The dimension block of a spherical fusion category

Abstract

Let $\mathcal{C}$ be a spherical fusion category with global dimension $D$. The Galois conjugates of the dimension character index a block of the $S$-matrix of the Drinfeld center, which we call the dimension block. We prove that for every character $\chi$ of $\mathrm{Gal}(\mathbb{Q}(D)/\mathbb{Q})$, the twisted sum ${\sum }_{\sigma }\chi (\sigma )/\sigma (D)$ has absolute value at most $e^{-1}\mathfrak{f}(\chi {)}^{-1/2}$, where $\mathfrak{f}(\chi )$ is the conductor of $\chi$ and $e$ is the order of the dimensional grading group of $\mathcal{C}$. As an application, we determine the possible global dimensions of spherical fusion categories below $\sqrt{5}$, extending the known classification up to $4\sqrt{3}/5$ of V.\ Ostrik and P.\ Etingof.

arXiv comment17 pages, auxiliary python file