Schur sector of Argyres-Douglas theory and -algebra
We study the Schur index, the Zhu’s algebra, and the Macdonald index of a four dimensional Argyres-Douglas (AD) theories from the structure of the associated two dimensional -algebra. The Schur index is derived from the vacuum character of the corresponding -algebra and can be rewritten in a very simple form, which can be easily used to verify properties like level-rank dualities, collapsing levels, and S-duality conjectures. The Zhu’s algebra can be regarded as a ring associated with the Schur sector, and a surprising connection between certain Zhu’s algebra and the Jacobi algebra of a hypersurface singularity is discovered. Finally, the Macdonald index is computed from the Kazhdan filtration of the -algebra.
It is important to understand moduli spaces of vacua of four dimensional superconformal field theories (SCFTs). An SCFT could have a Coulomb branch and a Higgs branch. The low energy effective theory on the Coulomb branch is solved by finding a Seiberg-Witten geometry. Almost every nontrivial four dimensional SCFT has a Coulomb branch, which is parameterized by expectation values of half-BPS operators 111 SCFT has a bosonic symmetry group , and the highest weight representation is labeled as , here is the scaling dimension, labels the representation, is charge, and are left and right spins. Short supermultiplets are classified in Dolan:2002zh (), and there are three types of half BPS operators which are important to us: a): with and ; b): with , ; c): with and . . These operators form a ring which is freely generated for almost all the theories we know222See Argyres:2017tmj () for the discussion on the possibility of nontrivial Coulomb branch chiral ring.. The important question is to determine the rational number of each Coulomb branch operator . In practice, one can often easily determine them using the Seiberg-Witten (SW) geometry.
It is also possible for a four dimensional SCFT to have a Higgs branch, which is parameterized by expectation values of half-BPS operators . These operators form a nontrivial ring called the Higgs branch chiral ring. Unlike the common appearance of the Coulomb branch, not all SCFT has a Higgs branch and in fact there does exist a large class of SCFTs which do not have a Higgs branch.
Given the asymmetry between the Higgs branch and the Coulomb branch, one might wonder whether a protected sector could exist for all non-trivial SCFT and contains the Higgs branch when the theory has one. Such sector indeed exists and is called the Schur sector Gadde:2011ik (); Gadde:2011uv (), which contains Higgs branch operators and operators . It is in general quite difficult to determine this sector as there is no powerful tool as the SW geometry of the Coulomb branch.
The understanding of the Schur sector becomes possible because of the following 4d/2d correspondence found in Beem:2013sza () (see Beem:2014rza (); Lemos:2014lua (); Buican:2015hsa (); Cordova:2015nma (); Buican:2015tda (); Song:2015wta (); Lemos:2015awa (); Cecotti:2015lab (); Lemos:2015orc (); Nishinaka:2016hbw (); Buican:2016arp (); Xie:2016evu (); Cordova:2016uwk (); Arakawa:2016hkg (); Lemos:2016xke (); Bonetti:2016nma (); Song:2016yfd (); Creutzig:2017qyf (); Fredrickson:2017yka (); Cordova:2017ohl (); Cordova:2017mhb (); Dedushenko:2017tdw (); Buican:2017uka (); Song:2017oew (); Buican:2017fiq (); Beem:2017ooy (); Pan:2017zie (); Fluder:2017oxm (); Buican:2017rya (); Choi:2017nur (); Arakawa:2017fdq (); Kozcaz:2018usv (); Costello:2018fnz (); Wang:2018gvb (); Feigin:2018bkf (); Niarchos:2018mvl (); Creutzig:2018lbc (); Dedushenko:2018bpp (); Bonetti:2018fqz (); Arakawa:2018egx (); Costello:2018swh (); Nishinaka:2018zwq (); Agarwal:2018zqi (); Beem:2018duj (); Kiyoshige:2018wol (); Buican:2019huq (); Pan:2019bor (); Beem:2019tfp (); Oh:2019bgz (); Dedushenko:2019yiw () for further developments). There is a map between the Schur sector of a 4d SCFT and a 2d vertex operator algebra (VOA). Once the 2d VOA for a 4d SCFT is identified, one can learn a lot about the Schur sector of the 4d theory from known properties of 2d VOA.
For a large class of 4d Argyres-Douglas type SCFTs engineered from 6d theories, we have identified their associated 2d VOAs as -algebras Xie:2016evu (); Song:2017oew (); Xie:2019yds () shown in figure 1. Such -algebra is derived from the quantum Drinfeld-Sokolov (qDS) reduction of an affine Kac-Moody (AKM) algebra with being a boundary admissible level 333Here is a simple Lie algebra, is its dual Coxeter number and k is an integer with following constraints: a) ; b) and coprime; c) and for , and for . , and is a nilpotent element of Lie algebra .
