JHEP11(2018)188 Springer September 7, 2018 November 19, 2018 November 29, 2018 : : : Received Accepted Published Published for SISSA by https://doi.org/10.1007/JHEP11(2018)188 -duality for four-dimensional gauge theo- S and principal nilpotent element the character agrees precisely A . 3 1809.01725 The Authors. Conformal and W Symmetry, Conformal Field Theory, Supersymmetric c

Families of vertex algebras associated to nilpotent elements of simply-laced , [email protected] Edmonton, Alberta T6G 2G1, Canada Research Institute for Mathematical Sciences,Kyoto Kyoto 606-8502, University, Japan E-mail: Department of Mathematical and Statistical Sciences, University of Alberta, Open Access Article funded by SCOAP Keywords: Gauge Theory ArXiv ePrint: with the Schur-Index formulasingularities. for For corresponding other Argyres-Douglas nilpotentIV theories elements Argyres-Douglas with they theories. irregular arevertex identified Further, operator there with algebras Schur-indices is that ofdiscussed a nicely by type conformal matches Buican the embedding and RG-flow Nishinaka. pattern of of Argyres-Douglas these theories as Abstract: Lie algebras are constructed.of These Feigin and algebras Tipunin areof and close vertex they cousins algebras are of appearing also logarithmicries. in obtained W-algebras the In as context the modifications of case of of semiclassical type limits Thomas Creutzig Logarithmic W-algebras and Argyres-Douglasat theories higher rank JHEP11(2018)188 – ]. 6 15 – 9 ]. 5 -duality for four-dimensional Nonetheless, one construction S 1 10 ]. For some further recent progress = 2 supersymmetric field theories [ 18 N 5 – ) 13 1 16 − 1) − – 1 – n ( N ,A 1 − N ]. Interestingly also modularity and Verlinde-like formulae ] is that interesting quantities of higher dimensional gauge A 2 12 , ]. There is no apparent relation known to me between these su- 8 11 ]. Moreover the construction makes the predicted automorphism 4 1 , 3 manifest and also gives a nice explanation of RG-flow via conformal -duality picture. 1 S N -algebra S Q ) × n ( Z ]. 2 B / 26 Z – 19 Argyres-Douglas theories are four-dimensional The slogan initiated in [ Enhancement of supersymmetry can be explained via RG-flow as in [ = 4 supersymmetric GL-twisted gauge theories. There the two-dimensional intersection 1 ] and they are very interesting from the vertex operator algebra perspective as their Schur Moreover indices of linevertex and operator algebra surface [ defectshave are nice related interpretations to insee characters the [ of gauge modules theory of [ the of this work isnaturally a in modification this of a semi-classical limit of8 vertex operator algebras arising indices are believed to coincide with characters of certain vertex operator algebras [ of three-dimensional topological boundaryvertex operator conditions algebras, parameterized is by describeddefects the gauge ending by coupling. on certain Moreover the categories familiesoperator boundary of algebra, of line conditions see are [ categoriespersymmetric of gauge theories modules and of Argyres-Douglas theories. the corner vertex point of view of logarithmic conformal field theories. theories are sometimes describedtations by of vertex this operator picture algebras. andN the There one are that I various am manifes- familiar with is The main pointcoincide of this with work Schur indices isBuican and to of Nishinaka construct corresponding [ vertexgroup Argyres-Douglas operator theories algebras asembeddings. whose computed At characters by the same time these vertex operator algebras are also interesting from the 1 Introduction 3 The 4 Another construction and more5 vertex algebras Conformal embeddings versus RG flow Contents 1 Introduction 2 Schur-indices of type ( JHEP11(2018)188 ], 34 – (1.1) 32 [ 2 . sl µ x 2 µ 2 2 n ]. − q 11 ) vertex operator alge- µ ( λ βγ m Argyres-Douglas theories λ ]. Let me now state a few + 1 factors through the Verlinde 38 Q A X 1 ∈ µ S ) ρ +2 λ,λ ( 2 n q ]. The first statement is a rewriting of and it reads 1 )) ρ 2 + λ ( – 2 – ] for