On $G$--equivariant modular categories

TL;DR

This paper explores $G$-equivariant tensor categories, introduces an extended Verlinde algebra, and establishes $SL_2(Z)$ action, providing new tools for understanding orbifold categories and their tensor structures.

Contribution

It introduces an extended Verlinde algebra for $G$-equivariant categories and relates it to orbifold quotients, offering new computational methods for tensor products.

Findings

Defined an analog of $s,t$ matrices for the extended Verlinde algebra

Proved that invertible $s$-matrix induces an $SL_2(Z)$ action

Showed $s$-matrix interchanges tensor product with convolution product

Abstract

In this paper, we study $G$-equivariant tensor categories for a finite group $G$. These categories were introduced by Turaev under the name of $G$-crossed categories; the motivating example of such a category is the category of twisted modules over a vertex operator algebra $V$ with a finite group of automorphisms $G$. We discuss the notion of "orbifold quotient" of such a category (in the example above, this quotient is the category of modules over the subalgebra of invariants $V^G$). We introduce an extended Verlinde algebra for a $G$-equivariant tensor category and give a simple description of the Verlinde algebra of the orbifold category in terms of the extended Verlinde algebra of the original category. We define an analog of $s,t$ matrices for the extended Verlinde algebra and show that if $s$ is invertible, then these matrices define an action of $SL_2(Z)$ on the extended Verlinde algebra. We also show that the $s$-matrix interchanges tensor product with a much simpler product ("convolution product"), which can be used to compute the tensor product multiplicities.