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 −h ∨ /6 − 1, which express the homogeneous Schur indices of 4d SCFTs studied by Beem, Lemos, Liendo, Peelaers, Rastelli and van Rees, as quasi-modular forms.

VOAs and Rank-Two Instanton SCFTs

“…As quasi-Lisse VOAs, the vacuum characters of the two-instanton VOAs V (2) g will necessarily be solutions of finite-order linear modular differential equations [50]. For their rank-one cousins, these differential equations can be expressed in a uniform way as a second-order modular differential operator whose free coefficient is a function of the dual Coxeter number, 29…”

Free field realizations from the Higgs branch

“…In [7] it was explained that this conjecture requires the vanishing of a certain ideal in R V whose definition refers to the R-filtration on V. Subject to this conjecture, one learns that the associated VOAs of four-dimensional SCFTs have associated varieties that are symplectic. 6 Such VOAs have been dubbed quasi-lisse by Arakawa and Kawasetsu [13]. This property suggests that in some sense, these VOAs should be more geometric than a generic VOA, with the finite-dimensional associated variety playing a central role.…”

“…In order to generate this VOA one needs to also invert e(z). One then has, for example, ∂(δ + ϕ) = NO(e −1 , ∂e), where NO(−, −) denotes conformal normal ordering 13. In equation (3.15) we use the convention of nested conformal normal ordering (A 1 A 2 .…”

VOAs labelled by complex reflection groups and 4d SCFTs

“…1 Introduction and summary Any four-dimensional N = 2 superconformal field theory (SCFT) contains a subsector isomorphic to a vertex operator algebra (VOA) [1]. This 4d/2d correspondence (see [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] for some further developments) promises to become an organizing principle for the whole landscape of N = 2 SCFTs. The aspiration is to combine the rigid associativity constraints of VOAs with additional physical requirements, such as unitary of the 4d theory, in order to constrain and ideally classify the set of N = 2 SCFTs.…”