itly. It can beexpressed checked in that terms of thishold free more field generally representation. from This the is discussion something of that was expected to by calculating the OPEsas of discussed in [ in appendix generators is All the higher The advantage of this orderingto is pass that to the the differentialsame right operators normalization are on as just the in left first thederivatives which order, to general we so the discussion need it right, of we is Miura find simpler transformation. for than Commuting the other the first orderings. three We use the Free field representation. terms of three bosons3rd using direction the and Miura one operators. associated to We the will second choose direction two with Miura the operators ordering in of the variety of possibletwo higher constraints spin coming charges from from (2free + the field 1)(1 + representation singular 1) of vectors.family In of the primaries next whoseany section, higher other spin using constraints charges on an higher satisfy explicit spin our charges constraints, that so could there reduce cannot the be dimension further. Although the result is complicated, we achieved what we wanted: we reduced the dimension JHEP05(2019)159 4 w (4.73) (4.76) (4.74) ... + ... 3 − + ) 3 2 4 q ∂ q ) by the simple 0 − 1 1 ) z q α ( ∂ − 1 h 1 )( 0 2 (2) 2 z α − ( h − R 3 2 )( differ by a set of null ) 2 ˜ z . h U − z h ( k 2 ( 2 3 + + q U 3 = 0 (1) 1 − zero modes are 2 , 2 2 − 3 3 j R + h ) k,N q − (3) 3 above. The generators of the h 3 2 ∂ 2 W + 2 J 3 2 0 1 − 0 + h , ) h 1) , . α 1 + (4.75) 2 2 , ∂ 1 2 − ) k,N 1 − )( 0 h h − z ), we find that they are identically 2 + Y 1 z (2) ( 1 α 2 2 h 1 ∂U ∂U ( + h q J 0 0 h 2 2 )( N (2) 2 ( 1 2 h is indeed null since its OPE is propor- ˜ α α z 4.69 U Y + h ( R 1 2 + ) + + + + − ˜ z U . To verify the identities one needs also U 2 (1) 2 3 1 1 (   q q and J 1 2 = − 2 1 + C ) , (1) 1 U U q 1) 3 3 2 1 = ∂ ) and ( , 1 – 35 – R h 0 − 1 1 1 ,N ) (3) (3) − 3 3 2 ) 2 2 α − (3) Y J J 2 3 h ∂ − h (3) 3 4.67 ,N )( 0 2 J   h 2 1 J z h − α h ( N + + + 2 charges. The situation is analogous to parametrization 1 3 )( 2(2 + Y 2 3 1 ˜ z h j U − 1 ∂ ( U U U 1 ) from the example + w 0 1 ) in terms of eigenvalues of the Cartan generators. 3 h 3 2 1 q α q U = = = ). This means that we have an explicit parametrization of N q 1 j ( − q 4.61 2 3 1 − + q 1) 1 ˜ ˜ ˜ 2 gl = 1 + U U U 1 2 1 h q ) = ( 1 − − − 1 2 − ) z 2 h 2 ( − h ∂ 2 h h 0 1 h − 2 (2) 2 q α ), i.e. 2(2 R − − z ) ( z = = generators ( ( ) = ( (3) 3 1 2 z can be then obtained by a multiplication of 3 (1) 1 ( R w w 1 R , ,U (2) 2 ) 1 2 , z 1 R ( . ) Y charges, but their expressions are too long. What is important is that plugging ,U 1 z , 1 (3) 3 ( 1 5 , U 1 R w Note for example that the spin one field The free field realization can be constructed simply by starting with the pseudo- Acting on the highest weight state, the eigenvalues of (1) 1 Y charges which give vanishing R j tional to with the generators algebra Miura factor differential operator with 4.7 A commentAs on null reviewed states above, of primary the generators. algebras Let usple illustrate appearance of such null states on the simplest exam- This parametrization linearizes theexample, variety as of we will highest see weights, later,q but for the is vacuum representation not there one-to-one. areof (2+1)! the For = characteristic 6 polynomial choices for ofof a the matrix Casimir in elements of terms of its eigenvalues or parametrization The spin 3 chargesand is given in thethese appendix explicit formulas insatisfied equations (for ( any valuesthe of variety of highest weights of allowed primary charges in terms of 3 free boson charges. JHEP05(2019)159 th j (4.77) automat- th and i 111 Y are not null in i U with the only local 000 Y . 6= 0 3 3 h 2 q generators vanish since they are always 1 parameters are specialized and the full h i with a generic exponential vertex operator − ˜ U specifying the singularity of the i − ) N U 3 κ 2 h ( j – 36 – 1 q x h − by a submodule generated by level 8 singular vector and 2 ) 1 h ∞ κ 1 q ( i h 1+ x (modulo the lattice), the next-to minimal Levi subgroup = 0. is generally proportional to − W ) 3  κ h parameters with quantized values of the U(1) charge. ( j + x (3) 3 2 3 N h h φ 3 + = q 1 h ) κ + i ( = x ) is preserved at the defect. Note that the value of the overall U(1) (2) 2 is preserved by the configuration. On the VOA side, these specializations N φ N 2 2 q − N + (1) 1 U(1) φ 1 , but a larger symmetry group can be preserved if the GW-parameters are specialized. q × N 1) supergroup is preserved at the boundary Chern-Simons theory by the defect and A maximal degeneration appears when We can see that the parameter space parametrizing generic modules is divided into On the other hand, the algebra itself is far from being trivial and  | charge does not affectthe the other structure hand, of maximally modules degeneratecorrespond and modules to breaking with of generic a values the nontrivialModules of gauge GW the associated symmetry. U(1) defect On charge to with still specializations line a of operators prescribed all the with singularity for a the trivial U(1) GW factor. defect correspond to maximal of such domain wallseration (where of more the parameters module. aresubgroups These decorated specialized), more by we complicated line expect representationson operators correspond further the in to degen- gauge a larger theory representation Levi side. of the preserved Levi subgroup gauge group U( different degenerate modules. Similarly,U(1 if parameters in differentone corners gets are different specialized, classes of degenerate modules as wedomains will with see a below. degeneration appearing at the boundaries of the domains. At intersections factors are equal U(2) are going to correspondparameters, to one degenerate can modules. stillof turn For the a on preserved fixed a U(2) value Wilson at of and each the ’t boundary. specialized Hooft GW oporator Different in choice some of representation the line operators will label 5.1 Surface defects preservingA Levi generic subgroups Gukov-Witten defect breaksU(1) the gauge group at theIn defect particular, to if the the maximal parameters torus and not the simple quotientfield which being would in the this identity. case be the trivial 5 Degenerate modules One can furthermore check thatzero there in exists the a free composite fieldically field realization realizes at as the level expected, quotient 8 i.e. of that the is free identically field realization of proportional to a generic module. Forexp example the OPE of Similarly, all the two-point functions of the JHEP05(2019)159 ) κ ( i x (5.4) (5.3) (5.1) (5.2) ). The Y states at 3 N + 2 -plane. The poles N u + generators of 1 j N f lifted GW parameters ) is a rational function. In ) in the 3 ), we conclude that we have u u i ( ( N and are Taylor coefficients of ψ ψ j + j hw 4.23 | e 2 ψ k N + j , + zeros and poles. In this way we also τ ψ 1 | 3 h N N − . hw j 1 k z + = j − 2 ) − h 3 ψ ) ), the σ σ N ( k u do we obtain a degenerate module. = x j ( – 37 – ] that the representation is quasi-finite (i.e. has ) + i X j ψ 4.23 − 1 κ 30 ( j ) hw N τ | x ( j j x e k f | ) is equal to the degree (i.e. number of zeros counted with u ( hw ψ h ] that the vacuum representation has exactly one zero and one -plane is determined by U(1) charge of the highest weight vector u 30 the quasi-finiteness is automatically satisfied. ). This is automatically consistent with the form of the generating collides with a pole of type u 3 ( j module with weights parametrized by ( ,N ψ 2 3 ,N ,N 1 ) which has generically 2 N ,N Y 1 4.23 N Y This also refines the statement it [ Applying the results of the previous discussion to the highest weight vector of the As we discussed in connection with ( generic a singular vector at level 1 if one of the following conditions isi.e. satisfied a zero of type and is translated under the spectral shift transformation. only a finite numberthe of case states of at each level) if and only if with highest weight charges multiplicities) of function ( rederive the result ofpole. [ The distance betweenposition them of is the fixed zero by in the parameters of the algebra. The absolute different version of this theorem because the coefficients but the result is the same: the number of vectors at level 1 in the irreducible module rank if and only if the associated generating function is a Taylor expansion ofis a equal rational to function. one Furthermore, plus the the rank degree of the of Hankel this matrix rational function. In our case we have a slightly Shapovalov form (where we used thematrix basic on commutation the relation right between Kronecker is which a tells Hankel us matrix that and (in we general can infinite use dimensional) a Hankel variant matrix of has the a basic finite theorem by ask is for which values of parameters 5.2.1 Singular vectorsThe at discussion level is 1 easythis at level. the We can first detect level. the appearance A of generic a module singular has vector by studying the rank of the Let us start withnext-to-minimal the Levi analysis subgroup. of domain walls of minimal degenerationscorrespond associated to to positions the ofare poles determined of up the to generating a function permutation of order of poles in each group. A natural question to 5.2 Minimal degenerations JHEP05(2019)159 1 ]. h pa- 54 and [ (5.9) (5.6) (5.7) (5.8) (5.5) j 2 , h 0 r,s , 0 Y ]. The singular ). The poles of . 2 63 N , rh Virasoro subalgebra 48 + ∞ 1 T if and only if the distance 1 are integers) are labeled sh rs ) 3 ≥ = h 2 the leftmost field in the Miura (3) 1 − r, s ) rh x . 1 2 J (3) 2 (3) 2 + − 3  x x 1 h (3) 2 (3) 2 − − sh 1) x x (where are known as well [ u u 2 = − − )( )( N h rs 3 j appears at level 3 1 h h (3) 1 W 2 (3) 1 h ) is an integer linear combination of , ) x 4 or x 0 + ( k , −  0 j − – 38 – − q Y 4.23 if and only if the highest weight equals ∆ (3) 1 − u j ) is ( x = 2 3 , u rs h 2 (still assuming the standard ordering and using the ( − + ( . (3) j ψ j rh k u 0 x , q ( q 1 , + ∆ = 1 − 1 Y j ) = q sh u ( = ψ label the (positive and negative) roots of SU( (3) 2 x N if and only if − )) by ≥ r,s ). Choosing the standard ordering ( (3) 1 k N 3.12 x 6= algebras j N ≤ W . Similarly for the other two types of poles. 2 h ) are related to U(1) charges u ( where 1 ψ conventions of ( The Kac determinant and singularvectors vectors (zeros of of the Kacby determinant) roots at of level SU( transformation), the equations for vanishing hyperplanes are Therefore, a singular vector of the algebra between two poles of theand 3rd type is a positive or negative integer5.2.4 linear combination of This is equal to ∆ We can extract the(decoupled conformal from dimension the ∆ U(1) field) with respect to the The first case is simplewhich — we we have are a interestedmodule known in classification: has singular a vectors for singular of genericThe vector the values generating at Virasoro of function level algebra the of for charges central charge the Verma is satisfied, we canlar learn vector more appears about andenough the the to relation corresponding look between distance at the betweenof the different case the level type of zero-pole where in the pair. such the zero-pole a case pair It singu- of of is5.2.3 the then same type in Virasoro the algebra algebra At higher levels the discussionevaluate is the not ranks so of simple Shapovalovof because matrices the the become commutation Shapovalov more relations matrices, involved.distance used we But to between expect from a the the zero structure highestrameters. and singular If vectors a this to assumption pole of appear of locality only ( (i.e. if pairwise the interaction between zeros and poles) 5.2.2 Higher levels JHEP05(2019)159 110 Y (5.11) (5.10) . We can , we have n N − W . -algebras) de- 1 n N . − 3 W ↔ nh n if + 2 n ). h u language determine which ( = . In this case, the parameter ψ N then label representations of 0 . 1 W (2) 2 , determine the distance between 1 x r, s Y r . − 2 (1) 1 rh and x s + ] or using the BRST construction of the 1 . Therefore, in the case of 2 30 The maximally degenerate modules of , sh h 5 or = running over all integers, but for non-positive 18 – 39 – and n (3) k 1 x , h 3 − nh (3) j − x 1 h − = (2) 2 0 , x 1, we have a leading singular vector at level 1 1). In other words, they can be thought of as pairs of partitions glued , − , 1 ≥ 2 Y , (1) 1 n x there is no difference between the maximally degenerate and minimally de- ) approach each other and the integers the level at which corresponding singular vector appears is 1 the anti-box in first direction is equivalent to a box in the second direction and vice versa, so u 0 ]. ( n , 110 1 1 ψ , 1 Y Y In A direct calculation (which we explicitly checked up to level 4) leads to the following In the gauge theory language, we see that (at least in the case of In general the box counting works only for so called covariant modules which have asymptotics made 18 direction). The degenerate modules are labeled by two integers, theof boxes heights (tensor of products asymptotic ofgeneral the class fundamental of representation). representations Incase where general of there it are is both importantthe asymptotic to simple consider boxes box a and counting more anti-boxes. picture Fortunately is in applicable. the generate modules. For(plane the partition) maximally interpretation degenerate ofin modules. modules this we picture can correspond usebox to at the position plane box (2 partitions counting together (with by possible the asymptotics) first column which (assuming have for no the moment that there is no asymptotics in 3rd Note that these twothus conditions use are only exchanged one ifvalues of we of the formally conditions replace with form (Kac determinant), using boxalgebra counting [ [ condition: given The remaining elementary case that wespace need of to generic analyze modules is is two-dimensional,with so a after decoupling one-dimensional the parameter overall U(1), space.there we is are Analogously left no to difference the between casefor minimally of degenerate and the modules maximally Virasoro degenerate in algebra, modules. at We least can three look possible ways: directly studying the Shapovalov to-minimal Levi subgroup isthe preserved. preserved The SU(2) parameters subalgebraat associated the to two the interfaces. corresponding line operators supported 5.2.5 Algebra these poles, quantized inan the independent units confirmation of ofmodules correspond the to fact pairwise that interactions the between poles leading of singulargenerations vectors appear in when degenerate the GW parameters are specialized in such a way that a next- This is exactly of thealgebra. same We form see is as that thepoles the condition of positive that or we negative found roots in in the the case of the Virsoro so we can rewrite the equations for vanishing hyperplanes as JHEP05(2019)159 (2) 2 φ (5.13) (5.12) × (1) 1 i in a given φ . i 0 (2) 2 , 1 φ , (2) 2 ) 1 φ 3 Y ) 3 nh h + = 2 can be realized as + 2 q , n q i + ( can be identified with the + ( (2) 2 n (1) 1 φ ) (1) 1 3 qφ h h = 1 and qφ ) can be also realized in terms of h n + q exp 5.11 k ) can be identified with one plus the exp m + (    5.11 (1) 1 J 1 (2) 2 qφ − J h k 1 ∂ parameters parameterizing the generic modules h – 40 – j exp − 1)! x 1  by constant shifts). It turns out that a half of the by a same constant corresponds to the spectral shift − (1) − 1 j ) (2) j 2 J k x κ ( 2 J ( j 1 appearing in ( ) that we found (together with their images under the h x  h  n k ∂ − m 5.11 k − 1 ! (1) 1 k 2 J  2 m h (2) 2 ) and (  n =1 0, one can realize the corresponding degenerate modules in terms of (related to J Y k 1 j n h q 5.10 = n > n − lie. Translating all j nm x (1) 1 + ]. In particular, the BRST analysis of the algebra have not found any other J into domains. The points lying on the union of these hyperplanes correspond 2 ··· 1 3 h + X 2  descendant. The descendants are generally given in terms of Bell polynomials ,N m 2  n ,N +2 2 1 1 1 Turning on a non-trivial asymptotics in 3rd direction decouples the pair ofThe Young same structure of maximally degenerate modules can also be seen from the BRST N − m Y Similarly, for any a level a free boson descendant ofordering, an the modules exponential in vertex the operator. 2nd direction For example, specialized to in the Let us briefly commentfree on field realization. the The realization highestgeneric of weight ones) the primaries can of degenerate be all modules realized theby representations as the of (including simple parameters the exponential vertexdegenerate operators modules with associated exponents to given the degenerations ( of degeneration of acorresponding given module dependstransformation on that the only number changesdoes of the not U(1) hyperplanes change charge on the of structure which the of the whole the5.2.7 singular module vectors. and in Free particular field representation of degenerate primaries 5.2.6 Summary The hyperplanes ( triality) therefore divide the full spaceof of to degenerate modules while the remaining points label the generic modules. The degree degenerate modules and the degenerateparameters. ones appear From exactly the for gauge thedifference theory above of values charges point of of of the generic U(1) view,on line the the operators Wilson value supported line at the operator boundary at 1 the and 2. boundary 3 Turning lifts the degeneration. diagrams so the box countingof predicts the a free boson generic character). moduleany The (i.e. conformal additional character dimensions zero equal of of to these modules the the also Shapovalov square don’t form, produce confirming theanalysis whole of box-counting [ picture. matters, the modules withcharge. the Finally, same the difference parameter difference of of heights the differ heights onlyinterpretation of predicts by the the the asymptotic correct overall level Youngand of U(1) diagrams. the the conformal null It vector, dimension. is the easy correct irreducible to character check that this Young diagrams in 1st and 2nd directions. But only the difference of these two integers JHEP05(2019)159 ]. 65 , (5.14) (5.17) (5.15) (5.16) 64 ]. ]. ) for such 30 66 u , , ] ( 5 5 ψ , 67 1 3 h 3 algebras [ N 3 − ,N . 2 2 3 ) . h h 1 ,N 2 3 ) 1 h 0 N N ) u N ψ 3 + Y 3 − h − u h 1 2 2 − )( h h h 0 1 2 1 u ψ h N N 3 )( 2 h − − − 2 h h u 1 u ) of a conjugate representation [ 1 − h − u h 1 ( – 41 – u = ( ( 1 N + ψ 0 − − ψ u ψ 3 ( = h 2 0 h ) = u ) = ψ 1 u 3 u ( h ( h ¯ 1 ψ 2 = 0. The general case is a bit more complicated because of +  h 3 1 u ψ h N . This is analogous to expressions for singular vectors in free field ) = u (2) ( 2 these reduce to Schur polynomials whose special case are the Bell • J 1 ψ ). Higher level specializations will be further discussed in the next section h 110 Y − 5.13 (1) 1 J 2 h = The generating function of the anti-fundamental representation can be obtained from The generating function for the vacuum representation has a single factor J and similarly for the fundamental representation in the otherthe two formula directions for [ the generating function On the other hand thedirection generating can function be for written the as fundamental representation in the first where we used the identity restrict to the casethe when appearance of continuouson families this of issue free later.representations. field Let realizations. us start We with will writing briefly down comment the generating function In this section, we mostlytogether concentrate with on the the free modulesanti-fundamental field representation. associated realization All to of the the thebe other identity line obtained operator maximally operators degenerate from representations the in can fusion the of fundamental these and two the (and a shift of U(1) charge). We will further hand, we will now discuss brieflyassociated free to field realization line ofmodules operators the maximally play supported degenerate an modules at importantcomplicated the role VOAs interfaces, by in extensions i.e. the of trivial gluing tensor GW products construction defects. of that5.3.1 allows These to engineer Identity, box more and anti-box at different corners. 5.3 Maximally degenerate modules In the previous section,algebra modules we and have concentrated mostly discussed on the the minimally general degenerate structure ones. of On the degenerations other of Y- representations of Virasoro algebra whichIn are the given in case termspolynomials of of ( Jack polynomials [ in the context ofalready maximally for degenerate the partially representations degenerate but modules associated note to that specializations of the GW issue parameters is present for JHEP05(2019)159 1 2 h N (5.20) (5.18) (5.19) ) contributes u with the same steps in the ), we have ( ) u 2 j ψ ( κ N ( j ψ x . ) 2 0 h ), we get the generating ψ 3 − h (2) i 2 x 5.16 h (2) i 1 − x generators transform in a more direction and h ) u j 0 − 1 )! + ψ ψ 2 h ! u 3 3 2 N h h 2 . To realize the generating function of 2 N 2 +1 ! N + 1 h + . One can draw such a combination as a 1 h + 1 i N 1 Y 1 u κ N h = N N i h )( ( + 0 1 + 1 ψ h ) = 3 i h steps in the κ – 42 – − h (1) i ( i ! 2 − generators. 1 x 2 . In the generating function x h i (1) i j u N 1 κ N ( − f x 1 h h u u + N by − + 1 2 u 2 and -plane. N h h 1 j u 2

