Norm estimates and asymptotic faithfulness of the quantum $SU(n)$ representations of the mapping class groups

TL;DR

This paper provides a direct proof of the asymptotic faithfulness of quantum $SU(n)$ representations of mapping class groups, utilizing peak sections in Kodaira embedding, and offers norm estimates for parallel transport on the Verlinde bundle.

Contribution

It introduces a new direct proof method for asymptotic faithfulness using peak sections, differing from previous Toeplitz operator and skein theory approaches.

Findings

Proof of asymptotic faithfulness using Kodaira embedding

Norm estimates for parallel transport on Verlinde bundle

Comparison with previous methods for faithfulness proof

Abstract

We give a direct proof for the asymptotic faithfulness of the quantum $SU(n)$ representations of the mapping class groups using peak sections in Kodaira embedding. We give also estimates on the norm of the parallell transport of the projective connection on the Verlinde bundle. The faithfulness has been proved earlier in [1] using Toeplitz operators of compact K\"ahler manifolds and in [10] using skein theory.