# Quasi-lisse vertex algebras and modular linear differential equations

###### Abstract.

We introduce a notion of quasi-lisse vertex algebras, which generalizes admissible affine vertex algebras. We show that the normalized character of an ordinary module over a quasi-lisse vertex operator algebra has a modular invariance property, in the sense that it satisfies a modular linear differential equation. As an application we obtain the explicit character formulas of simple affine vertex algebras associated with the Deligne exceptional series at level , which express the homogeneous Schur indices of 4d SCFTs studied by Beem, Lemos, Liendo, Peelaers, Rastelli and van Rees, as quasi-modular forms.

###### Key words and phrases:

Vertex algebras, Modular linear differential equations, Quasimodular forms, Affine Kac-Moody algebras, Affine -algebras, Associated varieties, Deligne exceptional series, Schur limit of superconformal index###### 2010 Mathematics Subject Classification:

17B69, 17B67, 11F22, 81R10Dedicated to the great mathematician Bertram Kostant

## 1. Introduction

The vertex algebra is called lisse, or -cofinite, if the dimension of the associated variety is zero. For instance, a simple affine vertex algebra associated with an affine Kac-Moody algebra is lisse if and only if is an integrable representation as a -module. Thus, the lisse property generalizes the integrability condition to an arbitrary vertex algebra.

It is known that a lisse vertex operator algebra has nice properties, such as the modular invariance of characters [Z, Miy], and most theories of vertex operator algebras have been build under this finiteness condition (see e.g. [DLM, Hua]). However, there do exist significant vertex algebras that do not satisfy the lisse condition. For instance, admissible affine vertex algebras do not satisfy the lisse condition unless they are integrable, but nevertheless their representations are semisimple in category ([AdMi, A5]) and have the modular invariance property ([KW2, AvE]). Moreover, there are a huge number of vertex algebras constructed in [BLL] from four dimensional superconformal field theories (SCFTs), whose character coincides with the Schur limit of the superconformal index of the corresponding four dimensional theories. These vertex algebras do not satisfy the lisse property in general either.

In this paper we propose the quasi-lisse condition that generalizes the lisse condition. More precisely, we call a conformal vertex algebra quasi-lisse if its associated variety has finitely many symplectic leaves. For instance, a simple affine vertex algebra associated with is quasi-lisse if and only if is contained in the nilpotent cone of . Therefore, by [FM, A3], all the admissible affine vertex algebras are quasi-lisse. Moreover, the -algebras obtained by quasi-lisse affine vertex algebras by the quantized Drinfeld-Sokolov reduction ([FF, KRW]) is quasi-lisse as well. The vertex algebras constructed from 4d SCFTs are also expected to be quasi-lisse, since their associated varieties conjecturally coincide with the Higgs branches of the corresponding four dimensional theories ([R]).

We show that the normalized character of an ordinary representation of a quasi-lisse vertex operator algebra
has a modular invariance property,
in the sense that it
satisfies a modular linear differential equation (MLDE) (cf. [MMS], [KZ],
[Mas], [Mil], [KNS] and [AKNS]).
This seems to be new even for an admissible affine vertex algebra.
Moreover,
using MLDE,
we obtain the explicit character formulas
of simple affine vertex algebras associated with the Deligne exceptional series
([D])
at level .
These vertex algebras arose
in [BLL]
as chiral algebras
constructed from
4d SCFTs^{1}^{1}1For types and the connection with
4d SCFTs is conjectural..
Thus [BLL], our result expresses
the homogeneous Schur indices of
the corresponding 4d SCFTs as (quasi)modular forms. This result is rather surprising
especially for types ,
,
and (non-admissible casees), since the characters of these vertex algebras
are
written [KT] in terms of non-trivial Kazhdan-Lusztig polynomials
as their highest weights are not regular dominant.

We note that in [CS] the authors have obtained a conjectural expression of Schur indices in terms of Kontsevich-Soibelman wall-crossing invariants, which we hope to investigate in future works.