=1 e molecules of length N i Y + Let us start with a discussion of possible realizations of the N 2 ). In the molecular picture, we want to connect the pole at in terms of a specialization of the parameters of the generic u + ( N 0 and a zero at , 1 ) = 2 ) 5.14 h = u i 1 ( ,N κ 1 and 1 ( i ψ ¯ N  x N 1 ψ Y h For such an analysis, it is useful to introduce a diagrammatic picture for zeros and poles Applying the conjugation to the generating function ( such paths (note thatthe the Weyl group molecules in of each one corner, type i.e. if are we indistinguishable identify if the we permutations mod of out by between zeros and poles in the Identity representation. identity representation ( zero with the zero at direction. There are clearly the vacuum, the fundamental orof the the generic anti-fundamental generating representation, function mostsuch must a of cancel. cancellation the At appears the factors when levelmolecule. a of the circular interaction In node of coincides ourrealizations molecules, with of diagrams, a various filled degenerate we node representations denote of thus translates a such into different a the analysis zero-pole of pair paths by a cross. The study of in the spectral parameterby plane. a pole Each at factor position diatomic in polar the molecule (with generating a circle function correspondingto to the zero) pole and separated amolecules full by dot of corresponding the length distance generating function and similarly for the other two directions. 5.3.2 Analysis atWe the will level now of generating identifyrepresentation) functions the for triple (identity, fundamental representation, anti-fundamental complicated way, mixing with function for the anti-fundamental representation eter and produces anspectral automorphism shift just is as necessary in into the order verify case to that of have the finite self-conjugate effect vacuumcharges. Yangians. of representation. The Note conjugation It additional is that is to easy there(not flip exists just the a acting sign conjugation on of all automorphism the odd of highest primary the weight highest state), whole weight affine but Yangian the This is a composition of the inverse anti-automorphism and the reflection in spectral param- JHEP05(2019)159 . 1 ). j N and N 1 5.16 (5.21) h ) whose with the 2 h 3 h + = 1 h u ) for the vacuum . If we treat the u + . In the picture of 5 ( 0 α α ) for the fundamental ψ u ), we need to connect ( = ψ 5.19 (3) k molecules of length , x 1 . One has 1 2 h N h + α with the generating function ( 1 = ) for any choice of h ! u is drawn in the figure )! solutions which nicely corresponds to ( 1 (2) j 2 0 , ψ leads to a more complicated story. In such 2 N − , 5 3 2 α, x + 1 Y N ,N 1 N 2 = – 43 – + N ,N 1 1 (1) i N N x 1 segments of length Y

