Asymptotic Faithfulness of Quantum $\mathrm{Sp}(4)$ Mapping Class Group Representations

TL;DR

This paper proves that quantum Sp(4) mapping class group representations are asymptotically faithful, providing a novel example outside the well-studied A_n family, using generalized skein theory methods.

Contribution

It introduces the first asymptotic faithfulness result for a non-A_n family, specifically for quantum Sp(4), expanding understanding of quantum mapping class groups.

Findings

Asymptotic faithfulness established for quantum Sp(4) representations

First example outside A_n family showing this property

Method generalized from SU(2)_k skein representations

Abstract

We prove asymptotic faithfulness for the quantum $\mathrm{Sp}(4)$ mapping class group representation. This provides the first example of asymptotic faithfulness lying outside of the $A_n$ family. The methods used are generalized from the proof of asymptotic faithfulness for skein $SU(2)_k$ mapping class group representations. In short, for any noncentral mapping class a comparison vector is found which allows for the mapping class to be detected.