Fermionic 6j-symbols in superfusion categories (Q1643112)

A supercategory is a category enriched over the category of super vector spaces with morphisms even linear mappings between them. A superfusion category over an algebraically closed field \(k\) of characteristic \(0\) is a semisimple rigid monoidal supercategory with finite many simple objects and finite dimensional superspaces morphisms. Using a construction in [\textit{J. Brundan} and \textit{A. P. Ellis}, Commun. Math. Phys. 351, No. 3, 1045--1089 (2017; Zbl 1396.17012)] enables one to describe the underlying fusion category of a superfusion category. The first main result of this paper is an explicit formula for the \(6j\)-symbols of the underlying fusion category in terms of the fermionic \(6j\)-symbols of the superfusion category, where these \(6j\)-symbols are shown to satisfy the pentagon equation for a monoidal category. A possible definition of the \(\pi\)-Grothendieck ring of a superfusion category is also suggested in [loc. cit.]. The second main result is a version of Ocneanu rigidity for superfusion categories claiming that the number of superfusion categories up to superequivalence is countable and the number of superfusion categories up to superequivalence with a given \(\pi\)-Grothendieck ring is finite.