an introduction, and it is good that they ]. These correspond to Argyres-Douglas theories ρ,w 27 ( ], the fractional level WZW theories of -BPS line defects was studied and a main claim is 37 − at admissible level or to the 2 1 q 31 2 ) – sl w ] and some progress is currently made on higher rank ( 28  35 W ∈ X w )-algebras [ + p P ( ∈ X B λ ]. It turns out that indices of surface defects can be identified with 2 ` ] observes for some eamples of type )-algebras [ ρ 2 34 ] denote relaxed-highest weight modules bilateral modules. ) p + – ) and it might serve as a nice example for further investigation of the 16 ( ) Argyres-Douglas theories [ q 3 ` 11 ( 1 12 B ]. If we restrict to 32 − η − q p 2 36 1) − ,A ] the operator algebra of n Buican and Nishinaka give a nice compact formula for Schur-indices of ) = -invariant vacua of the theory compactified on ( 1 r 17 N A ]. All these examples have appearances in higher dimensional super conformal N, n 36 ,A ( functions [ characters of relaxed-higest weight modules in certain examples [ algebra of the semisimplification of the category of modules of thenegligible chiral morphism, are algebra. the relaxed-highest weightsee modules e.g. of [ affine vertex algebras, bra, then characters of these modules are formal distributions times certain modular vertex operator algebra. that the homomorphism fromU(1) this operator algebra to the algebra of functions on that a limitular of data wild of Hitchin the characters semisimplification can of be the expressed module in category terms of of the the corresponding mod- 1 I Prototypical examples of negligible modules, i.e. modules on which the identity is a In [ The paper [ Personally, I find it very interesting that the vertex operator algebras appearing in this − The authors of [ 3. 2. 1. 2 N A resulting expression is the outcome of section Results. ( this formula that makes comparison with vertex operator algebra characters evident. The examples of the connectiontheory between data. representation theory data and super conformal field gauge theory. Note thatequivalent these simple representation objects theories and usually inhas have order to uncountable to look many go at in- toidentified the a semisimplification, with conjectural that the modular is tensor zeroalgebra the category context category morphisms. one on where negligible this For morphisms please details are see and work conjectures with Terry in Gannon the [ vertex operator the logarithmic cases [ field theories. Especially,ory I of would the like logarithmic of to type point ( out therelation work between representation on theoretic the data representation of the- the vertex algebra and properties of the gauge theory context arebut often of reducible logarithmic representations. type, Therearithmic that are is vertex various they operator difficult allow algebras,allow questions for for see indecomposable in gauge [ the theory context interpretations.algebras of So are log- far the the triplet best algebras studied [ logarithmic vertex operator JHEP11(2018)188 N S × ]. We Z 41 2 ]) of the / 40 Z -algebras are 1 is a positive Q ) the Weyl group n ( n > ] super conformal W B the root lattice and 42 -graded vector spaces Q T ) (1.2) . The idea is rather simple q, x 5 )( in the irreducible highest-weight 1 we get more new vertex operator µ − , f N, n prinipal nilpotent this is 2 N -modules as ( f 1) A I ) the quantum Hamiltonian reduction G c 24 − = , f is the Weyl vector, − g n and the compact Lie group of the simply-laced ( q Q ( ρ 2 , n N W G ρ 2 w sl its set of dominant weights, ) = – 3 – . − + = ]. g ` q, x -graded vector spaces are equivalent to categories 1 P g times the automorphism group of the corresponding ]( of T , 1 -algebra g = 2 − λ action to view W Q c N ) prinipal nilpotent we then recover the vertex operator A and n G ) ( f n B G ( )-modules with B , f ch[ . For g ( N is the rank of at a certain level and ]. The new vertex operator algebras are described as intersections sl W is the root lattice of a simply-laced Lie algebra and ` g 43 ⊗ = Q of