Twisted Modules of a Vertex Operator Algebra and Associators as Classifying Morphisms

TL;DR

This paper explores the structure of twisted modules of a Vertex Operator Algebra, showing how their monoidal category relates to 3-cocycles and $ ext{infty}$-group extensions, with implications for Moonshine and higher topos theory.

Contribution

It establishes a connection between twisted modules, 3-cocycles, and $ ext{infty}$-group extensions, providing a new perspective on their classification and structure.

Findings

The monoidal category of twisted modules reduces to a 2-group described by a 3-cocycle.

The 3-cocycle classifies an $ ext{infty}$-group extension of the symmetry group.

The skeletal 2-group with associator $ ext{alpha}$ presents the classified $ ext{infty}$-group extension.

Abstract

The monoidal category of twisted modules of a Vertex Operator Algebra $V$ is defined and reduced to its 2-group of invertible objects $G_\alpha$, which can be described by a 3-cocycle $\alpha$ on its 0-truncation $G$ with values in the group of units $A$ of the field of definition of $V$ serving as its associator. This cocycle also presents the classifying morphism of an $\infty$-group extension of $G$ by the delooping $BA$. Motivated by this, it is proven that the $\infty$-group extension classified by a 3-cocycle $\alpha$ is presented by the skeletal 2-group $G_\alpha$ with associator $\alpha$. The results are discussed in light of current developments in Moonshine and $(\infty,1)$-topos theory.