A complete classification of unitary fusion categories tensor generated by an object of dimension $\frac{1 + \sqrt{5}}{2}$

TL;DR

This paper classifies all unitary fusion categories generated by an object of the golden ratio dimension, showing they are related to Fibonacci categories and specific subfactor constructions.

Contribution

It provides a complete classification of such categories and introduces a classification of certain finite unitarizable quotients of Fibonacci tensor powers.

Findings

All categories arise as wreath products of Fibonacci or dual $2D2$ subfactor categories.

Classification of finite unitarizable quotients of $ ext{Fib}^{*N}$ with symmetry.

New connections between fusion categories and subfactor theory.

Abstract

In this paper we give a complete classification of unitary fusion categories $\otimes$-generated by an object of dimension $\frac{1 + \sqrt{5}}{2}$. We show that all such categories arise as certain wreath products of either the Fibonacci category, or of the dual even part of the $2D2$ subfactor. As a by-product of proving our main classification result we produce a classification of finite unitarizable quotients of $\operatorname{Fib}^{*N}$ satisfying a certain symmetry condition.