highest-weight G g λ a more general but conjectural construction is given. The idea is to first and a family of vertex operator algebras that are denoted ρ ) and ] and it is a modification of certain logarithmic W-algebras of Feigin and 4 3 f . The idea is now that one can replace Rep(G) by a braided equivalent N g 35 ]. The resulting vertex operator algebras are certain objects in a completion ( ]. The main result is then that for 3 su 39 ) the parity of the Weyl reflection the maximal torus of = w ( ) is the dimension of the weight space of weight T g  Another nice outcome is that our explixit construction allows to study conformal em- One nice outcome of this construction is that all vertex operator algebras have as In section In section µ . Essentially we forget the , ( g g λ beddings of resulting vertexand operator is algebras, taken from seeof [ section kernels of screeningis charge only on a considering free the field intersection algebra. of all Removing one screening such except screening, for that one gives a bigger vertex Hamiltonian reduction. manifest symmetry the Weyl group W-algebra. For example in theas case predicted of by Buican and Nishinaka [ of Fock modules ofthen the Heisenberg restrict vertex to operatoralgebra algebra, of see our e.g. previous sectionalgebras. construction 2 Their and of characters for [ canare general be labelled identified with by typeindices Young IV change Argyres-Douglas tableaux according theories and which to as the Euler-Poincar´eprinciple explained for in the section corresponding 4 quantum of [ associated to Lie algebra tensor category of modulesof for the Heisenbergwith vertex operator algebra of rank the rank expected chiral algebra for the Argyres Douglas theory. take a semi-classical limitmyself of in vertex operator [ algebrasof conjectured a by Davide category Gaiotto of and with the central charge of the and we check that this precisely matches the proposed central charge (see [ integer larger than one.and Wood The [ constructionTipunin is [ a generalization of joint work with Ridout of m representation constructed. Here Here JHEP11(2018)188 ] / ∈ ) 46 , ]. In 45 13 n, m , see [ 2 ] and Argyres- 3 sl examples corre- whose character = 1 4 F A g ) with gcd( 1 − m , ) ,A n ( 1 0 − 0 Φ n D A ] and one might wonder if the W 3 ⊗ ν − N Q ) -algebras and the two n ( W ]. There are a few natural questions that are – 4 – ⊗ B 1 ν Q of this are given in Theorem 10.5 and 6 point (2) ) n 1 ( A (5) ⊂ B B N Q ) n and ( 1 B A , eq. (6.7)]. 1 (4) are isomorphic to certain B ] and so here we see that RG-flow on the level of vertex operator algebras 1 1 − are unknown. The construction of this paper can however (at least conjec- N The present work gives a vertex operator algebra realization or interpretation ]. Quasi-lisse means that the associated variety (the maximal spectrum of the } A ) ]. 44 n ( 47 B -quotient) is symplectic with only finitely many symplectic leaves. Presumably the , n, m 2 1 of chiral algebras of these unexplored Argyres-Douglas theories. of [ { tural) be generalized. Ilevel-rank plan duality to in report the on future this and generalization hope as that well it as will its shed relation light to on the understanding coincides with [ be rather rich andof deserve further study. Thesponding expectation to is that Heisenberg cosets Douglas theories? and even superalgebras is possible,be though possible probably to difficult. construct a For example vertex it operator should algebra associated to for discussions of relations to Higgs branches. classical limits of vertexfour-dimensional operator algebras supersymmetric appearing gauge inconstruction theories the has a context [ physics explanation, of i.e. S-duality is for there a relation between [ property holds for characters ofbras ordinary [ modules of quasi-lisseC vertex operator alge- new vertex operator algebras ofindices this would work then are be quasi-lisse. solutions to Characters certain and modular thus differential Schur equations, see [ algebras found here converge into this meromorphic