Vertex operator algebras of ArgyresDouglas theories from M5branes
Abstract
We study aspects of the vertex operator algebra (VOA) corresponding to ArgyresDouglas (AD) theories engineered using the 6d theory of type on a punctured sphere. We denote the AD theories as , where and represent an irregular and a regular singularity respectively. We restrict to the ‘minimal’ case where has no associated mass parameters, and the theory does not admit any exactly marginal deformations. The VOA corresponding to the AD theory is conjectured to be the Walgebra , where with being the dual Coxeter number of . The Schur index of the AD theory is identical to the vacuum character of the corresponding VOA, and the HallLittlewood index computes the Hilbert series of the Higgs branch. We find that the Schur and HallLittlewood index for the AD theory can be written in a simple closed form for . We also conjecture that the associated variety of such VOA is identical to the Higgs branch. The M5brane construction of these theories and the corresponding TQFT structure of the index play a crucial role in our computations.
KIASP17032
1 Introduction
It has been found recently that for any 4d SCFT, there is a protected sector described by a 2d chiral algebra or more precisely a vertex operator algebra (VOA) Beem:2013sza . The VOA corresponding to 4d SCFT includes the operators in the Higgs branch chiral ring and more generally the socalled Schur operators Gadde:2011ik ; Gadde:2011uv . Correlation functions in this sector are meromorphic and do not change under exactly marginal deformations.
The 2d chiral algebra^{1}^{1}1We use the vertex operator algebra and chiral algebra interchangeably. is constructed from the 4d theory as follows. First, pick a twodimensional slice in the fourdimensional space with complex coordinates . Then choose a set of particular operators living on this plane annihilated by a combination of Poincare and conformal supercharges . At the origin these operators are the Schur operators. The operator product expansion (OPE) of these operators turns out to be meromorphic up to the exact piece. Therefore by passing through the cohomology, the operators form a meromorphic chiral algebra or VOA.
Under this 4d SCFT/VOA correspondence, the 2d Virasoro central charge is given in terms of 4d central charge as
(1) 
When there is a (nonR) global symmetry , the symmetry in 2d is enhanced to affine symmetry with the level given by the 4d flavor central charge as
(2) 
The (super)character of the VOA gives the Schur limit of the superconformal index
(3) 
where denotes the VOA or the vacuum module of the corresponding chiral algebra. We review the definition of the superconformal index and its various limits in appendix A.
Various aspects of the VOA corresponding to the 4d SCFT has been studied. The topological field theory (TQFT) structure of the chiral algebra for the class theory has been investigated in Beem:2014rza ; Lemos:2014lua . The chiral algebra for the (generalized) ArgyresDouglas theory Cordova:2015nma ; Xie:2016evu ; Creutzig:2017qyf and also for the theories Nishinaka:2016hbw ; Lemos:2016xke has been identified. Bounds on the central charges have been obtained using the chiral algebra structure Liendo:2015ofa ; Lemos:2015orc . The effect of the defect operators has been studied in Cordova:2016uwk ; Cordova:2017mhb . It has also been conjectured that it is possible to reconstruct the Higgs branch BRVOA and also Macdonald index from the chiral algebra Song:2016yfd . Moreover, the Coulomb branch index is related to the modular transformation of the VOA Fredrickson:2017yka .
Our primary focus in this paper is to understand the Schur sector and the Higgs branch of the generalized ArgyresDouglas theory Argyres:1995jj ; Argyres:1995xn that can be engineered from M5branes Gaiotto:2009hg ; Gaiotto:2009we ; Xie:2012hs ; Wang:2015mra . They are interesting because the corresponding chiral algebra/VOA for a subset of such AD theories are particularly simple Xie:2016evu . In this paper, we focus on the subset of the AD theories, where 1) there is no mass parameters associated to the irregular singularity , and 2) there is no exactly marginal deformations. It is widely believed that every exactly marginal operator in SCFT comes from the gauge coupling. Therefore the existence of a marginal coupling signals that this theory can be decomposed into decoupled SCFTs by setting the gauge coupling to zero. We will focus on the ‘‘nondecomposable” AD theories.^{2}^{2}2For example, theory has an exactly marginal operator. This theory can be decomposed into three copies of the theory (also called the theory in the literatures) coupled via gauge group. The associated VOA is conjectured in Buican:2016arp .
The main tools we use are the conjectural vertex operator algebra and the topological field theory structure suggested by the M5brane construction of the class theory theories. The Schur index for any class theory can be obtained by using the 4d/2d correspondence Gadde:2009kb ; Gadde:2011ik ; Gadde:2011uv ; Gaiotto:2012xa between 4d SCFT and 2d topological quantum field theory (TQFT). This correspondence has also been extended to the case with irregular punctures Buican:2015ina ; Buican:2015tda ; Song:2015wta ; Buican:2017uka , which allows us to obtain the index for the AD theory. It has also been found to be consistent with the vacuum character of the chiral algebra for the AD theory. The result also agrees with the IR computation in the Coulomb branch Cordova:2015nma ; Cecotti:2015lab ; Cordova:2016uwk ; Cordova:2017ohl done by using the BPS monodromy operator Cecotti:2010fi ; Iqbal:2012xm . The full superconformal index has been computed using the renormalization group flow from certain gauge theory to the AD theory Maruyoshi:2016tqk ; Maruyoshi:2016aim ; Agarwal:2016pjo , which gives a consistent result with the chiral algebra.
We consider a class of AD theories labeled as . This class of SCFTs is engineered using 6d theory of type on a Riemann sphere with one irregular singularity labeled by , and one regular singularity labeled by which also specifies a nilpotent orbit^{3}^{3}3In this paper, we use the Higgs branch label for a puncture, i.e. a full regular puncture corresponds to a trivial nilpotent orbit, while null regular puncture corresponds to the principal nilpotent orbit. of . When there is no regular puncture (or ), we use the notation to denote the corresponding theory.
The main results of this paper are the following (for the cases where is either full() or null(), the corresponding VOA is studied in Xie:2016evu ):

