On TQFT representations of mapping class groups with boundary

TL;DR

This paper investigates TQFT representations of mapping class groups for surfaces with boundary, demonstrating irreducibility at roots of unity and Zariski density of images for transcendental quantum parameters.

Contribution

It proves irreducibility of TQFT representations at roots of unity and Zariski density of the images for transcendental quantum parameters, advancing understanding of these quantum group actions.

Findings

Representations are irreducible at prime roots of unity.

Images are Zariski dense for transcendental quantum parameters.

Provides new insights into the structure of quantum representations of mapping class groups.

Abstract

We study the TQFT mapping class group representations for surfaces with boundary associated with the $SU(2)$ gauge group, or equivalently the quantum group $U_q(\Sl(2))$. We show that at a prime root of unity, these representations are all irreducible. We also examine braid group representations for transcendental values of the quantum parameter, where we show that the image of every mapping class group is Zariski dense.