The $F$-Symbols for Transparent Haagerup-Izumi Categories with $G={\mathbb{Z}}_{2n+1}$

Abstract

A fusion category is called transparent if the associator involving any invertible object is the identity map. For the Haagerup-Izumi fusion rings with $G={\mathbb{Z}}_{2n+1}$ (the ${\mathbb{Z}}_3$ case is the Haagerup fusion ring with six simple objects), the transparent ansatz reduces the number of independent $F$-symbols from order $\mathcal{O}(n^6)$ to $\mathcal{O}(n^2)$, rendering the pentagon identity practically solvable. Transparent Haagerup-Izumi fusion categories are thereby constructively classified up to $G={\mathbb{Z}}_9$, recovering all known Haagerup-Izumi fusion categories to this order, and producing new ones. Transparent Haagerup-Izumi fusion categories additionally satisfying $S_4$ tetrahedral invariance are further classified up to $G={\mathbb{Z}}_{15}$, and the explicit $F$-symbols for the unitary ones, including the Haagerup ${\mathcal{H}}_3$ fusion category, are compactly presented. The $F$-symbols for the Haagerup ${\mathcal{H}}_2$ fusion category are also presented. Going beyond, the transparent ansatz offers a viable course towards constructing novel fusion categories for new fusion rings.

arXiv comment25+9 pages, 1 table; v2: improved exposition; v3: adopted unitary gauge for pseudo-unitary fusion categories

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