The VOA for the theory is given by the Walgebra , where with being the dual Coxeter number of . Here is the Walgebra obtained via quantum DrinfeldSokolov reduction Drinfeld:1984qv ; Bershadsky:1989mf ; Bershadsky:1989tc ; Feigin:1990pn of the KacMoody algebra using the nilpotent orbit associated to :
(4) Here denotes the full puncture carrying the flavor symmetry .

The Schur index of the theory is given by the vacuum character of algebra . In particular, when and (the full puncture), the character has a surprisingly simple formula in terms of plethystic exponential,
(5) When there is no regular puncture, or equivalently , the Schur index is given by
(6) Here are the degrees of the Casimirs of and is the Weyl vector. The plethystic exponential is defined as
(7) We also give a similar expression for the general puncture of type in section 4.

The Higgs branch of theory is identified with the associated variety MR3456698 of :
(8) The associated variety is given in terms of the closure of a nilpotent orbit, which depends on the choice of and as:

If , the associated variety is
(9) where is the principal nilpotent cone, and is the Slodowy slice defined using the nilpotent orbit corresponding to .

If , the associated variety is given as
(10) where is the closure of a certain nilpotent orbit which depends on .

It was proven that our formula for the Schur index (5) for the theory is indeed identical to the vacuum character of the corresponding VOA Kac:2016aa .
This paper is organized as follows. In section 2, we review some basic facts about the AD theories we consider. In section 3, we describe the corresponding vertex operator algebras of our theories. In section 4, we describe the TQFT approach to the Schur index and thereby giving a physical derivation of the character formula for the associated chiral algebra. In section 5, we study the Higgs branch from three perspectives: the associated variety of the vertex operator algebra, the 3d mirror symmetry, and the TQFT approach. We find these different approaches give consistent results. Finally, we conclude with a remark in section 6. In the appendix, we review the definition of the superconformal index and its limits and also list the nilpotent orbits which can appear as the Higgs branch of the AD theories considered in this paper.
2 Generalized ArgyresDouglas theories from M5branes
One can engineer fourdimensional ArgyresDouglas SCFT by putting six dimensional theory of type on a twodimensional Riemann surface with the following configurations: a) Sphere with an irregular singularity; b) Sphere with one irregular singularity and one regular singularity. The regular singularity is classified in terms of the nilpotent orbits^{4}^{4}4We use the Higgs branch label, i.e. a trivial nilpotent orbit represents a full puncture. Gaiotto:2009hg ; Chacaltana:2012zy of the Lie algebra . The classification of the irregular singularity is related to the classification of positive grading of Lie algebra Xie:2012hs ; Wang:2015mra ; Xie:2017vaf .
2.1 Irreducible irregular singularity
Irregular singularity Let us start with the 6d theory of type and compactify it on a Riemann surface , to get a four dimensional theory. The Coulomb branch of the 4d theory is described by a Hitchin system defined on . The Hitchin system involves a holomorphic oneform which is called the Higgs field, and the irregular singularity is defined by the following singular boundary condition:
(11) 
Here is a regular semisimple element of . The allowed value of has been classified in Wang:2015mra and summarized in table 1. We label these irregular singularities as . The mass parameters are identified with the parameters appear as the coefficient of the first order pole. We list the type of irregular singularities that do not admit any mass parameter in table 2. We call them as irreducible irregular singularities. One can get an AD theory using a single irreducible irregular singularity of the above type, and the central charge for such theory is shown in table 3. One can also find the Coulomb branch spectrum by studying the SeibergWitten geometry.
Singularity  




