Non-trivially graded self-dual fusion categories of rank $4$

TL;DR

This paper classifies self-dual spherical fusion categories of rank 4 with non-trivial grading, showing their Grothendieck ring structure and identifying their braided cases as tensor products of Fibonacci and pointed categories.

Contribution

It completes the classification of such fusion categories by determining their Grothendieck rings and their braided structures, revealing a specific tensor product form.

Findings

Grothendieck ring is isomorphic to Fib tensor Z[Z_2]

Braided categories are equivalent to Fibonacci times Vec_{Z_2}^ω

Classification of rank 4 self-dual spherical fusion categories

Abstract

Let $\mathcal{C}$ be a self-dual spherical fusion categories of rank $4$ with non-trivial grading. We complete the classification of Grothendieck ring $K(\mathcal{C})$ of $\mathcal{C}$; that is, we prove that $K(\mathcal{C})\cong Fib\otimes\mathbb{Z}[\mathbb{Z}_2]$, where $Fib$ is the Fibonacci fusion ring and $\mathbb{Z}[\mathbb{Z}_2]$ is the group ring on $\mathbb{Z}_2$. In particular, if $\mathcal{C}$ is braided then it is equivalent to $\textbf{Fib}\boxtimes\textbf{Vec}_{\mathbb{Z}_2}^{\omega}$ as fusion categories, where $\textbf{Fib}$ is a Fibonacci category and $\textbf{Vec}_{\mathbb{Z}_2}^{\omega}$ is a rank $2$ pointed fusion category.