Classification of pivotal tensor categories with fusion rules related to $SO(4)$

TL;DR

This paper classifies all semisimple tensor categories with fusion rules similar to $SO(4)$, showing they are parametrized by two complex numbers, always braided, with exactly 8 braidings.

Contribution

It provides a complete classification of tensor categories with $SO(4)$-like fusion rules, including their braiding structures, parametrized explicitly by two complex numbers.

Findings

Categories are classified by two non-zero complex parameters.

All such categories are braided.

Exactly 8 braidings exist for these categories.

Abstract

In this paper we classify all semisimple tensor categories with the same fusion rules as $\operatorname{Rep}(SO(4))$, or one of the associated truncations. We show that such categories are explicitly classified by two non-zero complex numbers. Furthermore we show these tensor categories are always braided, and there exist exactly 8 braidings.