The Witt group of a braided Monoidal category

TL;DR

This paper introduces a Witt group for braided monoidal categories with duality, providing a new invariant for finite isocategorical groups and demonstrating categorical rigidity for groups under 64 elements.

Contribution

It develops the Witt group for certain braided monoidal categories and applies it to classify isocategorical groups and establish their rigidity.

Findings

Witt group explicitly described for braided fusion categories

Invariant for finite isocategorical groups established

All groups of order less than 64 are categorically rigid

Abstract

We develop the Witt group for certain braided monoidal categories with duality. In case of a braided fusion category over an algebraically closed field of characteristic zero, we explicitly describe this structure. We then use this description to prove that this tool provides an invariant for finite isocategorical groups. As an application, we show that all groups of order less than 64 are categorically rigid.