domain Jacobi and forms? canthis The case be answer characters meromorphically is also continued yes haveif for a the nice the product more case form general of and case so has a a second nice question product would form be as well. Schur indices of Argyres-Douglas theories of type ( Level-rank type dualities for the vertex operator algebras appearing here seem to From the vertex operator algebra perspective generalizations to non simply-laced The vertex operator algebras here have been constructed as modifications of semi- Setting the Jacobi variable to one, one expects modular forms as characters. This Is there a domain for the Jacobi variables so that the characters of the vertex operator 6. 5. 4. 3. 2. 1. Questions. of the findings ofinteresting Buican from physics and and Nishinaka mathematics [ perspective. with the last factorFock some modules in extension a of rank one ain isotropic section rank lattice. 4.2 two of This Heisenberg [ matchesis nicely vertex described the operator discussion by algebra of conformal RG-flow by embeddings. operator algebra. This results in the embedding JHEP11(2018)188 . in +1 i µ  − i  -deformed q = 1 is the rank i α − . ) . ), so we prefer to x N . . Namely consider (  N 1 µ 1  ( ). − − 2 µ x j su N N x N := h 2 ( i A µ µ h 2 n su i,j − ) be the multiplicity of F q of , . . . , q µ µ 2 n ) 3 ( . x µ − λ − 2 P ) , x ( λ q N µ µ λ m  ( x − ) is the eigenvalue of the quadratic m λ 2 R µ λ R , q m tr ) ( 2 n 1 + ) 2 1 λ − 2 − Q R + C − X ( N q ∈ 2 ) Q µ . − 1) X ∈ µ ) q λ nC − ( µ ρ  q λ n ( +2 N R m – 5 – N ) = − χ λ λ,λ 1 x k ( , then ) + ( 2 − n k λ λ Q ,A the Weyl vector, i.e. one half the sum of all positive q q X N 1 ∈ 1 ` ) µ ρ − − ) − k ` and then the simple positive roots are 24 q q N ∞ N = ( q ) (1 ) = tr η . So that we have rewritten i,j q −  A x δ ; =1 ( x ∞ Q q Y k j (1 λ ( 2 with 1 h = ) = χ i / − ) =1 h j Q k ρ ) =  N i i,j q, x  ( F ) 2 + 2 n q, x n is in the Cartan subalgebra. Let ( ) = ( R of highest-weight − ) ˜ q f x q n ( ( R R λ, λ  ˜ R ) Argyres-Douglas theories. I will rewrite them in a form that makes f 1 R C − tr 1) − n ] chose a standard realization of the simple roots of ( denotes an irreducible highest-weight representation of with product 1 denotes the Gram matrix of the weight lattice N R N i,j ,A the highest-weight representation of highest-weight 1 F ], Buican and Nishinaka found a nice compact formula for Schur-indices of 1 The second ingrededient is the coefficient − R N , . . . ,  of it. 1 A and here [  The last term needs explanation. The character is defined to be for the representation The root lattice is denoted by Here ` denote it by the correspondingCasimir, which highest-weight is ( roots. The element posal for simpleYang-Mills theory, wave functions for certain irregular punctures in SU(N) 2 Schur-indices of typeIn ( [ ( the comparison to vertex operator algebra characters obvious. They start with a pro- Most of all Inaka. appreciate I very am much also the gratefuland many to I instructive Matthew would discussions Buican like with for toissues. his Takahiro thank Nishi- interest Davide I and Gaiotto am his and supported comments Boris to by Feigin this NSERC for work Discovery many grant discussions number on RES0020460. related Acknowledgments JHEP11(2018)188 . (2.2) (2.1) i < j for j  1 is the rank. − − i  N  = 1 ρ . − 2 ` k − , N , i.e. minus for an odd ) q ) − 1 w N = − j ) and q, x k x ( i . q λ ) j x n , . . . , q  πiτρ ( R − 3 2 ˜ − f −  2 − ) )) (1 N e q ρ . 1 (1 ( − ) + − + 1 =1 2 = R λ =1 Y k N ( N , q N j C i

