On the tensor structure of modules for compact orbifold vertex operator algebras

TL;DR

This paper explores the tensor category structure of modules for fixed-point vertex operator subalgebras under compact group actions, establishing conditions under which these categories relate to group representation categories and applying results to specific algebra cases.

Contribution

It demonstrates that under certain conditions, modules form tensor categories equivalent to group representation categories, even without assuming rigidity, and applies this to vertex operator algebras and Virasoro algebra modules.

Findings

Modules generate tensor categories equivalent to Rep G.

Fusion rules match G-module intertwiner dimensions.

Virasoro modules at c=1 admit tensor category structures.

Abstract

Suppose $V^G$ is the fixed-point vertex operator subalgebra of a compact group $G$ acting on a simple abelian intertwining algebra $V$. We show that if all irreducible $V^G$-modules contained in $V$ live in some braided tensor category of $V^G$-modules, then they generate a tensor subcategory equivalent to the category $\mathrm{Rep}\,G$ of finite-dimensional representations of $G$, with associativity and braiding isomorphisms modified by the abelian $3$-cocycle defining the abelian intertwining algebra structure on $V$. Additionally, we show that if the fusion rules for the irreducible $V^G$-modules contained in $V$ agree with the dimensions of spaces of intertwiners among $G$-modules, then the irreducibles contained in $V$ already generate a braided tensor category of $V^G$-modules. These results do not require rigidity on any tensor category of $V^G$-modules and thus apply to many examples where braided tensor category structure is known to exist but rigidity is not known; for example they apply when $V^G$ is $C_2$-cofinite but not necessarily rational. When $V^G$ is both $C_2$-cofinite and rational and $V$ is a vertex operator algebra, we use the equivalence between $\mathrm{Rep}\,G$ and the corresponding subcategory of $V^G$-modules to show that $V$ is also rational. As another application, we show that a certain category of modules for the Virasoro algebra at central charge $1$ admits a braided tensor category structure equivalent to $\mathrm{Rep}\,SU(2)$, up to modification by an abelian $3$-cocycle.