Centers of graded fusion categories

TL;DR

This paper investigates the structure of the center of graded fusion categories, providing criteria for group-theoretical properties, analyzing Tambara-Yamagami categories, and exploring zeroes in S-matrices of modular categories.

Contribution

It establishes a canonical equivalence for the center of graded fusion categories and applies it to classify and find new modular categories, including criteria for group-theoreticality.

Findings

Center Z(C) is equivalent to a G-equivariantization of Z_D(C)

Criteria for C to be group-theoretical derived from the center structure

New series of modular categories identified from Tambara-Yamagami categories

Abstract

Let C be a fusion category faithfully graded by a finite group G and let D be the trivial component of this grading. The center Z(C) of C is shown to be canonically equivalent to a G-equivariantization of the relative center Z_D(C). We use this result to obtain a criterion for C to be group-theoretical and apply it to Tambara-Yamagami fusion categories. We also find several new series of modular categories by analyzing the centers of Tambara-Yamagami categories. Finally, we prove a general result about existence of zeroes in S-matrices of weakly integral modular categories.