12  
9  
8  

18  
14  

30  
24  
20 
No solution  
No solution  
Regular singularity One can also add a regular singularity on top of the irregular singularity to obtain ArgyresDouglas theories carrying a nonabelian flavor symmetry. The regular singularity is classified by the embeddings of into , or equivalently by the nilpotent orbits of the Lie algebra . The flavor symmetry associated to a puncture of type is given by the commutant of in . We list some special orbits in table 4.
Nilpotent orbit  Dimension  Flavor symmetry 

Maximal (Principal)  N/A  
Subregular  
Minimal  
Trivial  0 
We label our theory as Here represents the regular puncture (or its corresponding nilpotent orbit) and denotes the irreducible irregular puncture (has no mass parameter). We write the central charges^{5}^{5}5Here we use the normalized the flavor central charge so that a free hypermultiplet has . for such theories in table 5.
For , the regular singularities are classified by the partitions of or Young Tableaux with , and the corresponding flavor symmetry is given as
(12) 
For the classification of the regular punctures and the corresponding flavor symmetries of other Lie algebras, see Chacaltana:2012zy .
Theory  









Rank 1 SCFTs One interesting class of 4d SCFTs is the rank 1 SCFTs labeled by , , , , , , . They can be realized as the worldvolume theory of a single D3brane probing the 7brane singularities in Ftheory. They have the global symmetry , , , , , , respectively. A noticeable feature is that the Higgs branch of each theory is given by the minimal nilpotent orbit of the flavor group, which is the same as the centered 1instanton moduli space. These theories can be realized in our setup as ,,, , , , respectively.
2.2 Theory with 3d Lagrangian mirror
We can reduce 4d SCFTs to 3d and flow to IR to obtain 3d SCFTs. For a 3d SCFT A, it is often possible to find a mirror SCFT B. The essential feature of 3d mirror symmetry is that the Coulomb branch of theory A is identified with the Higgs branch of theory B, and vice versa. Since the 4d Higgs branch is the same as the Higgs branch of the 3d SCFT A obtained via dimensional reduction, we can use the Coulomb branch of the 3d mirror B to study the 4d Higgs branch. For example, if B admits a Lagrangian description, we can compute the Higgs branch index of A through computing the Coulomb branch index of B.
For , we find the list of AD theories that admit Lagrangian 3d mirrors, as given in figure 1. The idea for finding these theories is as follows: first, we use the method of GaiottoWitten Gaiotto:2008ak to find the 3d quivers whose flavor symmetry on the Coulomb branch is . Then we use the method of Xie:2012hs to find a Hitchin system realization where the Higgs field has integral order poles. From the Hitchin system, we can derive the 4d Coulomb branch spectrum. Finally, we search for a realization that fits into the class of theories we label as with .
2.3 Reducible irregular singularity
In the last subsection, we considered the theory defined by an irregular singularity which does not admit any mass parameter. For a general irregular singularity, it was found in Xie:2017vaf that it is useful to represent our theory by an auxiliary punctured sphere. Let us consider 6d theory on a sphere with the following irregular singularity
(13) 
Here are coprime and is regular semisimple, and or with an integer. One can also add a regular singularity of type. The range of is restricted to . Depending on values of , it is shown that one can have the following representations in terms of auxiliary punctured sphere Xie:2017vaf :

If , we get the usual class theory on a sphere with regular singularity.

