Classification of low-rank odd-dimensional modular categories

Abstract

We prove that any odd-dimensional modular category of rank at most 23 is pointed. We also show that an odd-dimensional modular category of rank 25 is either pointed, perfect, or equivalent to $\mathrm{Rep}(D^{\omega }({\mathbb{Z}}_7⋊{\mathbb{Z}}_3))$. Finally, we give partial classification results for modular categories of rank up to 73.

Final version, to appear in Journal of Algebra