Exotic Fusion Categories and Their Modular Data

Abstract

The majority of known examples of fusion categories come directly from classical structures -- vector spaces, groups, representations, and the like. In recent years the technique of constructing fusion categories as endomorphisms on Cuntz algebras was developed and has already lead to completely new examples of fusion categories. Somewhat surprisingly, the fusion categories found seem to belong to infinite families. We push the Cuntz construction further and find more examples within two potentially infinite families, the near group fusion categories and the Haagerup-Izumi fusion categories. For all of the newly cataloged fusion categories, we also compute their modular data. In the case of the Haagerup-Izumi series, we find that all new examples satisfy the conjecture of [Evans and Gannon, 2010], which posits an unexpectedly simple form for the modular data in terms of certain bilinear forms. In the case of near group categories associated to an odd ordered abelian group, we find that the modular data of new examples also satisfy a similar conjecture (found in [Evans and Gannon, 2014]). When the order of the group is even, no such conjecture existed; we provide a new conjecture which predicts the modular data for all current examples.