If , the theory can be represented by a sphere with marked points of the black type .

If , one can represent the theory by a sphere with marked points of the black type , and one extra red marked point representing the regular singularity.

If , one can represent the theory by a sphere with marked points of the black type, one marked point of the red type representing the regular singularity, and one marked point of the blue type.
For each marked point, one associate a Young Tableaux with varying size (except for the class case where the Young Tableaux has the fixed size determined by ). We refer to Xie:2017vaf for more detail.
In the previous sections, we have considered the case of and we will be only considering this case in the rest of the current paper (notice that in general this condition implies that the irregular singularity has no mass parameter and exact marginal deformation). Therefore our theory can be represented by the following three punctured sphere: one trivial blue marked point, one red marked point representing regular singularity, and one black marked point of type . We hope that the above representation of more general AD theories and the Sduality proposed in Xie:2017vaf can be helpful in studying the indices of those theories.
3 Vertex operator algebra of the AD theories
We have introduced a class of AD theories labeled as . Here represents an irregular singularity without any mass parameter, and is an arbitrary regular singularity. We also assume that the theory is ‘indecomposable,’ namely has no exactly marginal operators.
It was shown in Beem:2013sza that for any fourdimensional SCFT, one can associate a twodimensional chiral algebra or vertex operator algebra. The basic correspondence is as follows:

The 2d Virasoro central charge is given in terms of the conformal anomaly of the 4d theory as
(14) 
The global symmetry algebra becomes an affine KacMoody algebra and the level of affine KacMoody algebra is given by the 4d the flavor central charge as
(15) 
The (normalized) vacuum character of the chiral algebra/VOA is identical to the Schur index of the 4d theory:
(16)
The VOAs corresponding to the theories or with irreducible irregular puncture were discussed in Xie:2016evu . Here we conjecture VOA for the general choice of the puncture , along with the same line as in the case of the usual class theory Beem:2014rza :
Conjecture 1
The VOA corresponding to is given by the Walgebra . Here and algebra is the quantum DrinfeldSokolov reduction Drinfeld:1984qv ; Bershadsky:1989mf ; Bershadsky:1989tc ; Feigin:1990pn of the KacMoody algebra using the nilpotent orbit (or the embedding into ) associated with .
Two extreme situations where being (full) or (null) have been considered in Xie:2016evu , and the corresponding VOAs are the affine KacMoody algebra and the standard algebra associated with .
For a general choice of , we obtain the VOA for the corresponding theory via quantum DrinfeldSokolov (qDS) reduction. The central charge of the vertex operator algebra is given as deBoer:1993iz
(17) 
where is the dual Coxeter number, and is the longest root. Let us explain the notation: for a nilpotent orbit, one has a triple so that . Here is a semisimple element of the Lie algebra , and is a nilpotent element. This can be also obtained by choosing a embedding into given by the choice of . It gives . Now, has an eigenspace decomposition under the adjoint action of as:
(18) 
Let be the centralizer of in , which also has a grading by the adjoint action of :
(19) 
For us, is an element of the nilpotent orbit associated to . The algebra is strongly generated by fields , where runs over a basis of . The generator has a conformal weight if .
The level of an affine KacMoody algebra (for ) is called admissible if it can be written in the following way
(20) 
So only for , the level of our conjectural VOA is admissible. The vacuum character of the corresponding affine KacMoody algebra with admissible level and the algebra obtained from qDS are given in Kac:1988qc and Frenkel:1992ju respectively.
There is one simple consistency check we can do: the 4d central charge can be computed using the method presented in Xie:2012hs ; Xie:2013jc and can be used to check with 2d central charge formula (17).
Examples
Let us study several interesting class of examples. ^{6}^{6}6Here means that the corresponding BPS quiver is a product of two Dynkin diagrams for and . theories are a class of well studied ArgyesDouglas theories. The theory can be realized by the choice with being trivial. The theory can be realized from with being the full puncture. The theory can be realized from with , and the theory can be realized from with . The corresponding VOAs are listed in table 6. In the following, we will find VOAs for other AD theories in this class.
Example 1: Consider the case with , and . This theory is identical to the theory and has global symmetry (except for , which has enhanced symmetry). The corresponding VOA is .
Example 2: Consider the theory defined by data and . It gives the theory which has flavor symmetry (except for , which has enhanced symmetry). The corresponding VOA is .
Example 3: Choose and to be trivial. The same theory can be also obtained from , and a simple puncture . This also belongs to our list. The theory is identical to . Using the other description, we find that the corresponding VOA is .
The complete results are summarized in table 6. We are able to recover what was found in Creutzig:2017qyf . These theories can also be realized in other ways, for example theory can be realized using . In general, there are many different M5brane realizations of the identical 4d AD theory. Whenever this happens, we find isomorphisms between Walgebras.
VOA  

