Quasi-lisse vertex algebras and modular linear differential equations

TL;DR

This work generalizes the finiteness condition from lisse to quasi-lisse vertex algebras by requiring finitely many symplectic leaves in the associated variety $X_V$, and proves that ordinary module characters satisfy modular linear differential equations (MLDEs). It establishes finiteness of ordinary representations under quasi-lisse, and shows how MLDEs govern character behavior, enabling explicit character formulas for Deligne exceptional series at level $k=-h^06/6-1$ and connecting these to homogeneous Schur indices of 4d SCFTs as quasi-modular forms. The results cover broad classes, including admissible affine algebras and DS-reduced W-algebras, and provide a unified MLDE framework with concrete modular/form expressions for several Deligne-series cases. By tying VOA representation theory to MLDEs and SCFT indices, the paper offers new tools for analyzing non-lisse theories and expands the catalog of quasi-lisse examples with explicit modular phenomena.

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 $-h^{\vee}/6-1$, which express the homogeneous Schur indices of 4d SCFTs studied by Beem, Lemos, Liendo, Peelaers, Rastelli and van Rees, as quasi-modular forms.

Theorems & Definitions (27)

• Definition 2.1
• Theorem 2.2: EtiSch10
• Remark 3.1
• Remark 3.2
• Proposition 3.3
• proof
• Theorem 4.1
• proof
• Theorem 5.1
• Proposition 5.2