Twisted modules and $G$-equivariantization in logarithmic conformal field theory

TL;DR

This paper explores the structure of modules in logarithmic conformal field theory, demonstrating how $V^G$-modules relate to twisted $V$-modules and establishing a braided $G$-crossed category framework.

Contribution

It generalizes existing results by removing the assumptions of rigidity and semisimplicity, and applies these to the orbifold rationality problem in vertex operator algebras.

Findings

Every $V^G$-module with a unital $V$-action decomposes into twisted $V$-modules.

The category of twisted $V$-modules forms a braided $G$-crossed supercategory.

$V^G$ is strongly rational if $V$ is strongly rational and $V^G$ is $C_2$-cofinite.

Abstract

A two-dimensional chiral conformal field theory can be viewed mathematically as the representation theory of its chiral algebra, a vertex operator algebra. Vertex operator algebras are especially well suited for studying logarithmic conformal field theory (in which correlation functions have logarithmic singularities arising from non-semisimple modules for the chiral algebra) because of the logarithmic tensor category theory of Huang, Lepowsky, and Zhang. In this paper, we study not-necessarily-semisimple or rigid braided tensor categories $\mathcal{C}$ of modules for the fixed-point vertex operator subalgebra $V^G$ of a vertex operator (super)algebra $V$ with finite automorphism group $G$. The main results are that every $V^G$-module in $\mathcal{C}$ with a unital and associative $V$-action is a direct sum of $g$-twisted $V$-modules for possibly several $g\in G$, that the category of all such twisted $V$-modules is a braided $G$-crossed (super)category, and that the $G$-equivariantization of this braided $G$-crossed (super)category is braided tensor equivalent to the original category $\mathcal{C}$ of $V^G$-modules. This generalizes results of Kirillov and M\"{u}ger proved using rigidity and semisimplicity. We also apply the main results to the orbifold rationality problem: whether $V^G$ is strongly rational if $V$ is strongly rational. We show that $V^G$ is indeed strongly rational if $V$ is strongly rational, $G$ is any finite automorphism group, and $V^G$ is $C_2$-cofinite.