On odd-dimensional modular tensor categories

Abstract

We study odd-dimensional modular tensor categories and maximally nonself dual (MNSD) modular tensor categories of low rank. We give lower bounds for the ranks of modular tensor categories in terms of the rank of the adjoint subcategory and the order of the group of invertible objects. As an application of these results, we prove that odd-dimensional modular tensor categories of ranks 13 and 15 are pointed. In addition, we show that odd-dimensional tensor categories of ranks 17, 19, 21 and 23 are either pointed or perfect.