Local Systems of Vertex Operators, Vertex Superalgebras and Modules

TL;DR

This paper develops a framework for understanding vertex (super)algebras through local systems of vertex operators, establishing their structures, modules, and applications to Lie (super)algebras, with results on rationality and generalized intertwining operators.

Contribution

It introduces the notion of local systems of vertex operators, linking modules and homomorphisms to vertex superalgebra structures, and applies this to Lie superalgebras and affine algebra modules.

Findings

Local systems of vertex operators form vertex superalgebras with modules.

Modules for vertex superalgebras correspond to homomorphisms into local systems.

Certain modules for Lie superalgebras have natural vertex superalgebra structures.

Abstract

We give an analogue for vertex operator algebras and superalgebras of the notion of endomorphism ring of a vector space by means of a notion of ``local system of vertex operators'' for a (super) vector space. We first prove that any local system of vertex operators on a (super) vector space $M$ has a natural vertex (super)algebra structure with $M$ as a module. Then we prove that for a vertex (operator) superalgebra $V$, giving a $V$-module $M$ is equivalent to giving a vertex (operator) superalgebra homomorphism from $V$ to some local system of vertex operators on $M$. As applications, we prove that certain lowest weight modules for some well-known infinite-dimensional Lie algebras or Lie superalgebras have natural vertex operator superalgebra structures. We prove the rationality of vertex operator superalgebras associated to standard modules for an affine algebra. We also give an analogue of the notion of the space of linear homomorphisms from one module to another for a Lie algebra by introducing a notion of ``generalized intertwining operators.'' We prove that $G(M^{1},M^{2})$, the space of generalized intertwining operators from one module $M^{1}$ to another module $M^{2}$ for a vertex operator superalgebra $V$, is a generalized $V$-module. Furthermore, we prove that for a fixed vertex operator superalgebra $V$ and