Centers of graded fusion categories

Abstract

Let $\mathcal{C}$ be a fusion category faithfully graded by a finite group $G$ and let $\mathcal{D}$ be the trivial component of this grading. The center $\mathcal{Z}\left(\mathcal{C}\right)$ of $\mathcal{C}$ is shown to be canonically equivalent to a $G$-equivariantization of the relative center ${\mathcal{Z}}_{\mathcal{D}}\left(\mathcal{C}\right)$. We use this result to obtain a criterion for $\mathcal{C}$ 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 the existence of zeroes in $S$-matrices of weakly integral modular categories.