Such -algebra has been studied in physics and mathematics literature extensivelybouwknegt1993w (); kac2003quantum (); arakawa2017introduction (). The purpose of this paper is to extract important information of the Schur sector of the 4d theory from the knowledge of the 2d VOA. We obtain three main results:
The Schur index can be computed from the vacuum character of the -algebra , and the character has a very elegant product formula for the boundary admissible level kac2017remark (). We discovered that the index can be put as a very simple form
Here for a nilpotent element , one has an associated triple and the associated Lie group of has a subgroup with being the flavor symmetry group corresponding to . The adjoint representation of decomposes as under the subgroup , here is the spin representation of subgroup, and is the representation under the flavor group . is the character of representation of flavor group .This formula is a generalization of case considered in Song:2017oew () to the arbitrary simple Lie algebra. With this simple form, we can check many highly nontrivial properties of 4d and 2d theories, i.e. level-rank dualities and collapsing levels of 2d VOA, and more interestingly the S-duality conjecture proposed in Xie:2017vaf (); Xie:2017aqx ().
For a VOA, one can define a commutative and associative algebra called the Zhu’s algebra zhu1996modular (). The reduced ring from the algebra is identified with the Higgs branch chiral ring Song:2017oew (); Beem:2017ooy (); arakawa2018chiral (). Our new point of view in this paper is that the Zhu’s algebra is more important than its reduced version, and it can be thought as the associated ring of the Schur sector. For our cases, the Zhu’s algebra has a simple form and we gave a general proposal for its structure. If our 4d theory has no flavor symmetry, we conjecture that the algebra is actually isomorphic to a Jacobi algebra associated with a quasi-homogenous hypersurface singularity.
To compute the Macdonald index of a given 4d theory, one need to introduce another grading or filtration in the corresponding VOA. For our -algebra, there is a natural filtration called the Kazhdan filtration Arakawa2012Rationality () which we use to define the Macdonald index of the 4d theory. This filtration agrees with the filtration of theories considered in Song:2016yfd (), and gives a natural generalization to general models considered in this paper.
This paper is organized as follows: section 2 reviews the basic correspondence between the Schur sector of 4d theory and 2d VOA. Section 3 reviews known results between 4d Argyres-Douglas theories engineered from 6d theories and their associated 2d -algebras. Section 4 studies the Schur index from the vacuum character of the -algebra. Section 5 studies the Zhu’s algebra which might be thought of as a ring associated with the Schur sector. Section 6 introduces the Kazhdan filtration of our -algebra and it is used to define the Macdonald index. Finally a conclusion is given in section 7.
2 Schur sector and VOA
The representation theory of a 4d SCFT was studied in Dolan:2002zh (). SCFT has a bosonic symmetry group , and the highest weight representation is labeled as , here is the scaling dimension, labels the representation, is charge, and are left and right spins. Short supermultiplets are classified in Dolan:2002zh (), and there are three types of half BPS operators which are important to us: a): with and ; b): with , ; c): with and . We are interested in so-called Schur sector which contains operators satisfying the following condition
The Schur operators are contained in supermultiplets , , and . and multiplets will not appear in theories considered here Buican:2014qla (). is the supercurrent multiplet and multiplets contain Higgs branch operators. Notice that for type supermultiplet, the Schur operator is not the bottom component.
where represents the contribution from derivatives. The Schur index of the same multiplet is given by setting in above formula
Moreover, if the theory has a flavor symmetry, one may also add flavor fugacities in both indices, which keep track of the action of the flavor group. Such fugacities are crucial when considering modular properties of indices. Another important property is that Higgs branch operators form a ring and in most cases there is also a Hyperkhaler metric associated with this ring.
2.1 Quasi-lisse VOA
VOA arises as the chiral part of a two dimensional conformal field theory. Here we review the mathematical definition of a VOA. A vertex algebra is a vector space with following properties ( can be thought of as the vacuum module of the chiral part of 2d CFT) kac1998vertex ():
A vacuum vector .
A linear map
where . This is just the state-operator correspondence555In physics literature, the mode expansion of a field takes the form with the scaling dimension. In VOA literature, however, they use above convention of mode expansion so that they can consider VOA without the definition of scaling dimension. . Given a field , one can recover the corresponding state .
For our purpose, we need to consider the VOA with a conformal vector , which is nothing but the chiral part of the stress tensor. The modes in the expansion of satisfy the Virasoro algebra (using the standard contour integral and OPE of )
The normal order product of two fields and is denoted as , and its modes are
In current convention we have . Other properties of VOA can be found in kac1998vertex ().
Now let us review the definition of some special VOAs. A VOA is called rational if
V has finite number of irreducible representations .
The normalized character converges to a holomorphic function on upper half plane 666We use to denote the character of a VOA and to denote the character of a finite Lie algebra..
