On odd-dimensional modular tensor categories

TL;DR

This paper investigates the structure of odd-dimensional modular tensor categories, establishing bounds on their ranks and classifying certain low-rank cases as pointed or perfect, advancing understanding of their algebraic properties.

Contribution

It provides new lower bounds for the ranks of odd-dimensional modular tensor categories and classifies low-rank MNSD categories as pointed or perfect.

Findings

MNSD categories of ranks 13 and 15 are pointed

MNSD categories of ranks 17, 19, 21, 23 are either pointed or perfect

Lower bounds relate rank to the adjoint subcategory and invertible objects

Abstract

We study odd-dimensional modular tensor categories and maximally non-self 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 MNSD modular tensor categories of ranks 13 and 15 are pointed. In addition, we show that MNSD tensor categories of ranks 17, 19, 21 and 23 are either pointed or perfect.