Modular Categories of Dimension $p^3m$ with $m$ Square-Free

TL;DR

This paper classifies modular categories of specific dimensions, revealing that most are pointed except for certain cases related to orthogonal quantum groups, and also classifies related metaplectic categories.

Contribution

It provides a complete classification of modular categories of dimension p^3m with p prime and m square-free, including new insights into categories related to quantum groups.

Findings

All categories with odd p are pointed.

Categories with p=2 relate to orthogonal quantum groups at roots of unity.

Classified even metaplectic modular categories with specific fusion rules.

Abstract

We give a complete classification of modular categories of dimension $p^3m$ where $p$ is prime and $m$ is a square-free integer. When $p$ is odd, all such categories are pointed. For $p=2$ one encounters modular categories with the same fusion ring as orthogonal quantum groups at certain roots of unity, namely $SO(2m)_2$. As an immediate step we classify a more general class of so-called even metaplectic modular categories with the same fusion rules as $SO(2N)_2$ with $N$ odd.