Symmetric Invariant Bilinear Forms on Modular Vertex Algebras

TL;DR

This paper develops a theory of invariant bilinear forms for modular vertex algebras in characteristic p, introducing a new bialgebra framework and extending existing vertex algebra theories to the modular setting.

Contribution

It introduces a bialgebra $ ext{H}$ and a modular version of Frenkel-Huang-Lepowsky's theory, providing explicit descriptions of invariant bilinear forms in the modular context.

Findings

Explicit description of invariant bilinear forms on $ ext{H}$-module vertex algebras

Application to affine and Virasoro vertex algebras

Extension of vertex algebra theory to characteristic p

Abstract

In this paper, we study contragredient duals and invariant bilinear forms for modular vertex algebras (in characteristic $p$). We first introduce a bialgebra $\mathcal{H}$ and we then introduce a notion of $\mathcal{H}$-module vertex algebra and a notion of $(V,\mathcal{H})$-module for an $\mathcal{H}$-module vertex algebra $V$. Then we give a modular version of Frenkel-Huang-Lepowsky's theory and study invariant bilinear forms on an $\mathcal{H}$-module vertex algebra. As the main results, we obtain an explicit description of the space of invariant bilinear forms on a general $\mathcal{H}$-module vertex algebra, and we apply our results to affine vertex algebras and Virasoro vertex algebras.