and ρ 1 ) ) + ) times the Weyl denominator of α R + z z  X z z ∆ λ ( R ( . ( ( ) + ]. These correspond to type IV Argyres-Douglas ∈ – 7 – X R R 1 − R E = α m . χ χ ρ,w q χ (  + (1  − ··· − ) + ` )) = exp q z the number of boxes of the Young tableau. Another ∆ ) (   Q ) = P  -algebras. The example discussed above corresponds to ∈ , w q, z 1 α ( N ln(1 ( adj − `  W nm q, z ` f ) χ q 1) ( ( q . W − m to  =1 + 1 corresponding to SU(N) decomposing into n ∈ q ∞ E =1 ( q X ··· . with − n X w m ∞ ,N − X P 1 sl ··· + 1  n,m − R N P −  (1 C     ∈ . N X λ A E =1 . 2 I ` ∞ Y ρ ]. They are labelled by Young tableaux which can be interpreted as 2 m ) is nothing but the ) + q ` = exp = exp = ( 12 48 ) defined as q, z , = 1 + 1 + η ) = P q (  40 N q, z ··· q, z ( ( ) =    R ··· ··· C the character of the representation  N, n R  ( R R C I χ C So that with and the partition of the trivial representationoperator of algebra, SU(2). i.e. In no this reduction case is the performed. The Schur-index is then given by representations of SU(2) and thuster determining is an exactly embedding what of determinesvertex SU(2) operator in the algebra SU(N). different of This quantum lat- Hamiltonianthe reductions principal of embedding the of affine the SU(2) Young in tableau SU(N) consisting just which out corresponds of to one the column patition of height cide with these Schur indices.indices Before as doing discussed so in one the can introductiontheories, of obtain [ a see series [ of morepartitions general Schur of the integer interpretation is as decompositions of the standard representation of SU(N) into irreducible are parameterized by the positive Weyl chamber The main objective is to construct vertex operator algebras whose graded characters coin- here the sum is over all inequivalent finite dimensional simple modules of SU(N). These JHEP11(2018)188 . - ` ], c Q 24 for ) 42 − n (2.7) λ ) ) ( ρ ∨ B h , . . . , α +2 + 1 k α λ,λ )-Argyres- 2( ( 1 q -algebra is a − ) . The p 1) m n B q − p α by 2( the weight lattice. − ) is obtained from z p P ,A − 1 q, z ]. The ( A ) of highest-weight R 52 )(1 λ – ( C m k ] and I follow that notation. q 50 L and it is defined as the joint ) α 53 . z z . 1 ρ ( p > λ − − √ q χ Z / j (1 = α ∈ + − 0 z ∆ p e Q ∈ is the set of simple roots. Recall that associated to the simple roots α pQ ` p + √ ) √ V with be its root lattice and / m i q α Q ker Q ) 0 − − – 8 – p e and ∆ ` ( =1 -algebras were constructed. The construction is \ j p (1 N W B := sl =1 ∞ Y m Q = ) c ) to be this character up to the prefactor 24 p g ( − ) ) q, z ρ W ∨ ( h +2  + k ··· λ,λ ]. These have been investigated in [ 2( (  q R and the kernel of screening on the lattice vertex operator algebra. 39 C 0 1 -algebras named : )] = pA g W λ -algebra be the positive Weyl chamber. The higher rank analogue of the triplet − ( k Q 1 is the rank of √ + ] and it goes as follows. The triplet vertex operator algebra is defined as the ) L -algebra and it corresponds to the Schur-index of ( P p − n . Its U(1)-orbifold is called the singlet algebra [ ⊕ 13 ( B ], one has to replace this character by the Euler-Poincar´echaracter of the 1 ch[ ) by multiplying it with the supercharacter of a fermionic ghost vertex operator 0 1 denotes the lattice vertex operator algebra associated to the rescaled root lattice N B is given by 49 pA pA q, z pQ k = ( √ the rank of √ √ be a simply-laced Lie algebra and let ] a series of  ` V ` . To match notation with the previous section I will now replace ··· There are similar higher rank analogues to the triplet and singlet algebras constructed In the following sections these procedures will be realized on the level of vertex operator g 35 pQ  R Here √ algebra will beoperator defined algebra, namely as on the an tensoralgebra product intersection and of a of a Heisenberg certain the vertex isotropic operator lattice same vertex algebra. operator screenings The lattice on in question a is different isotropic, that vertex vertex operator algebra is denotedintersection of by certain screening operators with Douglas theories. by Feigin and Tipunin [ Let Moreover let lattice simple current extension of an U(1)-orbifoldoperator of the algebra, triplet that algebra is timesinside a a Heisenberg free vertex boson.This Namely, consider is the the isotropic diagonal sublattice 3 The In [ reviewed in [ kernel of a screening charge inside the lattice vertex operator algebra of the rescaled root C algebra and then specializing the Jacobi variable to algebra constructions. so that we recognize Similarly, if we pass to asee non-trivial also Young tableau, [ then ascorresponding explained quantum in section Hamiltonian 4.3 reduction. of [ For example the character of the irreduciblegeneric highest-weight representation Here JHEP11(2018)188 , λ ) Q (3.1) ) n and let ( . d W )) )) ρ ρ + + λ λ ( ( ρ,w ρ,w ( ( coincides with the − − 1 q q . We thus have ) − ) -orbifold of w ` Q N w . ( ( A 