Let us now move to the more complicated discussion − [Ψ]. We draw one example of the snake configuration. 0 2 , 2 , N 5 Y of 3 . The only possibility is to connect the pole at 2 h h but only [Ψ]. We draw one example of the snake configuration connecting the zero 1 0 , h 2 , 5 and then draw a snake connecting the other zero and the pole using Y 1 h − The structure of zeros and poles of the generating function The structure of zeros and poles of the generating function ). One such path for the algebra j = )! possible orderings in the Miura transformation. κ 2 u N Turning on all three parameters + molecules of length 1 2 -plane. In general, we get as many of these continuous moduli as is the minimum of N zero at segments of length of the fundamental representation inThe the discussion of direction the anti-fundamentalone. representation completely To mirrors find the the fundamental realizationthe in two terms poles of and theN two generating zeros function with ( two snakes composed of contribution cancels in any generatingmolecules, function such a factoru corresponds to a triangular loop that can be freelyFundamental moved representation. in the value of molecules to be distinguishable,( we get ( cases, we expect totinuous centre obtain of continuous mass families of of the realizations triple corresponding ( to the con- Figure 6. representation in direction Figure 5. representation of and the pole. JHEP05(2019)159 , j f are j (5.22) e ]. The 30 , is shown in 0 29 , 2 hexagon factor , 5 Y . 7 . that we expect to be ) ) , the eigenvalue of the 0 3 3 , j 2 h h f , 5 + − Y α α and ). j − − e u u 5.22 )( )( 2 2 h h + − α α − − α u u )( )( 1 1 – 44 – h h + − -plane as illustrated in the figure hexagon factor from ( α α u with ladder operators − − ∞ u u ( ( 1+ depend on the state on which we act. The number of such W α ) = Figure 7. α − ) changes by products of elementary factors u u ( ( ψ ϕ can produce singular vectors which are annihilated by all raising operators Generating function of the exponential factor in the realization of the fundamental . j e 6 Even though the attaching hexagon factors by itself does not ensure that the descen- In the generic modules the states that we get by acting with ladder operators It turns out that compared to the vacuum representation, the fundamental represen- operator. We can thereforepure find exponential vertex free operators field but are representatives dressed of by primary the action states of which bosonicdant ladder are field operators. is not singular, werealized will for now some sketch few ordering configurations of for free bosons. If we place the (the inverse) hexagon factor at never highest weight states.operators But in thei.e. case they of are degenerate primary. modules, Infree the the field irreducible representation, action modules these of we singularor ladder identify to vectors these a can states non-trivial be primary with mapped state zero. either which In is to the a an free identical boson zero, descendant of the exponential vertex These factors play apossible role values of of parameter thefactors structure determines the constants level of of theis the descendant. because affine Here it we Yangian forms will [ call a this hexagon factor in a the tation can be genericallywhen realized acting at on higher the states levelsgenerating in in function the bosonic Fock space. Recall that representation in the first direction at level one. possible configurations. An examplethe of figure such a realization for the algebra Figure 8. JHEP05(2019)159 ], and 40 with , 2 (5.23) 1 h 39 h with the 1 h with the zero 2 h ). Going to one 5.19 . ! 2 N 1 + for the identity and the fun- N algebra associated to different 1 ) 0 κ , N ( j 2 x

,N 1 = N Y ! 1 − 1 2 )! realizations and one can find them already − N 2 1 N + and create a snake between the pole at N – 45 – 1 + 2 1 h N N

+ and create a snake connecting the pole at ! 1 1 h − 2 1 N N . The other possibility is to connect the pole at . One can now realize this new configuration in various ways in 2 + )! free field realizations of any h 1 8 2 2 . The two possible snake configurations give the correct number of N 2 N N h

module since it is not clear that all of these have the correct fusion and + 2 + 1 by a molecule of type 0 N , 1 2 N 3 h + h 1 ,N 1 1 − N Generating function of the exponential factor in the realization of the fundamental N h 1 Y by a molecule of type N 3 h To determine the structure constants (and braiding and fusion in particular), one needs There are ( Moreover, it might be puzzling that we find more than one free field realization of Finally, let us discuss the situation of the fundamental representation in the third − braiding properties and leadit eaquivalent OPEs turns of out degenerate that all modules.screening the charges Following realizations (contour [ seem integrals to of be screening equivalent currents) if in we all workto the modulo determine examples insertions three-point bellow. of functions of all the degenerate modules. Choosing a particular possibilities realized in the context of the free fieldorderings representation. of the freethe bosons. level of It generating turns functions out are that realized not for all anythe the given same possibilities choice of discussed ordering. above at 5.3.3 Free fieldAfter realization the and identification OPE of ofdamental possible modules and values of the parameters anti-fundamental representation, let us discuss how are these different at zero at realizations levels are present for a given ordering of freedirection. fields. It turns out thatat there level are zero. ( There arethe two zero possibilities. at The firstthe option zero is at to connect the pole at is shown in theterms figure of the genericlevel higher, generating one function gets after offigure the the and division exponential by similarly the factor at tworealized ( hexagon higher for factors levels. a each combination ordering from We of the will the see free that bosons all but the only solutions some at of level the zero realizations are at higher Figure 9. representation in the first direction at level two. the origin, three nodes cancel out and three new ones are created. The new configuration JHEP05(2019)159 . )! 2 (2) j N φ (5.25) (5.24) + for box 1 σ N h of the pure j q and the direction ) τ is the left-most free ( , φ (2) i σ φ h and exponents in the Miura transformation. ) in ) j j φ κ ( j = 0 corresponding to ( i i x ], where j σ, τ, π (3) 2 (3) 2 q (2) i . The anti-fundamental field is given φ φ k φ κ 3 (2) i − − h h φ (3) 1 (3) 1 k

− 1 0 ∝ , ) 1 3 Y , (1) 1 1 w 0 ,

h , φ ( 1 Y , ) 1 ) i i 0 , 3 ¯ )  Y + ( w i h

( (2) 2 w ) M 2 ( 1 φ ) − (1) 1 (1) 1 z 1 3 ( z 1 ¯ φ φ 2  M h 2 ( 1 2 3 ) h 1 z h − h ( M 2  M ( ) h h h h

2 z 1 M ( 1 factor comes from the contraction with ,M

1 M 1 − 1 

= exp = exp = exp = = exp M h 1 2 2 1 1 1 1 1 ¯ ¯     Let us now show that the two-point function of both realizations of the anti- Let us first check that one-point function of the identity realized as M M M M M where the One gets the same expression from the other realization fundamental representation with the fundamental representations are also equal Similarly for the two-point function with two contour integrations, one gets vacuum amplitude Note that there is onlytions a (the single simple realization one) of of the the fundamental anti-box field is and at one level of one. the realiza- and the following screening current field in the first and second direction for the ordering gets the following realizations of the identity, the fundamental and the anti-fundamental JHEP05(2019)159 by with (5.36) (5.37) (5.38) (5.34) (5.35) 10a 10c , one gets 2 2 ) 2 h h κ h ( i that does not seem to + + x 2 + 1 1 h 1 h h h − 2 . Firstly, the generating 1 h 0 = 3 = 2 = h , with yet another hexagon 1 2 . , − = 2 2 h (2) 3 (2) 3 (2) 3 . 1 Y h − 2 h 10b (2) 3 2 , h x − h − 1 3 = 2 1 h h − = h − and (2) 3 1 + − 1 2 = h (2) 3 h u u h 3 3 2 2 − (2) 3 , x − h h − 2 , x = . If we multiply the configuration , x 2 u h 2 − − u 2 α h h (1) 1 h 1 1 1 , x + u x 1 h h h + − 1 ,, x x = 2 2 2 h 1 1 2 h u u – 49 – permuted. h h h − − − = (1) 2 = u u u = = = (2) 1 (1) 2 x (1) 2 (1) 2 (1) 2 (1) 2 , x 2 ) = ) = ) = h u u u , x and , x ( ( ( . In terms of the standard parameters 1 1 • + ) for some shift ¯   ψ (1) 1 1 = 0 , one gets the generating function from the figure 11 = 0 α ψ ψ x h 1 h (1) 1 − . From this figure, one can read of possible values of parameters (1) 1 = x x u ( , x (1) 1 ϕ 2 10a x possibly permuted. Finally, composing h + (1) 2 1 x ,, x x h charges for the vacuum, the fundamental and the anti-fundamental repre- example i = 0 = 0 = 0 19 and , ψ 1 , (1) 1 (1) 1 (1) 1 2 (1) 1 x x x Y x . One can thus realize the same module as a level one descendant of the module One can do a similar investigation for the anti-fundamental representation and recover One can also realize the same highest weight state as a free field descendant of a differ- Note that there exists also a configuration with 10b parametrizing the modules 19 i correspond to any simple realization. Similarly, we get one extra solution also at level two. the structure from the figure or with factor with the centerparameters at a hexagon with theure center at zero,with charges we get a structure of zeros and poles from the fig- as well as the solution with ent exponential primary. Chargeswith of such hexagon a factors module can be deduced from the composition The structure ofcaptured the in generating the function figure x in terms of zeros (dots) and poles (circles) is To illustrate thefundamental existence representation, of let us thefunction now simple of discuss realization the algebra ofsentation in the the first fundamental asymptotic and direction is the anti- 5.3.6 JHEP05(2019)159 ) ) ) 2 2 2 h h h ) 2 ) + − + h 2 ) 1 1 1 h 2 − Level two. Level two. h h , h , h 2 + − 2 2 , , h 2 − 1 2 , h h (c) (c) h 1 2 + 3 , h h , + 1 , h 2 1 1 1 h + , h , h , h 1 , h + h 1 h ) charges ) = (0 ) = (0 1 ) = (0 ) = (0 1 1 (2) 3 1 0) = ( h , h h , 2 0) = ( 2 , x 2 2 , h , , , h 2 1 2 1 , h (1) 2 h 2 + , h , h , h h , x + 1 2 1 h (1) (1) (2) 3 3 3 + (1) 1 , h R R R x 1 , h + ( h ) + (0 ) + (0 1 (1) (2) (1) 2 2 2 3 3 h h R R R ) + (0 , h − 3 Level one. Level one. ) + (0 ) + ( 0 – 50 – (2) (1) (1) 1 1 1 , 3 3 , 0 ) + ( , h R R R h h , 3 0 (0 − − , (b) (b) (0 , , , h (0 0 0 0 , , , (0 (0 (0 2 0 1 1 0 2 Level ¯ ¯ ¯       Level zero. Level zero. Representation Exponential factors of the anti-fundamental representation in direction 1. The two Exponential factors of the fundamental representation in direction 1. The two crosses (a) (a) In total, we found two realizations in terms of level zero, level one and level two Note that the level agreestype with on the our left proposal and about right the of number the left-most of and free the bosons right-most of free the boson other of the second type. corresponding ordering of free bosons, on recovers: Figure 11. crosses in the middle figure correspond to two possible paths. descendants. We will nowizations see of that the different 3! solutions free field can realizations. be identified Shifting with the simple simple real- exponentials according to the Figure 10. in the middle figure correspond to two possible paths. JHEP05(2019)159 ] -  5 th i ] and (6.2) (6.3) 71 ) trans- , Z 70 ] and at the , 72 , 67 ] constructed lattice -parameters. The 1) branes, one has h 69 , ) charges of the , ]. was conjectured in [ 68 5 p, q are universal parameters 20 [Ψ] by an SL(2 i (6.1) 3 .  . The resulting VOA is an  ,N [Ψ] 2 2 1 2 12 p p ,N A are the ( 1 − − − N i 1 -parameters and Y A Ψ Ψ A  1 2 1 − , q q ,N 2 2 − ,A ,N  1 4 ) and 3 A , N 2 The extension is then generated by fusions algebras are discussed for example in [ · ,N Y i ,  2 N 1 21 ⊗ A .  ,N W 2 1 – 51 –  = ]. N [Ψ] i = ( Y − 73 2 h ˜  A 1 -parameters are associated to each vertex and they are related  + h 1 − ˜ A , [Ψ] = 2 = 3 3 3 ˜ ) web of five-branes and D3-branes attached to them at A 3 ,A ,N ,N −  2 4 , 2 1 p, q ,A ˜ ,N ]. ,N A 1 2 1 − A N N with 75 as a building block, one can construct more complicated VOAs Y , Y i 3  )-brane configuration from the figure 74 ,N = 2 of the algebra can be easily determined from p, q i ,N i h 1 h [Ψ] is related to the standard algebra N 3 3 Y )-transformation which brings the vertex to the standard one. [ ,A ,N 2 2 Z , ,A ,N 1 1 A N Y by SL(2 Let us briefly review the gluing construction in the case of a single edge. The general- If we consider gluing of vertices, we need to distinguish There exists a large list of special examples appearing in various contexts in the literature. The story j  21 20 of extensions of VOAsextensions dates back of to the the freemany early other boson days places. of VOA. VOAs, More Extensionslevel where recently, of gluing of authors at quantum of the toroidal [ level algebras of appears affine in Yangians [ wasparameters initiated are in determined [ by Ψto while the and in the casethe of identification the standardof trivalent junction the tensor of product NS5,anti-fundamental of D5 representation the and associated fundamental to (1 representation the associated second to vertex the and first vice vertex versa. and where we have introducedinterface the with vector the arrow pointing out of the vertex. Note that where formation of parameters The parameters later. Consider a ( extension of the product realization of the glued algebra.via We bosonisation expect to some well-known of free-fieldKac-Moody the realizations, algebras free such [ field as realizations the to Wakimoto be realization related of ization to more complicated configurations is straightforward and will be briefly discussed Y-algebras associated to eachinternal vertex line by of bi-modules the (andbut web their no diagram. fusion) explicit Existence associated construction offield to of realization such OPEs each discussed an above between extension seems gluing tobi-modules. bi-modules provide In us was the with proposed. a two waycase. explicit to The examples Note determine free bellow, also OPEs that we of such will such construction indeed leads see to that an this algorithmic is way to indeed determine the a free field 6.1 Gluing usingStarting free fields with associated to anvarious arbitrary faces. ( The resulting vertex operator algebra is an extension of tensor product of 6 Gluing and generic modules JHEP05(2019)159 and (6.6) (6.7) (6.8) (6.4) (6.5) 3  ˜ M i 2 i (2) N 2 ) ) . φ (3) N 1 2 ¯  ˜ 1 φ , q , q h 2 M 1 2 ˜ p − h p = 0 and identify only -charges. All the other h h 4 and W = ( = ( N 1 2 3   ) exp ) exp ˜ A A ˜ 1 M M J J ( ( N 2 3 ⊗ f f N N 3 ¯  = = + + 1 2 M 3 3 ¯ ¯   (3) (1) N N ) ˜ ˜ field of the free boson. Even though ), they can be easily determined from 2 = R R M M ˜ q J 3 ¯  ( ··· ··· + f 6= 0, a general ordering and ‘non-simple’ N 1 2 4 4 +1 +1 2 2 , q N N N 2 (3) (1) N N p – 52 – ˜ R R ) and ,M Gluing of two vertices. 3 ¯ 2 2  J + ( (2) (3) N N ˜ 1 M f ˜ p R R ( ⊗ are primary fields of correct ) is a level ˜ J ··· ··· 3  3 ¯  ( i ˜ f M (2) (3) 1 1 i M (3) 1 ˜ 3 ˜ R R φ = Figure 12. (2) 1 ) ) N 2 1 φ 2  and ˜ and q ˜ q ˜ h 1 , , 1 M 3 h − 1  2 h h p N p are the primaries associated to the fundamental and the anti- M 3 ¯  = (˜ = (˜ 1 2 M = exp = exp ˜ ˜ A A 3 3   ˜ M M and ) is a level 3  J ( M f associated to the left vertex. The simple free field realizations in the given ordering In configurations with more internal finite interfaces, one can introduce corresponding The gluing fields are the generated from the fundamental and the anti-fundamental In the free field realization, the fundamental and the anti-fundamental representation 3 ¯  ˜ bi-fundamental fields can be constructed from the fusion of fundamental and anti-fundamental representationsextend associated the to tensor product each of finitethe Y-algebras segment gluing by procedure and fusion on of two all examples such generators. bellow. We will illustrate where we lack a closedthe form requirement expression that for fundamental module associatedM to the thirdare direction of of the form the right vertex and representation associated to lines supported at the internal interface generated by where in the left vertex.realizations is straightforward The but generalization the formulas for become more involved. the simple realization of thefollowing fundamental ordering and the anti-fundamental representation for the of free bosons in the right vertex and have a simple realizationFor in simplicity terms of of the an exponential discussion, vertex we operator will and restrict its to descendant. the case JHEP05(2019)159 (1) 1 ˜ and φ (6.9) (6.11) (6.10) (6.12) ≡ (2) 1 ˜ φ ∂φ = J . 2 ) w 1 − z (   3 are thus building blocks of 1 1  . The relative sign depends 1 2  N N 3

