Rigidity of $\mathbf{SU(2)}$ and $\mathbf{SO(3)}$ quantum representations of mapping class groups at prime levels

TL;DR

This paper proves the rigidity of certain quantum representations of mapping class groups at prime levels, using advanced mathematical tools like Ocneanu rigidity and Hodge theory, for surfaces of genus at least 7.

Contribution

It establishes the rigidity of $ ext{SU}(2)$ and $ ext{SO}(3)$ quantum representations at all prime levels for high-genus surfaces, extending previous results.

Findings

Rigidity holds for all prime levels

Applicable to surfaces of genus ≥ 7

Uses Ocneanu rigidity and Hodge theory

Abstract

We prove the rigidity of Witten-Reshetikhin-Turaev $\mathrm{SU}(2)$ and $\mathrm{SO}(3)$ quantum representations of mapping class groups at all prime levels for closed surfaces of genus at least $7$. The proof relies on Ocneanu rigidity of modular categories and harmonic representatives in Hodge theory.