Associated Varieties of Ordinary Modules over Quasi-Lisse Vertex Algebras

TL;DR

The paper studies the geometry of vertex algebra representations through their associated varieties, focusing on how fusion via intertwiners affects these varieties. It introduces a Hilbert-series-based bound on $\dim X_{M_3}$ in terms of $\dim X_{M_1}$ and $\dim X_{M_2}$, and derives that conical simple self-dual quasi-lisse vertex algebras have uniform associated varieties across simple ordinary modules, with $X_M= X_V$ when $X_V$ is irreducible. The results extend known statements for admissible-level affine vertex algebras to non-admissible levels and provide concrete instances, including $L_{-2}(G_2)$ and $L_{-2}(B_3)$, where nontrivial ordinary modules share the same associated variety as the algebra. These findings illuminate the geometric structure of representations and deepen connections between fusion, Poisson geometry, and the theory of quasi-lisse 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.

Theorems & Definitions (18)

• Proposition
• Theorem
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Remark 2.5
• Definition 2.6
• Theorem 2.7
• Lemma 3.1