On Frobenius-Schur exponent bounds

Here we study bounds on the Frobenius-Schur exponent of spherical fusion categories based on their global dimension generalizing bounds from the representation theory of finite-dimensional quasi-Hopf algebras. Our main result is that if the Frobenius-Schur exponent of a modular fusion category is a prime power for some prime integer $p$, then it is bounded by the norm of its global dimension when $p$ is odd, and four times the norm of its global dimension when $p=2$; these bounds are optimal. If one assumes in addition pseudounitarity, the categories achieving the optimal bound are completely described and we attain similar bounds and classifications for arbitrary spherical fusion categories. This proof includes an explicit classification of modular fusion categories of Frobenius-Perron dimension $p^5$; all examples which are not pointed are constructed explicitly from the Ising categories and the representation theory of extraspecial $p$-groups.