Singular support of a vertex algebra and the arc space of its associated scheme

Abstract

Attached to a vertex algebra $\mathcal{V}$ are two geometric objects. The associated scheme of $\mathcal{V}$ is the spectrum of Zhu's Poisson algebra $R_{\mathcal{V}}$. The singular support of $\mathcal{V}$ is the spectrum of the associated graded algebra $\text{gr}(\mathcal{V})$ with respect to Li's canonical decreasing filtration. There is a closed embedding from the singular support to the arc space of the associated scheme, which is an isomorphism in many interesting cases. In this note we give an example of a non-quasi-lisse vertex algebra whose associated scheme is reduced, for which the isomorphism is not true as schemes but true as varieties.

arXiv commentA few minor corrections, final version, to appear in Progress in Mathematics volume in honor of Anthony Joseph's 75th birthday

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• Chiral Homology of elliptic curves and the Zhu algebra
• Some remarks on associated varieties of vertex operator superalgebras
• Further $q$-series identities and conjectures relating false theta functions and characters
• Cosets of free field algebras via arc spaces
• Vertex Algebras and Commutative Algebras