Fermionic 6$j$-symbols in superfusion categories
Robert Usher
TL;DR
The paper clarifies how superfusion categories relate to fusion categories over sVect by constructing the underlying fusion category and giving an explicit formula for its 6j-symbols in terms of fermionic 6j-symbols. It introduces the Pi-envelope and Pi-complete framework to mediate between super and bosonic data, and proves that every superfusion category is equivalent to a Pi-complete one. A precise transfer of associativity data is established: fermionic 6j-symbols translate into ordinary 6j-symbols with parity factors, ensuring the pentagon constraint persists in the underlying category. The authors define the pi-Grothendieck ring, establish a version of Ocneanu rigidity for superfusion categories, and illustrate the theory with examples including Ising-type and level-k C_k(q) categories, highlighting implications for fermionic TQFTs and categorical classifications.
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 $π$-Grothendieck ring of a superfusion category, and prove a version of Ocneanu rigidity for superfusion categories.