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A remark on the $C_2$-cofiniteness condition on vertex algebras

Tomoyuki Arakawa

TL;DR

This work establishes that for finitely strongly generated, non-negatively graded vertex algebras, $C_2$-cofiniteness is equivalent to Beilinson–Feigin–Mazur lisse property, showing the two finiteness notions coincide in this setting. The authors develop and interconnect Li and standard filtrations, jet schemes, associated varieties, and singular supports to translate algebraic finiteness into geometric conditions, proving the main equivalence via the jet-scheme behavior of zero-dimensional varieties. They illustrate the framework with Virasoro and affine examples, demonstrating how $C_2$-cofiniteness aligns with classical notions like reducibility in Virasoro minimal models and integrability for affine algebras. The results provide a natural, geometric understanding of finiteness in vertex algebras and offer tools for studying a broad class of algebras, including $W$-algebras, through associated varieties and jet schemes.

Abstract

We show that a finitely strongly generated, non-negatively graded vertex algebra $V$ is $C_2$-cofinite if and only if it is lisse in the sense of Beilinson, Feigin and Mazur. This shows that the $C_2$-cofiniteness is indeed a natural finiteness condition.

A remark on the $C_2$-cofiniteness condition on vertex algebras

TL;DR

This work establishes that for finitely strongly generated, non-negatively graded vertex algebras, -cofiniteness is equivalent to Beilinson–Feigin–Mazur lisse property, showing the two finiteness notions coincide in this setting. The authors develop and interconnect Li and standard filtrations, jet schemes, associated varieties, and singular supports to translate algebraic finiteness into geometric conditions, proving the main equivalence via the jet-scheme behavior of zero-dimensional varieties. They illustrate the framework with Virasoro and affine examples, demonstrating how -cofiniteness aligns with classical notions like reducibility in Virasoro minimal models and integrability for affine algebras. The results provide a natural, geometric understanding of finiteness in vertex algebras and offer tools for studying a broad class of algebras, including -algebras, through associated varieties and jet schemes.

Abstract

We show that a finitely strongly generated, non-negatively graded vertex algebra is -cofinite if and only if it is lisse in the sense of Beilinson, Feigin and Mazur. This shows that the -cofiniteness is indeed a natural finiteness condition.

Paper Structure

This paper contains 15 sections, 21 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. Graded Poisson vertex algebras associated with vertex algebras
  3. Vertex algebras and their modules
  4. Vertex Poisson algebras and their modules
  5. Functions on jet schemes of affine Poisson varieties are vertex Poisson algebras
  6. Ideals of vertex Poisson algebras
  7. Li filtration
  8. Standard filtration vs Li filtration
  9. Example: universal affine vertex algebras
  10. Associated varieties of vertex algebras and their modules
  11. Filtration of $V$-modules
  12. Associated varieties of vertex algebras and their modules
  13. Singular support of $V$-modules
  14. Example: Virasoro vertex algebras
  15. Example: affine vertex algebras

Key Result

Proposition 2.3.1

Let $R$ be a Poisson algebra. Then there is a unique vertex Poisson algebra structure on $R_{\infty}$ such that for $u,v\in R\subset R_{\infty}$.

Theorems & Definitions (42)

  • Proposition 2.3.1
  • proof
  • Remark 2.3.2
  • Lemma 2.4.1
  • proof
  • Corollary 2.4.2
  • Proposition 2.5.1
  • proof
  • Proposition 2.6.1
  • proof
  • ...and 32 more