A unified construction of vertex algebras from infinite-dimensional Lie algebras

TL;DR

This paper presents a unified method to construct vertex algebras from various infinite-dimensional Lie algebras, simplifying and connecting previous approaches and broadening the understanding of their relationships.

Contribution

It introduces the concept of quasi vertex Lie algebra and establishes a canonical isomorphism linking modules of these Lie algebras to vertex algebra modules.

Findings

Unified construction of vertex algebras from multiple Lie algebra types

Introduction of quasi vertex Lie algebra concept

Establishment of an isomorphism between module categories

Abstract

In this paper, we give a unified construction of vertex algebras arising from infinite-dimensional Lie algebras, including the affine Kac-Moody algebras, Virasoro algebras, Heisenberg algebras and their higher rank analogs, orbifolds and deformations. We define a notion of what we call quasi vertex Lie algebra to unify these Lie algebras. Starting from any (maximal) quasi vertex Lie algebra $\mathfrak{g}$, we construct a corresponding vertex Lie algebra ${\mathfrak{g}}_0$, and establish a canonical isomorphism between the category of restricted $\mathfrak{g}$-modules and that of equivariant $\phi$-coordinated quasi $V_{{\mathfrak{g}}_0}$-modules, where $V_{{\mathfrak{g}}_0}$ is the universal enveloping vertex algebra of ${\mathfrak{g}}_0$. This unified all the previous constructions of vertex algebras from infinite-dimensional Lie algebras and shed light on the way to associate vertex algebras with Lie algebras.