 · 1 1 + + ,N 2 2 ) N N ], we defined the orientation of 2 2 (3) N (3) N 5 + + ˜ Kac-Moody algebras as shown in ,N 2 2 ∼ − φ A 1 . The relative orientation and the N N ) 1 Ψ N − ˜ q q w 2 Y 1 ( p (1) ˜ , . . . , φ ,..., A J + b ) 2 3 gl − q +1 +1 , . . . , q ,..., z h 1 2 2 ( i q ˜ (3) N (3) N +1 +1 n J + = ( ˜ 2 2 φ 2 ], where we introduced a vector of free fields  , φ , p N N = µ 1 2 2 ˜ · q i p , q , Φ (2) N (1) N 3 ˜ – 53 – q 2 2 µ ˜ , ˜ φ 1) A correspond to the same GW defect and the gauge N N 2  Q ˜ − ) q i − ), in particular i q 2 q w ˜ = 1 A , . . . , φ ,..., −  , . . . , q ,..., · (2) 1 (1) 1 z and ˜ 1 1 ( ˜ 3 φ φ ˜ i q q and 3   q A  1 is given by a product of such factors in the two vertices. In 1 1 be the free boson associated to the right vertex and  = = = = = A 22 3 µ µ µ µ h (2) 1 ˜ ˜ Φ Φ Q Q ∼ − φ 1 for the resolved conifold diagram and +1 for the toric diagram of ) (1)’s ≡ and w − b gl ( as a sign given by ( i φ J n 3 3 ) z ,A ,N ( 2 2 J ,A . Let ,N 1 1 A have the following OPE N . We will see later in examples that this condition is necessary for the gluing 13 Y C (2) 1 ˜ × φ ∂ 2 Note that inclusion of bi-fundamental fields might change the algebra of zero modes Note that the parameters The sign would be opposite if we have glued the fundamental representation of the first vertex with the Z = 22 / ˜ J fundamental representation of the second vertex and similarly for the anti-fundamental representation. 6.3 Gluing two Let us consider the firstthe example figure of gluing ofbe two the one associated to the second one. We normalize them such that that might become non-commutative.general Moreover, not we even will modules induced see from laterassociated the that modules to of the the the modules zero-mode commutative algebra. aremodules zero-mode in for GW algebra modules more of complicated algebras with non-commutative algebra of zero modes. on the relative orientationa of vertex the two gluedsign vertices. in In the [ aboveparticular, equation one gets C bi-modules to be local with the GW modules. theory setup suggests thatsupported they must at be the boundary identified (up to shifts induced by line operators for some integers A generic moduleexponentials of associated a to glued each vertex algebra in can the diagram. be then realized as a tensor product of such and similarly for the other vertex Let us discuss how toGW glue defects. generic modules The and highest itsas weight interpretation vector an in of exponential terms a of vertex generic theand operator module physics a exp of of [ dual a vector Y-algebra of can charges be realized 6.2 Gluing and generic modules JHEP05(2019)159 (6.15) (6.16) (6.17) (6.13) (6.18) (6.14) . i ˜ φ ... 2  + + ) φ w 1  ]( (1) Kac-Moody algebras. 2 − .  b gl h , 1 n i ) + − w ˜ φ ˜ w q q 1 , = − that has the following simple form 1 − = exp ) + ˜  z 3 2 ¯ z  is the well known bosonization. ¯ 3  (  − . In particular, requiring the OPE to be of n  1 ∼ qφ ), i.e. ¯ M  ¯ −  2 q h ) (  = 2 (1) Kac-Moody algebra. The relation between M M w  ˜ q ( , b 6.11 gl and – 54 – form the free fermion pair and the combination ¯  exp [  − − ,M  ¯ and  q M M 1 ∼ i ) ] = exp  ˜ M  ˜ φ 3 ) M ) z q  ( 2 1 w  1  M q,   [ ]( − − ˜ q and M ( M φ q, −  1 [  h M M are related by ( ) z ˜ β (  β, = exp M The simplest example of gluing of two  2. Moreover, the free field realization gives also an explicit realization / . Note that this is exactly the condition following form the locality of M n Figure 13. ] with the gluing bi-modules can be identified with the decoupled ˜ q ˜ Having an explicit description of the glued algebra in terms of free fields, we would One can immediately see that the BRST definition of the algebra is reproduced. In J q, [ (1) subalgebras and that the conformal weight with respect to the sum of the two stress- + for some integer M the following form where the parameters free fermions and the vertex operators like to discuss generic modulesexpect of the the correct glued algebra. GW-defect module According to to the be discussion generated above, by we descendants of particular, the added fields J with all the other OPEsexponents trivial. (with The the exponent metric was determined determined from by the the product normalization of of the the two free bosons) One can easily check thatgl the two generators haveenergy correct tensors charges with is respect 1 toof the the two OPE between the added fields Generators that need tofollowing be vertex added operators to realizing the the fundamental algebra and can anti-fundamental be representation identified with the fusion of the JHEP05(2019)159 . ˜ J + . The (6.21) (6.22) (6.19) (6.20) J 2 , -charges 1 , 0 ψ generators Y − (2) associated J b gl ) and only shifts and + 6.17 J , , . 2 2 2 ) ) ) (2) Kac-Moody algebra. w w w 1 1 1 b gl − − − . . z z z 3 ( ( ( i n J 3 2 2 ] are actually vectors of a single module.    ˜ (2) 1 2 q = 1 1 1 + 1 1 1 1 φ (1) charge of the decoupled current    2 3 q, 1 ˜ q [  gl  J 2 2 h   M + ∼ − ∼ − ∼ − 1 – 55 – − ) ) ) J were already found previously. For the purpose of 3 w w w q  ( ( ( = exp 1 2 ] and = 1 ,  2  1 ).  ,  (2) (3) (3) 1 2 3 J 0 J J J + M Y ) ) ) (1) field ˜ q 6.17 z z z , b ( ( ( gl 1  (3) (3) (2) 1 2 3 − J J J q [ M The web diagram associated to the First, let us construct the free field realization of the algebra . Fields . This example will serve as a prototype for a more general configuration -charges of the representation match. The anti-fundamental field is more n is precisely the charge predicted by the generating function of the 14 W 1 Figure 14.  is an integer, one gets a constraint (2) from gluing vertex. n b gl 2 Let us now discuss the free field realization of the fundamental and the anti- The generators of the algebra Note that the fusion with gluing fields preserve the constraint ( , 1 , 0 can be realized as Note that and all the fundamental module in the thirdafter direction tensoring that with will the play corresponding the modules role of of the other vertex. The fundamental field our discussion, let us recall the algebra has a free field realization in terms of three free bosons normalized as Let us now discuss theto structure the of figure glued genericwhose modules GW-defects give for rise the to algebra the VOA zero-modes modules induced algebra. from the Gelfand-Tsetlin modulesY of the coefficient The only parameter of the module is thus6.4 the where which is the same constraint as ( JHEP05(2019)159 (6.23) (6.24) (6.28) (6.25) (6.26) (6.27) . i (2) 1 φ 1  − h i exp (2) 1 ˜ φ  4 . 1 q . One finds )]  ∂J + z 3 2 (   M φ (3) 3 of the second vertex as 1 ˜ . J.  φ ∝ +  1 3  2 − . (2) 1 q 21 M ˜ 2 φ J + ) ∂J + ∂ = w J − 1 2 3 = (3) 2 21 1   − and fields with the exponential factor of the φ = exp [ ˜ n J J 2 z i − ¯ 3 1  ¯ ( q   J J ˜ 3 = M 3 M +  can be found from their OPE. One finds = 1  2 1 ,J  1  22 ∝ 4  ¯ (2) 1  21 (2) Kac-Moody algebra are reproduced. q – 56 – φ J 12 ∼ b gl − 1 M , + J ) q 3 ) 1 h  )] 1 3 w q  2 z ( and J  ( ˜ ˜ J J,J φ + ) 3 12 1 =  z 2 J  ( J ˜ 12 J = ( ] = exp 1 J (2) Kac-Moody algebra can be fixed by requiring the correct 4 J 11 b gl 1 , q J  3 = exp [ , q +  2 2 ˜ Generic modules can be now constructed from system of the Wakimoto realization in terms of two free bosons and M J , q 3 1 J q Having identified the fields and the relevant fundamental and the anti- [ . 2 β, γ  3 Let us normalize the free boson n are constrained by the condition at level one. We expect the two choices to correspond to the symmetric and  1 M  (2) in terms of two free bosons and parafermionic fields. Parafermionic fields 4 − ). One finds the following expression for the fundamental field q b gl − J (3) 3  φ and = and 1 ¯  vertex. + q system and we expect to recover our non-symmetric free field realization. Detailed J M 1 The fundamental and the anti-fundamental representations associated to the second , 0 , β, γ 0 where for some integer the first order poleanti-fundamental come field. from All the the OPE OPEs of of the Generic modules. The normalization of generators Note that the OPE of the exponential factors is trivial and both the second order and Glued algebra. fundamental representation of eachThe vertex, Cartan one elements can of now the OPE easily between construct them the and glued with VOA. the fields direction are then a discussion of the relation with Wakimoto realizationY is beyond the scope of this paper. both the asymmetric Wakimoto realizations. Therealization symmetric of Wakimoto realization is acan free be field bosonized and wecan expect bosonize to the find our symmetric free field realization. Similarly, one the left of Note that this asymmetric form ofto the our fundamental and asymmetric the choice anti-fundamental of field the is free-boson related ordering. The symmetric choice would to lead to complicated since it appears at level two (there are two of free bosons of the third type to JHEP05(2019)159 ¯ ].  + 4 M 3 , q  3 (6.29) (6.32) (6.31) + , q and 2 4 ]  q , q 1 3  − M  [ + + M 4 4 q (2) generators on , q ... − 3 b gl [ ]. + , q M 2 ] 43 1 w  , q 1 ]  − 4 − 4 z , q − 3 3 , q n  3 , q ) . 2 ) , q + w instead. We expect corresponding 2 . 2 w , q 4 ] − 3 − 1 q 12 n , q  −  ) 1 z J − = 1. In such a case both  ( [ z w + − n 4 4 + M − q ] 1 ) 1, one gets ! in the constraint above, one gets different ! (2). 4 , q z 3 q − shift the exponent of 3 n [ [ b  gl 22 22 n (( , q λ λ 3 , q + 21 (6.30) 11 M − M n > 2 ) ) ) ) J λ 4 , q 2 – 57 – 21 21 11 w w w w w  2 , q q ∝ ∝ O λ λ λ 1 ]( ]( ]( ]( w  ) ) − , q + + 4 4 4 4

