Building vertex algebras from parts

TL;DR

This paper develops a cohomological framework to construct vertex algebras from modules parametrized by abelian groups, identifying obstructions and conditions for existence and uniqueness, especially for simple current extensions in conformal field theory.

Contribution

It introduces a cohomological approach to building vertex algebras from modules, clarifies obstructions, and characterizes conditions for unique simple current extensions.

Findings

Obstructions to locality are described by a cohomology element that vanishes under certain conditions.

Simple currents organized into odd order abelian groups always produce vertex algebras.

In conformal field theory cases, the obstruction reduces to symmetry properties of a bilinear form.

Abstract

Given a collection of modules of a vertex algebra parametrized by an abelian group, together with one dimensional spaces of composable intertwining operators, we assign a canonical element of the cohomology of an Eilenberg-Mac Lane space. This element describes the obstruction to locality, as the vanishing of this element is equivalent to the existence of a vertex algebra structure with multiplication given by our intertwining operators, and given existence, the structure is unique up to isomorphism. The homological obstruction reduces to an "evenness" problem that naturally vanishes for 2-divisible groups, so simple currents organized into odd order abelian groups always produce vertex algebras. Furthermore, in cases most relevant to conformal field theory (i.e., when we have well-behaved contragradients and tensor products), we obtain our spaces of intertwining operators naturally, and the evenness obstruction reduces to the question of whether the contragradient bilinear form on certain order 2 currents is symmetric or skew-symmetric. We show that if we are given a simple regular VOA and integral-weight modules parametrized by a group of even units in the fusion ring, then the direct sum admits the structure of a simple regular VOA, called the simple current extension, and this structure is unique up to isomorphism.