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

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.