Schur index and the vacuum character
The Schur index is given by the vacuum character of the corresponding VOA. Here, let us consider the case and , then the VOA for the theory is the same as the minimal model. The vacuum character takes the following simple form Kac:1988qc ; Frenkel:1992ju ; Bouwknegt:1992wg ^{7}^{7}7For , the character is already considered in Cordova:2015nma .
(21) 
where is the rank of the Lie algebra , is the Weyl vector, is the affine Weyl group, and is the signature of the affine Weyl group element. and are principle admissible weights such that:
(22) 
We have the solution: and . Substituting into the character formula, we get the Schur index of our 4d theory .
In the next section, we will show that TQFT structure of AD theory dictates the Schur index of theory to be the simple form
(23) 
where are the degrees of the Casimirs of . We have checked that this formula matches with equation 21 up to a high power in .
It is viable to compute the vacuum character of hence the Schur index of by using the KacWakimoto formula. However, we propose a much compact formula for case,
(24) 
A derivation of this formula is given in section 4.
4 Schur index and TQFT
In this section, we derive a universal formula for the Schur index of the AD theory using the TQFT structure of the index, which provides a strong check for the identification of the corresponding VOA.
4.1 Wave function for the irregular puncture
As an intermediate step, we first write the Schur index of the pure YangMills theory in a TQFT form. Even though the superconformal index is properly defined only for a conformal theory, there is increasing evidence that the Schur index makes sense even for a nonconformal theory. See for example Cordova:2015nma ; Cecotti:2015lab ; Cordova:2016uwk .
The Schur index for the pure YM theory of gauge group is given by
(25) 
where is the rank of the gauge group and is the set of all roots. We write the measure factor as
(26) 
where denotes the order of Weyl group of . Here we used the shorthand notation .
The pure YM theory can be obtained from 6d theory. When is simplylaced, we pick type 6d theory and compactify on a sphere with two identical irregular punctures. The irregular puncture, which we denote as (here is the dual Coxeter number of ), realizes a singularity of the form
(27) 
From the TQFT structure of the index Gadde:2009kb ; Gadde:2011ik ; Gadde:2011uv ; Gaiotto:2012xa , the Schur index should be given by
(28) 
The wave function for the puncture is given by Song:2015wta
(29) 
where is the character (or Schur function) of for the representation . This is an analog of the Gaiotto state Gaiotto:2009ma ; Keller:2011ek in the AGT correspondence.
It is easy to verify that (29) reproduces the Schur index for the pure YM. Plugging in the wave function to the RHS of (28), we get
(30) 
Here we used the relation , where is the Haar measure.
The wave functions for the are evaluated when Song:2015wta . For , we get
(31) 
When , we get
(32) 
where .
The wave function for the irregular puncture is conjectured to be given by Song:2015wta
(33) 
Here, we further conjecture that the wave function for the irreducible irregular puncture (with ) with no flavor symmetry (where the condition is summarized in table 2) is given by
(34) 
In the following, we compute the Schur indices for a number of examples assuming the relation (34) and find that it agrees with the vacuum character of the VOA.
4.2 AD theories of type
Let us consider the theory of type , where is the dual Coxeter number of . As before, we can write the wave function for the irregular puncture realizing the analog of GaiottoWhittaker state for the pure YangMills as
(35) 
where is a Dynkin label for the representation of . From the TQFT structure, we should have