### Acknowledgments

The first named author thanks Victor Kac, Anne Moreau, Hiraku Nakajima, Takahiro Nishinaka, Leonardo Rastelli, Shu-Heng Shao, Yuji Tachikawa and Dan Xie for valuable discussion. He thanks Christopher Beem for pointing out an error in the first version of this article. Some part of this work was done while he was visiting Academia Sinica, Taiwan, in August 2016, for “Conference in Finite Groups and Vertex Algebras”. He thanks the organizers of the conference. He is partially supported by JSPS KAKENHI Grant Number No. 20340007 and No. 23654006. The second named author would like to thank Hiroshi Yamauchi for the helpful advice. He was partially supported by JSPS KAKENHI Grant Number No. 14J09236.

This research was supported in part by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Economic Development & Innovation.

## 2. Quasi-lisse vertex algebras

Let be a conformal vertex algebra, the Zhu’s -algebra of ([Z]), where . The space is a Poisson algebra by

Here denotes the image of in , and is the quantum field corresponding to . In this paper we assume is finitely strongly generated, that is, is finitely generated.

The associated variety [A2] of a vertex algebra is the finite-dimensional algebraic variety defined by

Since is a Poisson algebra, we have a finite partition

where are smooth analytic Poisson varieties (see e.g. [BG]). Thus, for any point there is a well defined symplectic leaf through it.

###### Definition 2.1.

A finitely strongly generated vertex algebra is called quasi-lisse if has only finitely many symplectic leaves.

Let be a quasi-lisse vertex algebra. The finiteness of the symplectic leaves implies [BG] that the symplectic leaf at coincides with the regular locus of the zero variety of the maximal Poisson ideal contained in the maximal ideal corresponding to . Thus, each leaf is a smooth connected locally-closed algebraic subvariety in . In particular, every irreducible component of is the closure of a symplectic leaf ([Gin, Corollary 3.3]).

For us, the importance of the finiteness of the symplectic leaves is in the following fact that has been established by Etingof and Schedler.

###### Theorem 2.2 ([Es]).

Let be a finitely generated Poisson algebra, and suppose that has finitely many symplectic leaves. Then

## 3. A necessary condition for the quasi-lisse property

A finitely strongly generated vertex algebra is called conical if it is conformal, and gives a -grading

on for some , for all , and , where . Note that if is a vertex operator algebra, that is, if is integer-graded, then a conical vertex operator algebra is the same as a vertex operator algebra of CFT type.

Let be a conical vertex algebra. The -grading induces the grading

on . In other words, the -grading induces a contracting -action on the associated variety .

###### Remark 3.1.

The associated variety of a simple conical quasi-lisse vertex algebra is conjecturally irreducible ([AM2]). The validity of this conjecture implies that the associated variety of a quasi-lisse vertex algebra is actually symplectic, that is, is the closure of a symplectic leaf.

###### Remark 3.2.

Conical lisse (-cofinite) conformal vertex algebras are quasi-lisse, since is a point in this case.

###### Proposition 3.3.

Let be a conical quasi-lisse vertex algebra. Then the image of the conformal vector of is nilpotent in the Zhu’s -algebra of .

###### Proof.

Since is conical, the -action on induced by the conformal grading contracts to a point, say , that is, for all .

Set . It is sufficient to show that the value of at any closed point is zero. Pick an irreducible component of containing . Note that is -invariant, and hence, . On the other hand, there exists a symplectic leaf such that , the Zariski closure of . Since it belongs to the Poisson center of , belongs to the Poisson center of . Hence is constant on as is symplectic, and thus, so is on . Therefore the value of at is the same as the one at , which is clearly zero. ∎

## 4. Finiteness of ordinary representations

Recall that a weak -module is called ordinary if acts semi-simply on , any -eigenspace of of eigenvalue is finite-dimensional, and for any , we have for all sufficiently large .

###### Theorem 4.1.

Let be a quasi-lisse conformal vertex algebra. Then the number of the simple ordinary -modules is finite.

