Associated Varieties of Ordinary Modules over Quasi-Lisse Vertex Algebras

TL;DR

This paper establishes that for certain classes of vertex algebras, the associated varieties of their modules match the algebra's variety in dimension, extending known results to broader levels.

Contribution

It proves the equality of associated varieties' dimensions for modules over quasi-lisse vertex algebras, generalizing previous results to non-admissible levels.

Findings

Associated varieties of modules have the same dimension as the algebra's variety.

If the algebra's variety is irreducible, the module's variety coincides with it.

Extension of known results to non-admissible levels in affine vertex algebras.

Abstract

We prove that if $V$ is a conical simple self-dual quasi-lisse vertex algebra and $M$ is an ordinary module then $\dim X_M=\dim X_V$. Hence, if moreover $X_V$ is irreducible then $X_M=X_V$. In particular, this applies to quasi-lisse simple affine vertex algebras $L_{k}(\mathfrak{g})$. For admissible $k$ it reproves a result in \cite{A2}, and it further extends it to non-admissible levels.