Mapping class group representations and Morita classes of algebras

TL;DR

This paper investigates the relationship between modular fusion categories and mapping class group representations, showing that irreducibility implies a unique Morita class of simple non-degenerate algebras, with implications for quantum gravity.

Contribution

It proves that irreducible mapping class group representations in certain categories imply a unique Morita class of algebras, strengthening previous results under stricter conditions.

Findings

Irreducible representations imply a unique Morita class of algebras.

The result improves upon previous work by Andersen and Fjelstad.

Motivated by questions in three-dimensional quantum gravity.

Abstract

A modular fusion category C allows one to define projective representations of the mapping class groups of closed surfaces of any genus. We show that if all these representations are irreducible, then C has a unique Morita-class of simple non-degenerate algebras, namely that of the tensor unit. This improves on a result by Andersen and Fjelstad, albeit under stronger assumptions. One motivation to look at this problem comes from questions in three-dimensional quantum gravity.