Centers of graded fusion categories
Shlomo Gelaki, Deepak Naidu, Dmitri Nikshych
TL;DR
The paper establishes that for a faithfully $G$-graded fusion category $\mathcal C$ with trivial component $\mathcal D$, the center satisfies $\mathcal Z(\mathcal C) \cong \mathcal Z_{\mathcal D}(\mathcal C)^{G}$, parsing the center via a relative center endowed with a braided $G$-crossed structure. This yields a practical criterion: $\mathcal C$ is group-theoretical iff $\mathcal Z(\mathcal D)$ contains a $G$-stable Lagrangian subcategory. The authors then apply this framework to Tambara–Yamagami categories, describing their centers as equivariantizations, deriving modular data, and obtaining a criterion for when such categories are group-theoretical; they also construct non group-theoretical semisimple Hopf algebras via TY equivariantizations. Beyond TY, they build new modular categories from non-degenerate quadratic forms, explicitly computing fusion rules, $S$- and $T$-matrices, and central charges in families $\mathcal E(q,\pm)$. An independent appendix proves a categorical analogue of Burnside’s theorem, showing zeros occur in $S$-matrices of weakly integral modular categories, linking number-theoretic methods with modular data. The results collectively advance structural understanding of centers of graded fusion categories and furnish concrete modular-category examples with controlled group-theoretical properties.
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.