 − ) ` W = W d . β mod ∈ ( ∈ n X X w Q  . w ` n + √ nµ ` n µ / β ` √ ) + j ` ) / − √ ⊗ H Z q j α 1 q β / = 1 √ ( ) ( j α − 0 . n η α ν η e − 0 ( µ ) Z ) ) e − 0 0 P z ν µ z ⊗ F e ) + D ( ) ( nµ for . We then define the isotropic lattice µ n Q λ Q λ V √ ( ( ( ⊕ · · · ⊕ ) P Q n χ F λ 0 P m ) 2 ∼ = √  1 2 D ) ) m V ρ β λ ⊕ · · · ⊕ ρ ker W n, µ P + ( Z
+ F n, ν λ ∈ ker ( ` λ ) ( =1 − 1 X ( µ ker – 9 – \ n = j 2 in the instance of M n 2 ( β n ` W =1 q 0 \ ` P P q j =1 Q ) \ + := j ) ∈ ) D ) = M ν nP M µ µ ∈ z n + + 1 Q :=  ( + λ ( ) := β Q ∼ = λ , then the corresponding vertex operator algebra is Q ( B Q √ ( χ λ ) Q ∩ := Q ∩ ) X Z in the highest-weight representation of highest-weight + ) X n, µ = ) + n ( P n n ( P µ ( n, ν (  ∈ ∈ ( B 0 P λ the Gram matrix of M B λ ) := D D 2 n ). It remains to compute the character. Firstly, W n 2 n ) be the Heisenberg vertex operator algebra of rank 2 ρ 2 G ( ρ 2 d W q q 0 P q, x (2 in the characer of D )( and H ] = ] = µ Q z Q ) i,j ) N, n ( nG n, ν I ) and n, µ (  g ( W and = M ) the multiplicity of  j P ch[ µ ch[ β ( ∈  i λ = rank( ν β m d be the Fock module of weight λ By construction is the coefficient of so that The character of the subspace (actually it is a module for the U(1) with Let The claim is thatSchur-Index the of character of and Let F with is every vector has norm zero. For this let JHEP11(2018)188 . 1 . − and )) N (3.3) µ ρ g z A + and let 2 λ ( µ ]) of the g = 2 n . We can ρ,w 40 − ( Q ψ q − )) q ρ ) + w λ ( (  ρ,w W ( ∈ − X . w q ) µ )) and such that z w λ ) (3.2) 2 ( ∨ (  ` µ h ) 2 . n L be a simply-laced Lie algebra, q, x λ W (
+ − ( ∈ 0 g )( ` principal nilpotent by the main q ` X ,f be a nilpotent element of w L ) L q f ) µ = DS f ( , µ ⊗ 0 N, n 2 λ ( 1 H ) ( ψ λ − λ m 1) I ⊗ N ( m λ c be the dual Coxeter number of A k 24 ) 2 − n. ) + ) λ L and − n ∨ ρ n Q ( ( q = 1 and = ] h X ( + Q k ∈ B ∨ λ 0 µ 2 n ∩ ( n h L nµ ]. For this let 1 ) ρ 2 ψ 2 + n 3 − ρ ) = Q + P q M – 10 – √ ∩ − + ) ∈ +2 k ) = tr + µ λ F ` q, x 1 ψ P M + = λ,λ ]( ∈ ( Q 1 q, x λ ( ]ch[ 2 n = 2 ψ − ] := ∩ )( q Q )). Then we can for example apply this functor to the c X N + ) g + A P ( ] := ) -algebra is P ∈ k -algebras. For this let G, ψ n N, n ∈ Q [ λ X n, µ L ( ( ) λ n ( ( W P I n B ) A ∈ ( ,f 2 M X ` µ n B 2 G, f, ψ ch[ DS ` [ ) ] to get 2 a positive integer. Let ch[ n q 2 (1+ n ) H ( ρ 2 2 n q A P n η ρ 2 ( q ∈ G, ψ η q X µ [ ) = n = = the corresponding quantum Hamiltonian reduction functor with resulting , f A ) = g ,f ( k q, z DS W ]( H Q ) n ( B be complex numbers, such that its root lattice and -algebra ch[ We again expect the existenceobject, of but the presently structure this of is a simple only vertex known operator for algebra on this can be givengeneralize the this structure by of including denote a by simple vertexW operator algebra forsecond generic factor of Then one of our basic proposals is that There is a second point ofas view explored and by this Davide one relates GaiottoQ to and S-duality myself for [ vertex operator algebras k, ` or put differently 4 Another construction and more vertex algebras and we check thatexpected this chiral algebra precisely for matchesWe the see the that Argyres proposed Douglas central theory charge in (see the instance [ that The central charge of the and so JHEP11(2018)188 ) ]. of G i 58 ]. -limit ,W ψ i 55 , V )-module ) into the 3 G prin , f . In the case of g G ( ` . In other words, and sets i of . The large W i so that we obtain W λ -action and just keep ⊗ )) W g λ G V,W G ( ` of ]. Let me explain this in L ( G 57 ) becomes in the limit just a , ,f be a positive integer as before g ( 56 DS )) n k but the action of the zero-modes equivalent to it and consider the λ H L ( ` C Z to the dual