and 3 2 1 w w z 4 2  − +  , q , q , q , q q q ]( ]( 12 3 3 3 3 3 z 4 4 + 1 (2) are parametrized by a triple of complex parameters J  )( +  , q , q , q , q 4 4 gl , q , q 2 2 2 2 2 q + q 3 3 q In this section, we show that the above action of zero modes 4 − , q , q , q , q + [ , q , q q 3 3 3 3 . + 2 2 1     − Z 1 q M [ , q , q q ( + + + + 4 1 1 1 ∈  M q 4 4 4 4 q q [ [ q q q q n are vectors of the same module. For generic values of parameters, ∼ ∼ ∼ − ∼ − − − − − M M n [ [ [ [ ) ) z z + M M M M ] for ( ( ) ) ) ) 1 z z z z 21 12 11 ( ( ( ( n J J λ 22 12 21 11 − J J J J 4 and , q 3 1, the singularity is present in the OPE with 11 λ , q 2 Gelfand-Tsetlin modules for The most interesting situation appears when For each such module, it is simple to compute the action of the n < , q 1 where the Gelfand-Tsetlin module is spanned by vectors with Gelfand-Tsetlin table of the form generate a generic Gelfand-Tsetlin module of We can see that theThe zero modes representation of ofn the zero-mode subalgebra isGelfand-Tsetlin modules. thus spanned by have a simple pole in the OPE with generic modules and one obtains For modules to be a special type of the irregular modules discussed in [ each such vector. Dependingstructure on of the the number modules. For example, for JHEP05(2019)159 i q (6.34) (6.33) by an integral 4 q by an integer as , 11 ! λ , shifts 22 ! λ + 1 21 22 J 11 21 λ . 11 λ λ -algebras associated to their -algebras λ 22 21

λ ) Kac-Moody algebra. W and λ ) W 1 N

↔ 22 ( 12 0) interfaces, i.e. they differ by a 1 ) − , N λ J b gl 21 11 λ − N . λ 3 11 ,  − λ 1 ! −  21 , )( , 3 2 λ 1 22 , 21  q q λ ) and their 4 ! 1 11 λ + act on such vectors as  q 1 − − λ 22 with the lifted Gukov-Witten parameters 21 N − – 58 – 22 λ 21 ( − λ = = = λ 11 b

gl ,J 11 11 21 λ 11 21 22 λ λ ( 12 11 , λ λ λ λ − λ ,J 22 ) can be realized in terms of a web diagram in the fig- In the previous section, we have described the structure 22 = = = = (1 + , λ N ) Kac-Moody algebra and ( ,J 21 ! ! ! ! N gl λ ( 11 22 22 22 22 (2) Kac-Moody algebra. Let us now comment on the structure b J gl λ λ λ λ 11 11 11 11 b gl λ λ λ λ 21 21 21 21 λ λ λ λ The web diagram associated to the

23 12 21 11 22 J J J J . Generators , this corresponds exactly to the shift of parameter 1 Z  ∈ Figure 15. ), one gets n . The lifted GW parameters associated to internal faces must be again equal up 6.31 ) Kac-Moody algebras. The Kac-Moody algebra Comparing parameters There are actually two solutions related by an exchange of 15 N ( 23 b Drinfeld-Sokolov reduction. ure to shifts induced by line operators supported at the (1 6.5 Gelfand-Tsetlin modulesgl for of generic modules for the of generic modules for any Note that fusion of amultiple vector of of the generic moduleexpected. with Note alsocorrespond that to the Gelfand-Tsetlin parameters parameters associated of a to given a row given of face the of Gelfand-Tsetlin table. the toric diagram from ( for each JHEP05(2019)159 with (6.36) (6.37) (6.35) complex i i > j corner pa- N 1 for N ij J . Have the following ) Kac-Moody algebra shifted by any integer. 16 N (1) algebra. ( 11 b gl λ b gl × (2) 3 ]. The corresponding Gelfand- ). Except of the W 5 i N    entries. For example, in the case of 2 . . 33 n λ !    3 +1) + 1 2 33 33 N 3 ( 22 λ λ n 22 N λ λ + 32 32 11 32 11 1 λ (2) Kac-Moody algebra, one should be able to λ λ λ λ – 59 – 11 n 21 b gl λ λ 31 31 + λ λ (1) associated to the diagram b 31 21

