Modular Virasoro Vertex Algebras and Affine Vertex Algebras

TL;DR

This paper investigates modular Virasoro and affine vertex algebras over fields of characteristic p>2, classifying their irreducible modules and establishing their finiteness and cofiniteness properties.

Contribution

It introduces quotients of universal vertex algebras related to p-centers and explicitly classifies their irreducible modules, advancing understanding in modular vertex algebra theory.

Findings

Classified irreducible a0a0-graded modules via Zhu algebras

Proved these vertex algebras have finitely many irreducible modules

Established that these algebras are C_2-cofinite

Abstract

In this paper, we study Virasoro vertex algebras and affine vertex algebras over a general field of characteristic $p>2$. More specifically, we study certain quotients of the universal Virasoro and affine vertex algebras by ideals related to the $p$-centers of the Virasoro algebra and affine Lie algebras. Among the main results, we classify their irreducible $\mathbb{N}$-graded modules by explicitly determining their Zhu algebras and show that these vertex algebras have only finitely many irreducible $\mathbb{N}$-graded modules and they are $C_2$-cofinite.