The function span a invariant space.
A VOA is called finitely strongly generated if there is finite number of elements such that the whole VOA is spanned by following normal order products
Notice that the choice of generators may not be unique and in general there are relations between the above basis. It is interesting to find a minimal generating set of a finitely strongly generated VOA.
For a VOA , there exists a Li’s filtration Li2005 () which is a decreasing filtration
in which each is spanned by following states
then there is a graded sum of VOA
It is obvious that , and is generated by . Zhu’s algebra is defined as
is a Poisson algebra and is finitely generated if and only if is strongly finitely generated. Moreover the image of generators of in generates as well. Notice that is in general not reduced, namely the ideal defining it would contain a nilpotent element 777A nilpotent element of an ideal is an element not in but for some .. The product and Poisson structure on are defined as
We have now an associated scheme and an associated variety defined from Zhu’s algebra
Recall that is a Poisson variety. If is smooth, one may view as a complex-analytic manifold equipped with a holomorphic Poisson structure, and for each point , there is a well-defined symplectic leaf through , which is the set of points that can be reached from by going along Hamiltonian flows. If is not necessarily smooth, let be the singular locus of , and for any define inductively . We get a finite partition
where the strata are smooth analytic varieties. It is known that each inherits a Poisson structure. So for any point of there is a well-defined symplectic leaf . In this way one defines symplectic leaves on an arbitrary Poisson variety.
A lisse VOA is defined as the VOA such that . A rational VOA has to be lisse, but it is an open problem to prove that lisse VOA has to be rational. A quasi-lisse VOA is defined as the VOA whose associated variety has finite number of symplectic leaves. Quasi-lisse VOA has many interesting properties Arakawa2016Quasi (); Beem:2017ooy ():
The VOA is strongly finitely generated.
The Virasoro vector is nilpotent in .
There are finite number of ordinary modules, and they transform nicely under modular transformations. A weak -module is called ordinary if acts semi-simply on , any -eigenspace of of eigenvalue is finite-dimensional, and for any , for all sufficiently large .
The character satisfies a modular differential equation.
2.2 4d/2d correspondence
There is an AKM subalgebra () in 2d VOA, where is the Lie algebra of four dimensional flavor symmetry .
The 2d central charge and the level of AKM algebra are related to the 4d central charge and the flavor central charge as
The (normalized) vacuum character of 2d VOA is the 4d Schur index .
2.3 Comments on constraints of 2d VOAs corresponding to 4d SCFTs
It is conjectured that the VOA corresponding to a 4d SCFT is always a lisse VOA Beem:2017ooy (). However, not all lisse VOA has a 4d SCFT counterpart. We do have some constraints based on 4d unitarity:
The 2d central charge is negative and has to satisfy the constraint for interacting 4d SCFTsLiendo:2015ofa ().
If 4d theory has a flavor group , its level is bounded from below Beem:2013sza (), so the corresponding 2d AKM level is also constrained.
The minimal conformal weight of primary fields of VOA is constrained to be Beem:2017ooy ().
These constraints come from considerations purely on the Schur sector. On the other hand, there are some very mysterious relations between the Schur sector and the Coulomb branch data:
If we know the common denominator of Coulomb branch operators, the flavor central charge seem to be bounded by a number which depends on the denominator Xie:2017obm (). This bound is different from the minimal bound found from Higgs branch data only.
So from this perspective, the bound from purely Higgs branch data seems to be not enough on constraining the set of quasi-lisse VOA which can be VOA of 4d theory. With input from Coulomb branch data, one can get much stronger constraint on VOA, and we plan to study this further in the near future.
3 Argyres-Douglas theories and -algebras
In this section we review known results on the classification of AD theories from branes and their corresponding VOAs. We focus on AD theories whose VOAs are -algebras at boundary admissible levels.
3.1 AD theories correspond to algebras
One can engineer a large class of four dimensional SCFTs by starting from a 6d theory of type on a sphere with an irregular singularity and a regular singularityGaiotto:2009we (); Gaiotto:2009hg (); Xie:2012hs (); Wang:2015mra (); Wang:2018gvb (). The Coulomb branch is captured by a Hitchin system with singular boundary conditions near the singularity. The Higgs field of the Hitchin system near the irregular singularity takes the following form
where is determined by a positive principle grading of Lie algebra reeder2012gradings (), and is a regular semi-simple element of . is an integer greater than . Subsequent terms are chosen such that they are compatible with the leading order term (essentially the grading determines the choice of these terms). We call them type irregular puncture. Theories constructed using only above irregular singularities can also be engineered using a three dimensional singularity in type IIB string theory as summarized in table 1Xie:2015rpa ().
One can add another regular singularity which is labeled by a nilpotent orbit of (We use Nahm labels such that the trivial orbit corresponding to a regular puncture with maximal flavor symmetry). A detailed discussion on these defects can be found in Chacaltana:2012zy ().