of ⊗ L to the dual of i ( λ i i i V ρ V ,f . Then under certain assumptions on W W Q DS to infinity. Then we expect that we can ∩ W ⊗ ⊗ 2 are completely understood in [ + , i i H -modules this implies that there is a braid- ψ ⊗ P M V V . So let us construct a category equivalent to ∈ ⊗ W C λ = 1 V i i ) mapping – 11 – λ mapping M M . If we forget about the n ⊗ ρ Q ( W ) to and = = ) W Z , then this forgetful functor embeds Rep( C F n C modules such that G and A A ( V G Q 0 B ] = 2 ∩ of of and W sl + and P M T V = W = V ∈ C C C λ g , G, f, ψ [ V ,W n C 0 A V ) and replace it by a category = G →∞ . Nonetheless we still expect a braid-reversed equivalence to hold [ lim V ψ G -graded vector spaces, namely the Heisenberg vertex operator algebra of . The identification goes as follows. Let g T . This construction is conjectural but we are working on understanding it ]. Let us now take the limit Q ) -graded vector spaces. But there is a vertex operator algebra that has tensor 54 p ( T W is the equivalence from Rep( the irreducible highest-weight representation of highest weight ⊗ ). Actually we only construct one that is equivalent to the subcategory of Rep( F λ Z ρ G Now, I want to employ the picture that has been developped with Kanade and McRae, the prinipal nilpotent element the right-hand side coincides as a can recover certain subalgebrasthe of action of the maximalcategory torus of categories inside rank the rank of where Rep( whose objects have weights lying in the root lattice and we do this in such a way that we can be given thereversed equivalence structure between of awe can simple now vertex take operator Rep( extension algebra only if there is a braid- is a semiclassical limitcompact Lie and group there weOn have the to other replace hand one again under vertex certain operator assumptions algebra on by the involved the module categories is a vertex operatorthe underlying algebra categories extensionreversed of equivalence between namely that such vertexbraid-reversed operator equivalence algebra of extensions involved area module possible few categories if words. [ and only Assumeinequivalent simple if that one we has have two a vertex operator algebras with f with and for example the cases of scale fields in suchlarge a commutative way Heisenberg that vertex the operatorsurvives limit algebra and exists integrates and to that an action of the compact Lie group Theorem of [ JHEP11(2018)188 . ] ∨ f h 39 (4.1) (4.2) (4.3) . We g with 1 n ))-modules and weight differs from + g ( ∨ Q ` and otherwise f . Proving this h L Y ) Q ( Q ) − n ,f , while the one of ( z, q = G DS B ` )]( . We immediately see H λ Q ( ⊗ ` ) . L G ) )) Q )) )) z, q . According to the principle ) λ ( λ ch[ contains λ ( n ( ,Q ( z, q µ .  ` ` ( 0 ` ( with root lattice be the corresponding quantum x ··· ) L L  L nµ 2 f,Q ( ( g n ( µ λ ,f ) ··· − (  ρ ⊂ B 2 n ,f ,f n ,f √ B DS ( C −  , F ) )) DS DS q DS 1 H λ ) W C − H H ( H µ ` 1) q, x ⊗ ( ( ⊗ ⊗ − L λ ⊗ ) in the root lattice ) ( n ) µ λ ( n ) ( m ( λ R ρ λ P ,f λ ˜ ,N λ f ρ ρ be a positive integer and and let 1 Q ( m ( , then we define the module + DS + − ∩ n g λ C Q F P N F – 12 – H + X ρ as group of automorphisms as these permute the ∈ ∈ A + X + P µ M λ µ I ⊗ P P ∈ G in ) and for the trivial nilpotent element 0 one obtains M + ∈ c c ) λ ). ∈ M 24 24 M λ P λ λ λ µ of − − ∈ 2.7 X ρ ρ λ q q ( ( : = ) := := : = W λ F )–( F = = 1 the central charge of ρ Q ( f,Q − f,Q 2.4 f,Q ) ) : = ∩ ) F ) be the Young tableau corresponding to the nilpotent element n + N n n ( ( P M ( f z, q ∈ ) is trivial for weights that lie in the root lattice B Y B W )+ ]( λ λ ))-modules have the structure of a simple vertex operator algebra is principal nilpotent that ρ g ,Q ( ( 0 nN ` f ) and F L n − ( n ( ,f sl B be a nilpotent element of denotes the Heisenberg vertex operator algebra of rank the rank of 1)(1 DS = f ch[ H − ) be the multiplicity of g H 2 µ ( N λ . Let H ⊗ contains the Weyl group m P = ( ] that the super conformal index associated to the Young tableay c f,Q Let now Let me summarize this