Fermionic 6$j$-symbols in superfusion categories

TL;DR

This paper explores superfusion categories by relating them to fusion categories over super vector spaces, providing explicit formulas for associators via 6j-symbols, and establishing foundational properties like the $\

Contribution

It introduces a construction of the underlying fusion category of a superfusion category and derives explicit 6j-symbol formulas for associators, advancing the mathematical understanding of superfusion categories.

Findings

Explicit formula for associator in superfusion categories

Construction of the underlying fusion category from superfusion categories

Proof of Ocneanu rigidity for superfusion categories

Abstract

We describe how the study of superfusion categories (roughly speaking, fusion categories enriched over the category of super vector spaces) reduces to that of fusion categories over sVect, in the sense of Drinfeld, Gelaki, Nikshych, and Ostrik. Following Brundan and Ellis, we give the construction of the underlying fusion category of a superfusion category, and give an explicit formula for the associator in this category in terms of 6$j$-symbols. We give a definition of the $\pi$-Grothendieck ring of a superfusion category, and prove a version of Ocneanu rigidity for superfusion categories.