On $G$--equivariant modular categories

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 $S{L}_2(Z)$ 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.

LaTeX,33 pages, many figures

References in corpus (3)

• Group Cohomology and Gauge Equivalence of some Twisted Quantum Doubles
• Fusion of twisted representations
• Twisted Modules over Vertex Algebras on Algebraic Curves

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• G-crossed Modularity of Symmetry-Enriched Topological Phases
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• Orbifold deconstruction: a computational approach
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• Representation theory of vertex operator algebras and orbifold conformal field theory
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• A New approach to the construction of braided T-categories
• Equivariant categories from categorical group actions on monoidal categories
• The construction of braided $T$-category via Yetter-Drinfeld-Long bimodules