discussion. Let ) 42 n ( . Comparing with equation ( with of [ f as well. Here see that by constructionB the autoorphism group of Fock modules appearing in each and the the dual Coxeter number oflattice the simply-laced Lie algebra Hamiltonian reduction functor. Then conjecturallyhave the the following structure of a simple vertex operator algebra can be given the structureThis of can a be simple viewed vertex as operatorconjecture would a algebra amount for natural to any generalization proving nilpotent a ofto element generalization our any of construction quantum the of Hamiltonian ideas of reduction. Feigin and This Tipunin is [ difficult. We conjecture that The braiding on the only depend on the cosetthat of in the the weight case lattice that and let JHEP11(2018)188 (4.4) (5.1) 0 Φ . n ) 1 − ) and we have √ ,A 1 Z ⊕ z, q . ) A 0 ) ν ( Φ , section 4]. They N − ))]( 1  n ⊕ x λ N ( ) ( √ + ` 1 its weight lattice. We − L Nν ⊕ 1 ( 1) ··· ν − . = ,f − − ) 0 N + 0 n n N DS )( A ( ν 0 ) +1 H 0 Φ nP ν − =  ) N 3.1 − ( D ch[ ( . N ν ) µ √ P . ⊕ z, q 0 x ,A ( − ) 2 ⊕ 1 Φ f z, q λ ) µ n N ( Y − ν ρ ( ν 2 n ν and f ⊕  ν − C Q − − ,Y ) − ν N q N 1 + N N − ⊂ ) − 0 sl P µ A N ν nP 1) q, x ( ( ··· ( D P − − λ ) √ n ( ⊕ n N + ⊕ ⊕ ( m R ⊕ ) – 13 – ˜ 1 ,N Q ) f ν λ 1 1  ν n + P + − − ⊕ ( )( Q P N ν 1) ν nP ν X ⊂ ∈ 0 ∈ P . A X − µ λ − Q I n − D N ( + f f f,Q √ ν 24 24 c c P ). So that the dual lattice is Φ ) P N ⊂ vertex operator algebra, that is the chiral algebra of the ν − − ∈ n ⊕ X ) ,A ( λ q q 1 ν − n = ( B by replacing characters by the corresponding Euler-Poincar´e ( βγ − = = ν N N 0 nP ( 0 P A Y ) : = √ ( D Nν ⊂ z, q → denote the root lattice of ]( = ) x, x N 1 1 2 Z − − f,Q x nP ) 1) N − n − A ( Φ = n √ ( B = N ⊕ ch[ N N ,A 1 Q the central charge of )-theory by a larger vertex operator algebra. − nP 1 f N c √ Let A ,A ( 1 A and via restriction to the isotropic sublattices defined in ( the embedding of dual lattices This induces embeddings of and the orthogonal complement is denoted by We compute that ( then have the embeddinng I want to conclude thisside work almost with embeds the observation conformallythat that we in the have the chiral to algebra one replace of of the the the left-hand right-hand side. Here almost means 5 Conformal embeddings versus RGBuican flow and Nishinaka also discuss RG-flowfind of that Argyres-Douglas theories RG-flow [ interpolates between the Argyres-Douglas theories as with characters we immediately get the identification the one of the trivial one JHEP11(2018)188 , , (5.2) ]. ]. ]. ) and 10 336 1 ) brane n − ( 0 SPIRE SPIRE 5 1) 0 Φ SPIRE − IN IN [ [ M D n IN JHEP ( ν , W ]. ][ ⊗ ,A 1 by the construction  − ) n ν SPIRE n √ ( IN A 0 / superconformal field j 0 Φ ][ α D − 0 supercharges = 2 e W  Commun. Math. Phys. 32 N ⊂ ) ]. superconformal field theories in supersymmetric gauge theory , hep-th/9511154 n arXiv:1708.00875 arXiv:1703.00982 ( to  [ ]. , ν , n βγ − = 2 New 16 √ N / SPIRE 0 P SU(3) j N D α IN SPIRE − 0 W ][ IN e hep-th/9603002 (1996) 71 [  ][ ker ) n ( 1 0 +1 Study of . − ν 0 Φ ) \ ]. D ]. = N n B 461 Flowing from – 14 – j ( 0 W 0 Φ ⊗ ⊗ ) (1996) 430  D n  n ( SPIRE SPIRE n W ν √ n New phenomena in IN / IN − √ j √ ⊗ / Chiral algebra of Argyres-Douglas theory from N / ][ ][ j α hep-th/9505062 0 arXiv:1705.07173 P j ν On irregular singularity wave functions and superconformal ), which permits any use, distribution and reproduction in α B 471 − 0 [ D [ α Vertex algebras for S-duality − ]. Nucl. Phys. Vertex algebras at the corner e − 0 − 0 N , W e  e Q ⊗  ) ) ) ) n n ) n ( ( n ( n ( ( SPIRE N ν N ν ) Argyres-Douglas theories. But the last factor is different 0 0 0 P P P IN 0 P 1 [ (1995) 93 D D D D − (2017) 066 ⊗ B CC-BY 4.0 W W W Nucl. Phys. W 1) ν , Infinite chiral symmetry in four dimensions − 09 Q This article is distributed under the terms of the Creative Commons ker ker ker n ) ker 1 1 1 arXiv:1312.5344 )( n ν ν 1 B 448 arXiv:1807.02785 [ ν − − − ( =1 =1 =1 6= 6= − =1 \ \ \ [   j j j \ − j N N ν B N vertex operator algebra. 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