###### Proof.

Let be the Zhu’s algebra of . By the Zhu’s theorem [Z], it is sufficient to show that the number of the simple finite-dimensional -modules is finite.

The algebra is naturally filtered: There is a natural filtration of induced by the filtration of ([Z]) that makes the associated graded a Poisson algebra. Moreover, there is a surjection map

of Poisson algebras [DSK, ALY]. Therefore, is a Poisson subvariety of , and hence has a finitely many symplectic leaves. Hence, thanks to Theorem 1.4 of [ES] that follows from Theorem 2.2, we conclude that has only finitely many simple finite-dimensional representations. ∎

## 5. Modular linear differential equations

Let denote the Serre derivation of weight

where is the normalized Eisenstein series of weight . Let be the -th iterated Serre derivation of weight with . Recall that a modular linear differential equation (MLDE) of weight is a linear differential equation

with a classical modular function of weight for each .

In this section, we prove the following theorem. Let denote the complex upper-half plane.

###### Theorem 5.1.

Let be a quasi-lisse vertex operator algebra, the central charge of . Then the normalized character

satisfies a modular linear differential equation of weight .

Let be a quasi-lisse conformal vertex operator algebra with the weight grading . Y. Zhu introduced a second vertex operator algebra associated to ([Z]), where the vertex operator is defined by linearly extending the assignment

and . We write and for every .

Set , . Here, is a formal variable and , where () are the Fourier coefficients of the Eisenstein series of weight , that is, . Let be the -submodule of generated by

Let denote the -span of elements , , in .

###### Proposition 5.2.

The -module

is finitely generated.

###### Proof.

We set

Also, put , , and

Define an increasing filtration of the -module by

This induces the filtration on and : , , , and

Since is a Noetherian ring, it follows from Proposition 5.2 that is a Noetherian -module. Hence, we have the following lemma.

###### Lemma 5.3.

For an element of , there exist and () such that

Let be an ordinary -module. Define the zero-mode action by linearly extending the assignment

For any , define the formal -point function by

For each -series and , define the formal Serre derivation of weight by

Let and be non-negative integers. For each and of weight , define the formal iterated Serre derivation by

and . Here, is said to be of weight if is a homogeneous element of weight of the graded algebra , where the weight of is for each .

###### Lemma 5.4 (cf. [Dlm, (5.9)]).

Let be an element of and an ordinary -module. We have

###### Proof.

The assertion follows by [Z, Proposition 4.3.5] with and . ∎

Let be a primiary vector of of weight , that is, and for .

###### Lemma 5.5 (cf. [Dlm, Lemma 6.2]).

For each , there exist elements () such that for any ordinary -module ,

(1) |

###### Proof.

The proof is similar to that of [DLM, Lemma 6.2]. We prove the assertion by induction on . When , it follows from Lemma 5.4 that , and therefore (1) follows. Suppose that . Then by Lemma 5.4, we see that

By using the relation of the Virasoro algebra, we have

with a scalar for each and if . Therefore,

By the induction hypothesis, we have (1), which completes the proof. ∎

Let and be elements of .

###### Lemma 5.6 ([Z, Proposition 4.3.6]).

For every ordinary -module ,

###### Theorem 5.7.

Let be a quasi-lisse vertex operator algebra, primary with . For each ordinary -module , the series converges absolutely and uniformly in every closed subset of the domain , and the limit function has the form with some analytic function in . Moreover, the space spanned by for all ordinary -module is a subspace of the space of the solutions of a modular linear differential equation of weight .