gl    λ λ 1) are charges of the finite interface of the given face. ×    m, (2) 3 ( − W . These shifts are generated by the fusion with bi-modules coming The web diagram associated to the 3 , n , where 2 2 , n 1 m The same structure of modules is expected also for similar configurations n + Figure 16. 1  For example the algebra -algebras. -algebra associated to the Drinfeld-Sokolov reduction of the (3), the Gelfand-Tsetlin table will be of the form b The full modules is spanned by such vectors with the parameter rameters in the upper-rightbi-modules face, added to all the the algebra. other parameters canGelfand-Tsetlin be table shifted parameterizing by generic fusion modules with with different ranks ofW gauge groups. The correspondingpossibly algebra with can extra beTsetlin symplectic identified modules with bosons are a as parametrized bynumbers discussed a associated in generalized to Gelfand-Tsetlin [ table each with face with gauge group U( for any integers from line operators at each internal face. W The parameters in eachto line a will given be face. shifted The and full renormalized modules is GW then parameters spanned associated by the vectors gl multiple of In the same waychoose as of the in shifts thegeneric of case the modules of lifted have GW the OPEparametrized parameters such by with that a a the Gelfand-Tsetlin OPEs simple table of pole. of Generic modules are then going to be JHEP05(2019)159 ]. 83 , 5 ]. 44 ]. It would ]. It would 43 30 , , 5 42 ]. 87 ] to the categorification of 97 – 88 ]. It would be interesting to extend the ]. 86 , 1 Gukov-Witten defects were originaly intro- ] based on previous work of [ 85 , 44 – 60 – ], Gukov-Witten defects play the role of generic 66 (1). The specializations of the affine Yangian are 84 , gl ]. In the recenly introduced corner approach to the 4 Recently, partition functions of theories coming from 33 In this paper, we restrict our attention to algebras as- ]. . Considering quotients of the algebra by such null fields 5 i The web algebras studied in this paper are associated to h as proved in [ ]. 3 4 ,N 2 Apart from modules induced from the algebra of zero-modes dis- M ]. If one restricts to the toric three-fold case, one can relate the ,N 1 82 As we have seen in the case of the Ising model, null-fields appear at N – Y There exists a DIM algebra approach [ 76 ) gauge theories. There should exist an analogous story associated to N DT-invariants associated to toricdeformation Calabi-Yau of three-folds. the The affineisomorphic DIM Yangian to of algebra the itselfbe is nice a to q- find aDIM precise algebra relation and between the the gluing construction of of [ intertwining operators of the Ortho-symplectic algebras. sociated to U( ortho-symplectic groups as briefly sketched in [ DIM algebra. Rational levels. rational values of parameters lead to new constraints oncontext. modules. Many It properties would be ofin be the terms interesting non-generic of to case an address existence might this of have issue a line in operators gauge our in theory the explanation bulk. analysis also to non-toric surfaces. Irregular modules. cussed here, one canbe consider more interesting complicated to irregular extendrole modules the in from discussion the [ to wild such ramification modules. of the This Geometric might Langlands also program play [ important in this context. More general VOA[ toric surfaces in toric Calabi-Yaumore threefolds. general as There discussed are for many example hints in that [ the story is much Ramification of geometricduced Langlands. in the context ofgeometric the Langlands ramification program (inclusion in ofgeometric [ punctured Langlands Riemann program surfaces) frommodules of [ the for kernel VOAs. This work sets foundations for the inclusion of the ramification corresponding geometric setup to theGeneric one modules considered in discussed this here paper canthe along be moduli the then lines space identified of of with [ to spiked the AGT instantons equivariant correspondence cohomology with for of aaffine spiked geometric Yangians instantons. origin and This the of issue, cohomological the together Hall VOA with algebra action the will leading relation be with further discussed in [ Finally, let us mention few possible extensionsAGT of the for discussion from spikedbranes this instantons. paper. wrapping various four-cyclesseries in of Calabi-Yau papers four-folds [ have been considered in a 7 Outlook JHEP05(2019)159 - in W j ) U 1 U -matrix 0 1 R U ( 0 α 2) ]. This − 98 N N , 2 to the generators 1)( 31 , j − W 27 ]. Moreover, we do not give N ( 40 − )) 1 -matrix [ Given a free field realization of the U 1 R U ( 1 U ( 00 1 2) U 0 1 2 0 − – 61 – U α 0 2 N 2) α N − 1) 1)( 3 2 − − N 12 We have sketched that Gelfand-Tsetlin modules of N N 1)( ( ( − − ) + ) N 1 2 ( U U 1 1 − U U 0 2 ( ( U 1 2 0 , we have various realizations of maximally degenerate modules. When − − N α N 3 2 2) N N ,N 2 2 The different free field representations corresponding to different ordering of − + + ,N 1 1 2 3 N N ( U U U Y − − + − = = = 2 3 1 -matrix. W W W A Transformation between primary andWe list quadratic first bases few formulasquadratic relating basis. the primary basis generators by the Government ofnomic Canada Development through and the by Department theInnovation and of Province Science. of Innovation, Ontario Science through and the Eco- Ministry of Research, & Gukov, Andrew Libor Linshaw, Kˇriˇzka, Alexey Litvinov,Yan Soibelman, Faroogh Yaping Moosavian, Yang, Ben Ivo Webster, Sachs, Kris Gufang Thielemans Zhao for for his useful Mathematica discussions. packageby OPEdefs. the We thank The DFG research Transregional of Collaborative TPexcellence Research was Origin Centre supported and TRR Structure 33 of andPerimeter the the Institute Universe. DFG for The cluster research Theoretical of of Physics. MR was Research supported at by Perimeter the Institute is supported hand and the theory of quantum integrable models and Yangians onAcknowledgments the other hand. We would like to thank Mikhail Bershtein, Thomas Creutzig, Davide Gaiotto, Sergei R free fields in thesatisfies Miura transform the are Yang-Baxter related equationthe by which quantum an is integrable the models.relation starting between More point the detailed of algebraic exploration structures many of of developments this two-dimensional in should quantum field strengthen theory the on one inserted in aexamples. correlator, It they is all desirablea to give closed-form explore an expression this for equivalent issueIt the result further would descendant as (if be realization in nice of non-vanishing) [ to maximally in find degenerate all an modules. explicit the formula. algebras naturally appearprove in that the the context gluedzero-modes of algebra modules gauge and are explore theories. the indeed relation induced It further. from wouldFree field Gelfand-Tsetlin be realization modules interesting of to ofalgebra degenerate the modules. Gelfand-Tsetlin modules. JHEP05(2019)159 ) (3) 1 ) 1 2 0 U U α 00 1 1) U ( − ) + 2) 2) 2 0 N ) )) 0 α 1 N − 1 ( ))) U 1 U 0 N 1 +10 N 1 U + 2) ,U U ( N − +5 1 ( 2 0 0 1 N U 3) N α ( ( U 0 2 0 )(4 2 3 , ( 00 − 2 2 0 N 1 α U − α 4 U 1) U 0 2 ) − 2 N ( 2 0 α 2 0 − u , − N 2 0 α 1 α 2 0 3) ) )) )(4 + 1) + 3) α U 2 2 N 2 N α 2 0 1) ( ) 2 0 17) 2 0 − U U α 2) + 3 N 2 0 0 N 14 1 α − α 1 ( 1) 17) 2 0 2 α − 7 N − U 1 U − ( α − N ( ( N − − N 2 u − N ( + 1) 1 . Since this choice of normalization 2 2 0 N 2 0 2 17) 11 5 − 5) ( 17) N U + 2) α N N N α N ( 4 0 ) + N N 2 0 ... 5 ) − − 2 0 − − ( 3 (1 − − 4 0 α 2 2) N α − 2 1) α + 17) ( 3 1 − α N U ( N − 1)(2 × N N ,U j N 2 − u − 2 0 N 5 2 0 3 2 0 − 1 N − 17) 5 )(1 2 0 − N U 2 α − U − α α U 2 0 2) 2 0 ) N N ( − α 6 3 − ( 5 N − − − 2 0 N 17) 2 0 N α α (1 5 2 0 N + 1) − 5 2 0 α 5) 5 N α N 2 0 3) − 17) + 7 1)( = α N − N + 6) 2 0 α × − 2 0 1 )(4 α 107) – 62 – 2 0 3 − 2 1) 5 2 N j − N − 10 2 0 α − − u 2 α − − + 4) N α N N 5 17) 3 5 α 0 N N − − N − − − W 2 5 N 5 N N 2 0 2 0 4)( 17) 1)( 3 N N 4 N α 2 0 2 − N 2 4 0 0 N − ( − ( + 6)( 5 α N α ( N 4 − + 1) N 2 N − 0 α α α − − − 1) N 2 0 N 1)(5 3 ( 7 2 0 ) 5 + 1) 3 α 2 4 0 (5 (5 N 2 0 N 2 − N − N 2 0 N α α N N 0 5 − N 4 0 N α N − 6 N N − ) + α α 2 1 (5 N − 2 0 5 N − + 7 2 0 ( N 0 ( − α 2 5 α N 2 2 0 2 0 u ( 0 α 3 − − 4 2 α 2 3)( 0 N N U − α 3)( 24 0 1) 1)(5 1)( 5 ( α N − − 1 α N N + 6)( 3 (5 N N α 4 N − + 1) (5 5 2 0 2 2 0 Nα − U 2 4 2 0 0 1) − − 1)(2 − − 2)( )(16 )(1 2 0 − 2 ( − u 7 N 2) α 2 2 0 0 α N α 3 α N − − − N 2 1 1 0 2 N 0 N − N N N N α α ( 5 2 Nα (5 − u u − 3 α α N 4 N N 5 2 0 1 2 2 0 2)( N N − N 2)(1 2)( N 2)(2 2) 2)( 2)( 2)(5 2)(5 2)( (5 N N α α 4 2 0 3 − 3 − − N + 3) + 3) 2)( 2)( − − 3 2 N − − − − − − − − − α 3)( 2 (5 0 2 N N 1)( − − N N N α N N N 4 2 0 (5 N N N − N N N N N N N ( ( N 3 2 − N N α (7 2 2 1)(1 + + 3 N N 1)( N N N 1)( 1)( (5 3)( 3)( 3)( 3)( 3)( 3)( 3)( 1 2 3 ( N − 3)( 3)( N ( (5 − − u u u − N − − − − − − − − − 2 − − + 2 N − − − + 10 (9 (9 N ( 4 N N N N N N N N N N N N ( 1 2 ( 4 ( ( ( ( ( ( ( ( ( ( × × N = = = U 1 2 3 − − + + + − + + − − = = = = = w w w = 0 0 0 0 0 55 33 44 11 22 4 C C C C C W Acting on the highest weight state, the relation between charges becomes somewhat simpler is rather arbitrary, werelative should normalization of also the specify charges: the values of structure constants that fix the We choose the normalization such that JHEP05(2019)159 . 0 , 1 , 1 (B.2) (B.3) Y 3 u 0 α 1 -charges 2 1 3) u u algebra, it is 2 w 2 u − 3 1 (B.4) ) u W + 1) N + 1) 3 2 2 2 x u 3( N N 29) u 2 0 2 0 3 1 + 1) ) − α u − α 5) 2 3 = 3 (B.1) 4 1 − x h + 1) 0 u − N 6 1 + 7 2 17) N + + − 1) 2 15 0 N 2 1 3 N 2 0 − − α 2 x 2 0 x − N N ) , ψ − α
