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 $χ$ of $\mathrm{Gal}(\mathbb{Q}(D)/\mathbb{Q})$, the twisted sum $\sum_σχ(σ)/σ(D)$ has absolute value at most $e<sup>{-1}\mathfrak{f}(χ)<sup>{-1/2}$, where $\mathfrak{f}(χ)$ is the conductor of $χ$ 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.