###### Proof.

The proof is similar to those of [Z, Theorem 4.4.1] and [DLM, Lemma 6.3]. By Lemma 5.3, we have where and for each . It then follows by the definition of and Lemma 5.6 that . By Lemma 5.5, we obtain a differential equation

(2) |

for the formal series with . Since converges absolutely and uniformly on every closed subset of , and (2) is regular, it follows that converges uniformly on every closed subset of . By using (2) again, we see that the space spanned by for all ordinary -module is a subset of the space of the solutions of the MLDE

where is the limit function of with and . The remainder of the theorem is clear. ∎

## 6. Examples of quasi-lisse vertex algebras

Let be the universal affine vertex algebra associated with a simple Lie algebra at level , and let be the unique simple graded quotient of . We have , where is equipped with the Kirillov-Kostant-Souriau Poisson structure, and is a conic, -invariant, Poisson subvariety of , where is the adjoint group of (see [A2]).

Let , the nilpotent cone of , which is identified with the zero locus of the argumentation ideal of the invariant ring via the identification . It is well-known since Kostant [Kos63] that the number of -orbits in is finite.

###### Lemma 6.1.

The affine vertex algebra is quasi-lisse if and only if the associated variety .

###### Proof.

The “if” part is clear since the symplectic leaves in are the coadjoint orbits of . Conversely, suppose that . Since is closed, there exists a nonzero semisimple element in . As it is conic, contains infinitely many orbits of the form , . ∎

Recall that is called admissible if it is an admissible representation ([KW2]) as a module over the affine Kac-Moody algebra associated with . All the admissible affine vertex algebras are quasi-lisse, since their associated varieties are contained in ([FM, A3]). In fact, the associated variety of an admissible affine vertex algebra is irreducible, that is, for some nilpotent orbit of (see [A3] for the explicit description of the orbit ).

Highest weight representations of an admissible affine vertex algebra are exactly the admissible representations of of level whose integral Weyl groups are obtained from that of by an element of the extended affine Weyl group ([AdMi, A5]). Let be the Cartan subalgebra of . The modular invariance of the normalized full characters

of those representations, where is some domain in , has been known since Kac and Wakimoto [KW1, KW2], and was extended in [AvE] to that of the general (full) trace functions. Here it is essential to consider their full characters, since an admissible representation is not an ordinary representation in general, and thus, the normalized character is not always well-defined.

Theorem 5.7 states the modular invariance of the normalized character (instead of the normalized full character) of an admissible representation that is ordinary. As far as the authors know, this fact is new.

Here are more examples of quasi-lisse affine vertex algebras.

###### Theorem 6.2 ([Am1]).

Assume that belongs to the Deligne exceptional series

and let . Then , where is the minimal nilpotent orbit of .

In Theorem 6.2, the affine vertex algebra is admissible for types , , , , and so the statement is contained in [A3]. However, is not admissible for types , , , . These non-admissible quasi-lisse affine vertex algebras have appeared in [BLL] as main examples of chiral algebras coming from 4d SCFTs. In fact, the labels , , , also appear in Kodaira’s classification of isotrivial elliptic fibrations, and the corresponding 4d SCFTs are obtained by applying the -theory to these isotrivial elliptic fibrations. By construction [BLL], the character of the above non-admissible quasi-lisse affine vertex algebras are the (homogeneous) Schur indices of these 4d SCFTs obtained from elliptic fibrations. In mathematics, such a non-admissible affine vertex algebra was first extensively studied in [Per13].

In the next section we derive the explicit form of the characters of these non-admissible quasi-lisse affine vertex algebras.

Now let us give examples of quasi-lisse vertex algebras outside affine vertex algebras. Let be the -algebra associated with and a nilpotent element at level , defined by the quantized Drinfeld-Sokolov reduction

where denotes the cohomology of the BRST complex with coefficient associated with the Drinfeld-Sokolov reduction with respect to . This definition was discovered by Feigin and Frenkel [FF] in the case that is principal as a generalization of Kostant’s Whittaker model of the center of ([Kos78]), and was generalized to an arbitrary by Kac and Wakimoto ([KRW]).

By [A3], the natural surjection induces a surjective homomorphism of vertex algebras, and moreover,

where is the Slodowy slice at , that is, . Here is an -triple and is the centralizer of in . Therefore, we have the following assertion.