u 2 2 0 2 0 3 α 6 3 19 N 2 1 17) − x 1 N α α 17) − N x 5 u √ 2 0 2 0 2 + 7 − u 2 which is the condition for 1)( − 17) − − )( − α − x α 2 2 1) )( 17) 3 − = 2 − u 1)(6 we can determine the 3 − 17) N N x − − N N N 17 3 − N is a special case of h − 17) 5 = )( N 2 0 2 0 5 2 0 2 − 3 N − A 2 = 2 + 0 N − α α 2 0 α x N + − − x 5 N ∞ N α − 2 ∞ 0 5 1 2 2 0 5 N α ( c N 5 2 2 0 x 2 0 x 17) 0 α − − 5 N N x 1+ α 2 N − N − − α 2)(5 2 0 − 5 α 2 0 2 0 , h u − − 2 − 3 − 2 W − N α − α α 3 2 − − 2 − 2 0 )( 2 5 N N 5 1 N u N N 1 x 2 0 2 algebra with parameters N α 4 0 , ψ 2 0 2 0 – 63 – x + 6)( + 6) r 5 1 N x 2 0 N − N )( 0 5 α + 2 6 − α 3 2 0 α N α x 2 0 3 α 5 − = 3 and 3 α 6 )( ( 3 2 0 3 − − α N N √ h − 5 α 3 0 (5 W 3)( 3 − α N − 5 = N − 3 u x = N α λ N + 2 0 + − 2 3 ( N 2 0 2 0 2 − 3 4 2 2 0 3 − 2 N α 3 x 1 1) 1)(5 1)(5 1)( 3 α α h 2 α 0 N x + 6)( − 3 x N N (5 5 2 0 N + α (5 4 0 − − − − ( 2 2 2 0 (15 N − α − 2 2 α N x − (5 0 2 0 N truncation of α N N N N + N 2 x u 4 (5 3 α α 4 , h x + ( N 3 0 (5 , + 2 3 2 N 2 u 1 N 2)( 2)( 2) 2)( 2)(6 2)( 2)(5 (5 1 , − 1 2 2 1 2 0 3 x 1 N ( u 1 r x − − − − − − − α ) = Y 2 N x 3 u 4 1 (5 = ( N N N N N N N 1 3 27 − 2 − 3 1 ψ N N h 3)( 3)( 3)( 3)( 3)( 3)( 3)( = = = N W 1 2 3 − − − − − − − + w w w 4 N N N N N N N ( ( ( ( ( ( ( u from − + + − − + − − 0 , = ). From this and the formulas of appendix 1 , 4 1 , the generating function of higher spin charges of a highest weight representation is w 3 Y 4.23 W In of the form (see and as usual This choice in particular means that Since it turns outinstructive that to have ahigher spin look charges. how Let this us start happens with at the level of the generating function of As one can see fromis these unfortunate that expressions, no they closed-form are expression becoming for increasingly the complicated primary and charges it isB known. JHEP05(2019)159 110 Y (B.5) (B.6) (B.7) 2 ) 0 2 33 ) ) 1 C h 0 33 become bound C + ) 3  2 2 x 1 6 x x , − − − j 154512)( 2304)( x 2 and u u ) − 2 − 3 )( )( 2 ) c 1 3 x c x 3 5 x h 2 34 ) ) − C 188 − + 0 2 ( 33 5 34 2 2 1 x u 26404 C ). We can now impose the C ) ) − 0 ( 33 , i.e. ( x ( 3 1 2 5 4 , 34 33 C − remains a zero of the third type. c 4 0 4)( 33 − , W 2 C C 1   1 C ( ( c 3 u − x 2 1 6 2 Y 2 ( h 4 ) 33 c 2304) − 4 C 0 33 = 33 = 2 + 1076 − C C ) 6  3 2 c 3 1 2 2 3 c 4 + 114) ) + 114) 33 0 33 √ + 46628 ) x 64) c c 0 C 33 C 3 + 13)(5 q 4 33 0 0 33 188 + 114) 33 − 5 c = 34 4 33 − C c (7 (7 C c C ( C 1 – 64 – 2 + 2 3 C c − C + 114)( ) 2 x 5 5 34 x 2 34 6 2 4)( 2 4) 4 1) ( 1) 33 c 0 33 + 22) x c + 26) C x C − C − + 13)(85 − c − C − (7 − c 2 4 33 − + 22) 2 2 + 6979 2 − c ) 0 c c c   33 c + 22) + 22)( c + 10) x C c 4 u + 22)(7 u 6 1 4 1) c c C 33 (5 (5 c c (2 (2 1)(5 2 c 2 2 2 2 )( C − + 68) − + 1076 + 114) c c 1   + 114) − + 45 c 6 c 2 3 c x 1)(5 1)(5 c + 13)(7 + 10)( 1 c 1)( ) (259 c 3 2 √ 1)(5 , + 2)( 2 2 + 114) (7 c c c − − − + 10) in the primary basis 1 + 13)(5 + 10) + 10) 2 − x c , c − + c c 1)(7 u c 0 c c c c 1)(7 ( 1)( c 1 − ) which leads to a sextic equation in zeros Y 1)( + 7)(2 012 − x − 1 + 7)(2 (2 − + 10) 1)( 1)(7 Y c c c x + 10) c + 10)( − ( c c c 0 − 55 ( + 4)( ( 2 + 10)(85 − + 2)( + 2)( + 10)(5 − (2 c c c 4.51 + 7)(2 + 7)(2 (2 c c 0 C u  c c 33 + 7)(2 c c c c c c c c c  c 5 C 34 ( ( 90( ( (2 c c 72( 120( 720( 1800( 1800( c 30( 60( C c 12( 12( 36( 0 = ) ======u ( 0 4 5 4 3 3 0 4 55 55 45 35 45 34 44 44 ψ (33) (33) (34) (33) 55 35 45 44 C C C C C C C C C C C C Structure constants of which is exactly whattogether and we behave expect as a from zero the of algebra the firstC type while More details of (and other five permutations of this) with the corresponding generating function truncation relation ( which has a solution (and higher primary charges vanishing since we are in JHEP05(2019)159 , , ]. 2 3 q 2 SPIRE Adv. Theor. q , IN [ 3) 2 3 3 − q 1) 0 33 2 3) C h − 2 2 − (2 1) h 2 ]. 2 (2019) 160 1) − 5 2 1 34 h (2 1) 0 33 arXiv:1009.6032 − 2 C 01 (2 , C hq − h 2 145776) 3 2 4 33 ) h 2 1 SPIRE 6 (2018) 109 − 2 1) q C h 5 2(2 34 IN q ( 3 c 1 3(2 − 1) C q − 11 ][ arXiv:1708.00875 q JHEP − 1 + 2 , 3 870) , q − + 2568) 3 1) ]. h q 2 q 2 2 2 c 00 1) 2) − 2 1 2( 32276 1) q − 3) h 870)( 1) q c − − 2 JHEP 2) − (2 0 − (33) − − 33 55 − 1) h − 2 2 , ]. h SPIRE 2 2 2 00 C C c 2 h h − 2 c 2( 2 2 − h q 0 h IN 33 h 2 ) + 2412 h (2 + 533 2 C (33) (2 ][ h 6) 1)( 55 3 + 2 h 4 (2 33 h 2 ( 2 870) h c ( 2 C SPIRE c 3 1)(2 h − C − + 533 q 6 5 − IN hep-th/0604151 34 2 2 2 870) 2 + + + 3222 − 2 [ c – 65 – q h c h C ][ 3 2 2 3 1 ( − 1) c 2 2 q q 4 + 914 q 33 (5 h c 2 2 1 2 ( 3 − C − 1) 1) q q 1) deformation, branes, integrability and Liouville theory c + + 114)( 2 2 3 − + 533 1) 1) − − c − 3 q Ω h + 114)(29 2 2 (2007) 1 1 q 3 2 2 2 c c − − 2 q q + 533 (2 + 1146 Webs of W-algebras h h h 1 q ]. ]. 2 2 ), which permits any use, distribution and reproduction in + 114)(29 2 4 1)(7 + 114) 3) 3) Vertex algebras for S-duality The Electric-magnetic duality and the geometric Langlands program (2 (2 c h h c 4( Vertex algebras at the corner (29 arXiv:1106.4789 c 2 4) c ( ( Knot invariants from four-dimensional gauge theory 1)(7 2 − [ − − 1) + 13)(35 1) 1) − − c 2 2 − + + c (1) charges in the free field representation − 2 1)(7 2 SPIRE SPIRE − 2 h h − 1)(7 u c 2 2 2 h q 2 arXiv:1002.0888 2 IN IN 2 1 (2 (2 − + 10) h (2 [ − 1) 1) h 1) h 2 2 q c ][ ][ c c c + 10)(29 3 3 + 7)( + 10)(49 2 1 (2 (2 + 7)(2 (2 1) − 1) − 1) 1)(3 − CC-BY 4.0 q q c h (2 c c 2( (2 h 2 2 c 2 2 c (2012) 935 2 c ( − − − − h h 8( h h h This article is distributed under the terms of the Creative Commons c 3 2 2 2 2 16 600( 300( (2 h h h h − + ( 3(2 ( ( 3(2 ( A new look at the path integral of quantum mechanics (2010) 092 = = + − − + − + ]. 09 = (44) (35) 55 55 3 C C w SPIRE arXiv:1711.06888 IN arXiv:1703.00982 [ Commun. Num. Theor. Phys. [ JHEP Math. Phys. [ A. Kapustin and E. Witten, E. Witten, D. Gaiotto and E. Witten, T. Creutzig and D. Gaiotto, T. and Proch´azka M. Rapˇc´ak, D. Gaiotto and M. Rapˇc´ak, N. Nekrasov and E. Witten, [6] [7] [3] [4] [5] [1] [2] References Open Access. Attribution License ( any medium, provided the original author(s) and source are credited. Spin 3 charge in terms of JHEP05(2019)159 ] , ] , B (2012) ]. A 10 (2015) B 236 07 -algebras SPIRE 340 W IN ]. ), JHEP Nucl. Phys. ][ super Yang-Mills arXiv:1411.7697 , , arXiv:1211.1287 ]. ∞ [ hep-th/9410013 , ]. . [ Phys. Lett. = 4 super Yang-Mills SPIRE , ]. ]. (1 + IN N ∞ SPIRE ][ W = 4 SPIRE Int. J. Mod. Phys. IN W , IN ][ N SPIRE (2015) 116 SPIRE ][ (1995) 94 ]. IN IN 09 ][ ][ arXiv:0804.2907 ]. algebras [ Commun. Math. Phys. SPIRE B 343 , IN arXiv:1202.2756 algebras with infinitely generated W [ JHEP , five-branes, five-dimensional field theories , algebra and second Hamiltonian structure SPIRE ) W ) IN Instanton moduli spaces and arXiv:0807.3720 [ [ ∞ p, q hep-th/9312049 ( ( (2010) 097 [ hep-th/9405116 ]. -algebra and vertex algebras of type and the Racah-Wigner algebra [ W Phys. Lett. ∞ 06 hep-th/9710116 , ∞ arXiv:0804.2902 – 66 – [ W ]. [ ]. W The complete structure of A new higher spin algebra and the lone star product A class of SPIRE (1992) 713 Triality in minimal model holography Webs of IN (2009) 721 [ JHEP (1994) 409 arXiv:1101.3216 SPIRE , (1996) 113 13 ]. ]. SPIRE , Cherednik algebras, W algebras and the equivariant IN IN Branes and supergroups in the quadratic basis ][ Quantum groups and quantum cohomology (1998) 002 B 373 [ (2009) 789 175 ]. S-duality of boundary conditions in Janus configurations, chern-simons couplings, and the theta-angle Supersymmetric boundary conditions in Universal Drinfeld-Sokolov reduction and matrices of complex ∞ B 420 SPIRE SPIRE 01 ]. W IN IN ]. 135 Coset realization of unifying ][ ][ Representation theory of the vertex algebra Nonlinearly deformed SPIRE ]. SPIRE JHEP IN (1990) 401 [ SPIRE IN , [ Nucl. Phys. algebras of negative rank IN arXiv:1710.02275 , [ Universal two-parameter Nucl. Phys. Exploring ), SPIRE , W hep-th/9406203 IN [ B 242 Fivebranes and knots [ super Yang-Mills theory ]. ]. ]. Adv. Theor. Math. Phys. J. Statist. Phys. ,...,N , , = 4 3 Commun. Math. Phys. arXiv:1205.2472 arXiv:1410.1175 (1990) 191 , , [ [ N SPIRE SPIRE SPIRE (2 IN IN IN theory cohomology of the moduli space of instantons on A2 [ arXiv:1406.2381 [ W [ (1995) 2367 127 classical limit size Phys. Lett. hep-th/9512150 of KP hierarchy 339 (1990) 173 in theory 699 and grid diagrams V. Mikhaylov and E. Witten, E. Witten, D. Maulik and A. Okounkov, A. Braverman, M. Finkelberg and H. Nakajima, T. Proch´azka, A.R. Linshaw, O. Schiffmann and E. Vasserot, R. Blumenhagen et al., M.R. Gaberdiel and R. Gopakumar, J. de Boer, L. Feher and A. Honecker, B. Khesin and F. Malikov, K. Hornfeck, V. Kac and A. Radul, F. Yu and Y.-S. Wu, C.N. Pope, L.J. Romans and X. Shen, C.N. Pope, L.J. Romans and X. Shen, C.N. Pope, L.J. Romans and X. Shen, D. Gaiotto and E. Witten, D. Gaiotto and E. Witten, O. Aharony, A. Hanany and B. Kol, D. Gaiotto and E. Witten, [9] [8] [27] [28] [24] [25] [26] [22] [23] [19] [20] [21] [17] [18] [14] [15] [16] [12] [13] [10] [11] JHEP05(2019)159 , , ]. 223 JHEP (2015) , (1988) 255. SPIRE IN , Erratum ibid. 2015 22 (2016) 077 Commun. [ ][ ]. 