Classification of super-modular categories by rank

TL;DR

This paper classifies low-rank super-modular categories, identifying unique categories up to rank 6 and extending classification techniques from modular to super-modular categories.

Contribution

It provides a complete classification of super-modular categories up to rank 6 and adapts existing modular category constraints for super-modular categories.

Findings

Exactly one non-split super-modular category of rank 2, 4, and 6

Classification of spin modular categories up to rank 11

Extension of Verlinde and Frobenius-Schur constraints to super-modular categories

Abstract

We pursue a classification of low-rank super-modular categories parallel to that of modular categories. We classify all super-modular categories up to rank=$6$, and spin modular categories up to rank=$11$. In particular, we show that, up to fusion rules, there is exactly one non-split super-modular category of rank $2,4$ and $6$, namely $PSU(2)_{4k+2}$ for $k=0,1$ and $2$. This classification is facilitated by adapting and extending well-known constraints from modular categories to super-modular categories, such as Verlinde and Frobenius-Schur indicator formulae.