###### Lemma 6.3.

Let be non-critical and suppose that is quasi-lisse, that is, . For any , is quasi-lisse, and hence, so is the simple quotient of .

Lemma 6.3 implies admissible affine vertex algebras produce many quasi-lisse -algebras by applying the Drinfeld-Sokolov reduction. For instance, if is a non-degenerate admissible number (see [A4]), then , and hence,

which is irreducible and therefore symplectic ([Pre]). In particular, if is a subregular nilpotent element in types , has the simple singularity of the same type as ([Slo]). In type , it has been recently shown by Genra [Gen] that the subregular -algebra is isomorphic to Feigin-Semikhatov’s -algebra ([FS]) at level .

## 7. The characters of affine vertex algebras associated with the Deligne exceptional series

In this section, we give the explicit character formulas of the quasi-lisse affine vertex algebras associated with the Deligne exceptional series appeared in Theorem 6.2 by using MLDEs.

Let be a Lie algebra in the Deligne exceptional series and the simple affine vertex algebra with . The Deligne dimension formula (7) below implies that the central charge of is given by

###### Lemma 7.1.

The square of the Virasoro element of is in the Zhu’s -algebra .

###### Proof.

Let be the ideal of generated by the image of the maximal submodule of , so that . We need to show that , where is the Casimir element of .

If is not of type , then this result has been already stated in Lemma 2.1 of [AM1], see (the proof of) Theorem 3.1 of [AM1]. So let be of type , in which case the maximal submodule of is generated by a singular vector, say ([KW1]). For , the assertion follows immediately from a result in [FM], which says that the image of in coincides with up to nonzero constant multiplication, see the proof of Theorem 4.2.1 of [FM]. Finally let . Then the vector has degree , cf. [Per]. Let be the -submodule of generated by the image of in . Proposition 3.3 of [GS] (which is valid for type cases as well) says that contains a submodule isomorphic to . On the other hand, Kostant’s Separation Theorem ([Kos63], cf. Proposition 3.2 of [GS]) implies that is the unique submodule of isomorphic to . Thus, , and the assertion follows. ∎

As is isomorphic to , it follows that

for some , where and , , are -homogeneous elements of such that . On the other hand, it follows from the definition of that

for . However, and for as the -weight of is . Therefore, we get that

with . By using Lemma 5.5, we see that the formal characters of all ordinary -modules satisfy a second order differential equation of the form with . Hence, the characters of the ordinary -modules satisfy a second order MLDE of weight 0.

The second order MLDEs (of weight 0) have the form

(3) |

with . Here,

A function is called of vacuum type if has the form with and for each , where . Let be a solution of (3) of the form with . Then by substituting into (3), we see that or . If , it follows that is one of the following numbers [KNS, (3.12)]:

(4) |

On the other hand, if , we have [KNS, (3.16)]

(5) |

Since is of CFT-type with the central charge , the character is of vacuum type and has the form . Therefore, (4) and (5) imply that the MLDE must be the following one:

(6) |

The vacuum type solutions of (6) are also given in [KK] and [KNS]. As a result, we conclude that

Here, , , ,

with the Legendre symbol , , , and .

In particular, it follows that the characters of , , , and are modular forms, while those of , , and are quasimodular forms of positive depths ([KNS, pp.450]). Moreover, if is the dual Coxeter number of , , or , then MLDE (6) has a solution with a logarithmic term (see [KK, section 5] and [KNS, Remark 3.8]). Note that the above formula for and follows also from the recent remark [KW4] by Kac and Wakimoto.

###### Remark 7.2.

It should be notable that the coefficient of in (6) is a non-constant rational function in , as such phenomena are often observed for the Deligne exceptional series. In fact, the dimensions of specific modules over any Lie algebra in the Deligne exceptional series satisfy the so-called Deligne dimension formulas, which are rational functions in . For example,

(7) |

and ([CdM], [D] and [LM]). Here, is the irreducible highest weight module of weight over