To get non-simply laced flavor groups, we need to consider the outer-automorphism twist of ADE Lie algebra and its Langlands dual. A systematic study of these AD theories was performed in Wang:2018gvb (). Denoting the twisted Lie algebra of as and its Langlands dual as , outer-automorphisms and twisted algebras of are summarized in table 2. The irregular singularity of regular semi-simple type is also classified in table 3 with the following form
Here is an element of Lie algebra or other parts of the decomposition of under outer automorphism. , and the novel thing is that could take half-integer value or in thirds (). One can also represent those irregular singularities by 3-fold singularities as in table 3.
|with twist||SW geometry at SCFT point|
We could again add a twisted regular puncture labeled also by a nilpotent orbit of . If there is no mass parameter in the irregular singularity, the corresponding VOA is given by the following algebraWang:2018gvb ()
where is the dual Coxeter number of , is the number listed in table 4, and is restricted to the value such that no mass parameter is in the irregular singularity.
In this paper, we are going to focus on the choice of and such that the corresponding algebra takes the following form
There are some further constraints on value : a) ; b) and coprime; and c) for , and for .
4 The character of -algebra and the Schur index
Now we discuss Schur indices of AD theories from their corresponding -algebras. The index can be written in a simplified form which implies many interesting properties of the SCFT and the VOA.
4.1 The -algebra from the qDS reduction
We first set up the notation for Lie algebra datas. Let be a simple finite dimensional Lie algebra, and let be the Cartan subalgebra of , and let be the set of roots, where is the dual space of . Let be the root lattice and let be its dual lattice. We also use to denote the set of positive roots, and be the set of simple roots with be the rank of . We denote as the half of the sum of all positive roots. The bracket is the invariant bilinear form on with the normalization for the long roots. is the dual Coxeter number. We use to denote the fundamental weights of Lie algebra .
Now for AKM algebra , its Cartan subalgebra is . The bilinear form on AKM algebra is extended from the bilinear form of as follows
We can use this bilinear form to identify the dual space with . Roots of AKM are denoted by three sets of eigenvalues. The imaginary root has the label and simple roots are with being simple roots of . Furthermore we have the zeroth simple root with being the highest root of . The set of real roots are , and the set of positive real roots is denoted as . Affine fundamental weights are and with being the comark which is for simply laced Lie algebra. We also define . One has following important set of roots
which is used in defining principle admissible weights.
For a , one define a translation with the following formula
An element in the extended affine Weyl group can be written in the form with an element in Weyl group of lie algebra .
Now is called a principle admissible weight if the following two properties hold
The level is a rational number with denominator , such that
where takes 1 for of type ADE, and 2 for of type B, C, F, and 3 for .
All principal admissible weights are of the form
where , are such that , is an integrable weight of level , and dot denotes the shifted action .
Starting with an AKM algebra , one can get a large class of algebras by using the quantum Drinfeld-Soklov reduction kac2003quantum (). Given a triple with the nilpotent element , and the commutation relation is defined as
The corresponding algebra is denoted as . The universal algebra has following properties: it is finitely strongly generated by the following fields with scaling dimension . Here with . Let’s explain the notation now: Given a triple with a semi-simple element, we can decompose as: with . is defined as the elements in which also commutes with nilpotent element . There is a symmetry between such that .
4.2 Character of -algebra modules
For admissible modules of AKM and corresponding W-algebras at boundary level, their characters decompose in products in terms of the Jacobi form kac2017remark (). This result provides an elegant closed form formula for Schur indices of the AD theory discussed in this paper.
Starting with AKM at boundary level , all boundary principal admissible weights are of the form
where , are such that . The character of the module corresponding to admissible weight can be expressed in products of theta functions kac2017remark ()
Convention of and are summarized in appendix A. In particular the vacuum module has the weight , and its character is
The Schur index of the corresponding AD theory is obtained simply by setting and normalizing the character such that the Schur index goes to one when goes to zero.
For W-algebra from the vacuum -module of level by the qDS reduction, there is a reductive functor which maps principle admissible modules of AKM to either zero or an irreducible module of . The character of the irreducible -module is
where forms the -triple in , is the eigenspace decomposition for , is the set of roots of the root spaces in and . If the reduction of gives zero, automatically. If and lead to the same module in the W-algebra, . In particular the vacuum module of the W-algebra is with the character
It also gives the Schur index of the corresponding AD theory after normalization.
4.3 The simplified form
Using the product formula in the previous section, we can put the index of in a even simpler form. If is regular principle, the index is thus
where is the dual Coxeter number, the plethystic exponential is defined as
and is the set of exponents of Lie algebra . On the other hand, if is trivial, the index becomes
where is the character of the adjoint representation of