On Haagerup-Izumi fusion categories

Abstract

For every odd $N$, we construct a Haagerup-Izumi (HI) category for $\mathbb Z/N$, using Barnes double-sine functions. Equivariantization gives near-group categories of type $((\mathbb Z/N)^2,N^2)$ for every odd $N$. We classify connected separable algebras up to Morita equivalence in such HI categories and describe their dual categories. These duals are often HI for noncyclic groups with nontrivial pointed associators. We prove the Evans-Gannon conjecture that the modular data of the center is always a smashed sum involving a finite metric group of order $N^2+4$ and recover this group and its quadratic form directly from the data of the HI category. We also give the HI equations and reconstruction in arbitrary characteristic and prove a Frobenius symmetry in characteristic two. Using this, we obtain the complete list of (untwisted, subfactor type) HI categories over $\mathbb{C}$ for odd $N\le 43$. Other applications include finding three pairwise non-Morita-equivalent $\mathbb Z/15$ categories whose centers have the same modular data. We show that the center of a $\mathbb Z/5$ HI category cannot be defined (i.e., admits no split ribbon form) over a cyclotomic field, disproving a conjecture of Davidovich-Hagge-Wang.