2, 10 − Phys. Rept. ]. ]. -cofiniteness of , 2 = c PTEP c , SPIRE (2017) 583 Commun. Math. Phys. JHEP IN (1989) 215 SPIRE SPIRE , WZW model , [ IN IN 304 + 3 [ SCFT ][ H B 317 = 1 algebras realization of minimal model CFT: Funct. Anal. Appl. arXiv:1509.01000 -algebra and , c algebra with [ 2 . N . c W (1984) 312 3 SH ]. Higher spins and Yangian symmetries D (1988) 507 W Adv. Math. 2 ]. ]. , Cohomological Hall algebras, vertex Irreducible generic Gelfand-Tsetlin modules A 3 B 240 SPIRE Nucl. Phys. (2015) 168 Classical , SPIRE IN arXiv:1203.1052 ]. [ Plane partitions with a “pit”: generating [ IN 11 revisited ][ Zhu’s algebra, 1 – 67 – algebras commuting with a set of screenings ]. gl SPIRE ]. arXiv:1207.3909 ]. symmetry in conformal field theory arXiv:1512.08779 IN , W JHEP , Nucl. Phys. ][ , , Solitons: differential equations, symmetries and infinite W The models of two-dimensional conformal quantum field Irregular singularities in the (2012) 050 Conformal algebra and multipoint correlation functions in arXiv:1409.8413 SPIRE On SPIRE [ SPIRE IN ]. IN 12 IN ]. ][ Irregular singularities in Liouville theory and Argyres-Douglas Int. J. Mod. Phys. ]. [ ][ ]. Yangian associated with , ]. Gauge theory, ramification, and the geometric Langlands program ]. arXiv:1810.10402 SPIRE , JHEP SPIRE arXiv:1702.05100 IN SPIRE , SPIRE [ (2015) 018 IN , Cambridge University Press, Camrbidge U.K. (2000). SPIRE IN ][ IN [ ][ SPIRE IN (1998) 113 symmetry [ 11 -symmetry, topological vertex and affine Yangian IN The affine Yangian of ) arXiv:1609.06271 [ W n [ Quantization of the Gel’fand-Dikii brackets ( Classification of irreducible modules of 195 hep-th/9210010 Z BRST approach to minimal models [ (2017) 152 SIGMA arXiv:1504.04150 (1989) 548] [ [ 04 ), n (1991) 543 ( (2016) 138 gl arXiv:1404.5240 arXiv:1512.07178 triality, poset and Burge condition (1993) 183 dimensional algebras algebras and instantons 140 type gauge theories, I arXiv:1301.5342 two-dimensional statistical models B 324 of Math. Phys. parafermion vertex operator algebras functions and representation theory 11 theory with JHEP hep-th/0612073 [ [ 093A01 P. Bouwknegt and K. Schoutens, T. Miwa, M. Jimbo and E. Date, S. Lukyanov, P. Di Francesco, C. Itzykson and J.B. Zuber, M. Fukuda, S. Nakamura, Y. Matsuo and R.-D. Zhu, D. Gaiotto and J. Teschner, D. Gaiotto and J. Lamy-Poirier, M. Rapcak, Y. Soibelman, Y. Yang and G. Zhao, G. Felder, V. Futorny, D. Grantcharov, and L. E. Ramirez, W.Q. Wang, T. Arakawa, C.H. Lam and H. Yamada, V.S. Dotsenko and V.A. Fateev, A. Litvinov and L. Spodyneiko, V.A. Fateev and S.L. Lukyanov, M.R. Gaberdiel, R. Gopakumar, W. Li and C. Peng, S. Gukov and E. Witten, M. Bershtein, B. L. Feigin, and G. Merzon, T. Proch´azka, R.-D. Zhu and Y. Matsuo, A. Tsymbaliuk, [48] [49] [45] [46] [47] [42] [43] [44] [40] [41] [37] [38] [39] [35] [36] [32] [33] [34] [30] [31] [29] JHEP05(2019)159 ] 9 , ]. 322 J. , 6), , JHEP , SPIRE Int. J. , 345 B 222 , 5 , IN , ]. ]. 4 J. Alg. ][ , , Commun. (3 , Lett. Math. SPIRE , SPIRE hep-th/9503043 IN and IN [ [ Phys. Lett. ][ Proc. Nat. Acad. Sci. 5) , , , J. Amer. Math. Soc. 4 , ]. , , Springer, Germany (3 ]. hep-th/9308153 algebra (1995) 347 [ SPIRE Commun. Math. Phys. IN Orbifolds and cosets of minimal (3) , ]. SPIRE ][ W IN ]. [ B 449 arXiv:1903.10372 ]. arXiv:1610.09348 SPIRE (1995) 447. , (1993) 429 [ IN SPIRE Excited states of Calogero-Sutherland model ]. ][ IN The supersymmetric affine Yangian 174 algebra with negative integral central charge 157 SPIRE ][ IN ∞ ]. Conformal field theory ][ Nucl. Phys. Vertex operator algebras and the Monster 1+ SPIRE , – 68 – (2017) 339 IN W arXiv:1111.2803 [ SPIRE ][ W-algebras related to parafermion algebras ]. IN arXiv:1806.02470 355 ][ algebra ], 4 ) Cosets of affine vertex algebras inside larger structures hep-th/9212104 Poisson-Lie group of pseudodifferential symbols M N SPIRE [ [ ( Singular vectors of the virasoro algebra in terms of jack hep-th/9312088 IN (2013) 269 [ [ W The associative algebras of conformal field theory arXiv:1407.8512 VOA ]. Commun. Math. Phys. Commun. Math. Phys. [ 319 Quasifinite highest weight modules over the Lie algebra of differential , , hep-th/9811239 Instanton moduli spaces and bases in coset conformal field theory [ ]. (1993) 237 SPIRE . (1995) 475 (1991) 787 algebras with set of primary fields of dimensions Vertex algebras, Kac-Moody algebras and the monster IN A Mathematica package for computing operator product expansions arXiv:1711.07449 [ The structure of the Kac-Wang-Yan algebra Invariant theory and the Instanton R-matrix and W-symmetry Determinant formula and unitarity for the [ SPIRE W (2019) 396 171 Commun. Math. Phys. B 407 C 2 IN , [ (1999) 379 ]. Modular invariance of characters of vertex operator algebras 517 47 (1986) 3068 (2018) 200 SPIRE -algebras IN 83 Academic Press, U.S.A. (1988). Commun. Math. Phys. [ 05 (1989) 226 symmetric polynomials and singular vectors of the (2009) 2366. Mod. Phys. W operators on the circle Nucl. Phys. Algebra (2016) 545. (1996) 237. (2012). arXiv:0811.4067 Math. Phys. Phys. I. Frenkel, J. Lepowsky and A. Meurman, A.A. Belavin et al., B. Feigin and S. Gukov, M.R. Gaberdiel, W. Li, C. Peng and H. Zhang, R.E. Borcherds, K. Mimachi and Y. Yamada, H. Awata, Y. Matsuo, S. Odake and J. Shiraishi, C. Dong, C.H. Lam and H. Yamada, K. Thielemans, S. Mizoguchi, V. Kac and A. Radul, K. Hornfeck, T. Creutzig and A.R. Linshaw, A.R. Linshaw, T. Arakawa, T. Creutzig, K. Kawasetsu and A.R. Linshaw, P. Francesco, P. Mathieu, and D. S´en´echal, A.R. Linshaw, T. Proch´azka, D. Brungs and W. Nahm, Y. Zhu, B. Khesin and I. Zakharevich, [69] [70] [66] [67] [68] [64] [65] [61] [62] [63] [59] [60] [56] [57] [58] [54] [55] [51] [52] [53] [50] JHEP05(2019)159 ] , , JHEP n , , gl , ]. Lett. , (1998) 91 Nucl. Phys. ]. 2 , SPIRE IN ]. -characters ][ SPIRE qq arXiv:1310.7281 [ IN affine Yangian ][ SPIRE arXiv:1106.4088 Commun. Math. Phys. , IN = 2 [ ]. (2007) 123520. arXiv:1306.4320 N , 1(1), 48 (2016) 29 A ]. SPIRE ]. IN ]. 106 ]. ][ arXiv:1701.00189 SPIRE Adv. Theor. Math. Phys. [ SPIRE , IN SPIRE [ ]. ]. IN SPIRE arXiv:1608.07272 IN ]. ][ [ IN ]. arXiv:0710.0631 ][ Branching rules for quantum toroidal J. Math. Phys. ][ , -models and defects in gauge theory , ]. SPIRE σ SPIRE IN IN (2018) 863 SPIRE Vertex algebras and 4-manifold invariants [ SPIRE [ Coupling of two conformal field theories and – 69 – IN IN Lett. Math. Phys. [ ][ algebra SPIRE (2017) 503 358 Quantum algebraic approach to refined topological , Twin-plane-partitions and arXiv:1512.05388 IN Fivebranes and 4-manifolds ∞ [ 21 ][ Spiked instantons from intersecting D-branes 1+ arXiv:1712.08128 W (1997) 181 Affine Kac-Moody algebras and semiinfinite flag manifolds , arXiv:1112.6074 arXiv:1711.11011 ]. ]. arXiv:1309.2147 [ (1990) 161 [ ]. Quantum Langlands dualities of boundary conditions, D-modules 41 [ Generalization and deformation of Drinfeld quantum affine Branes and toric geometry (2016) 181 ]. 128 SPIRE SPIRE arXiv:1805.00203 03 SPIRE , IN IN [ [ IN arXiv:1611.03478 SPIRE [ (2012) 041 analog of the (2019) 579 ][ Notes on Ding-Iohara algebra and AGT conjecture IN (2016) 229 ) arXiv:1807.11304 [ Fock representations of the affine Lie algebra Commun. Math. Phys. JHEP 03 [ Magnificent four BPS/CFT correspondence III: gauge origami partition function and BPS/CFT correspondence IV: BPS/CFT correspondence V: BPZ and KZ equations from BPS/CFT correspondence: non-perturbative Dyson-Schwinger equations and BPS/CFT correspondence II: Instantons at crossroads, moduli and compactness , , 109 q, γ 300 ( Gauge theory and wild ramification Lett. Math. Phys. Adv. Theor. Math. Phys. A ]. ]. ]. , , (2017) 257 JHEP , (1986) 605 (2018) 192 SPIRE SPIRE SPIRE -characters IN IN hep-th/9711013 IN [ vertex [ algebras and conformal blocks arXiv:1705.01645 arXiv:1711.11582 [ B 914 qq Math. Phys. qq-characters theorem Adv. Math. 104 Commun. Math. Phys. Nakajima-Yoshioka blow-up equations [ 11 K. Miki, H. Awata et al., H. Awata, B. Feigin and J. Shiraishi, A. Gadde, S. Gukov and P. Putrov, E. Witten, J.-t. Ding and K. Iohara, E. Frenkel and D. Gaiotto, M. Dedushenko, S. Gukov and P. Putrov, N. Nekrasov, N. Nekrasov, N.C. Leung and C. Vafa, N. Nekrasov, N. Nekrasov, N. Nekrasov, N. Nekrasov, N. Nekrasov and N.S. Prabhakar, M. Wakimoto, B.L. Feigin and E.V. Frenkel, M.R. Gaberdiel, W. Li and C. Peng, B. Feigin, M. Jimbo, T. Miwa and E. Mukhin, M. Bershtein, B. Feigin and A. Litvinov, [89] [90] [91] [86] [87] [88] [84] [85] [81] [82] [83] [79] [80] [76] [77] [78] [74] [75] [72] [73] [71] JHEP05(2019)159 07 d , 5 JHEP , ]. -matrix and ]. R SPIRE IN -character for SPIRE ][ qq (2016) 703 IN ]. ]. instanton partition ][ 345 = 1 SPIRE SPIRE ]. N IN IN d ][ ][ 5 algebra and SPIRE ∞ IN 1+ arXiv:1703.10759 ][ W [ arXiv:1606.08020 [ Ding-Iohara-Miki symmetry of network matrix ]. Commun. Math. Phys. – 70 – , (2017) 034 arXiv:1603.05467 arXiv:1608.05351 [ [ 11 SPIRE IN (2016) 123B05 arXiv:1611.07304 ][ [ ]. JHEP , (2016) 196 -webs of DIM representations, (2016) 047 ) 2016 10 p, q SPIRE Coherent states in quantum IN (2017) 358 B 762 ][ PTEP , Toric Calabi-Yau threefolds as quantum integrable systems. Anomaly in RTT relation for DIM algebra and network matrix models Explicit examples of DIM constraints for network matrix models JHEP , On the instanton R-matrix B 918 arXiv:1604.08366 [ Phys. Lett. , relations arXiv:1302.0799 functions and qq-characters [ Nucl. Phys. Super Yang-Mills models (2016) 103 RT T A. Smirnov, H. Awata et al., J.-E. Bourgine et al., J.-E. Bourgine et al., ( H. Awata et al., H. Awata et al., A. Mironov, A. Morozov and Y. Zenkevich, [98] [95] [96] [97] [93] [94] [92]