Quantum $SU(2)$ faithfully detects mapping class groups modulo center

Abstract

The Jones–Witten theory gives rise to representations of the (extended) mapping class group of any closed surface $Y$ indexed by a semi-simple Lie group $G$ and a level $k$. In the case $G=SU\left(2\right)$ these representations (denoted $V_A\left(Y\right)$) have a particularly simple description in terms of the Kauffman skein modules with parameter $A$ a primitive $4r$th root of unity ($r=k+2$). In each of these representations (as well as the general $G$ case), Dehn twists act as transformations of finite order, so none represents the mapping class group $\mathcal{ℳ}\left(Y\right)$ faithfully. However, taken together, the quantum $SU\left(2\right)$ representations are faithful on non-central elements of $\mathcal{ℳ}\left(Y\right)$. (Note that $\mathcal{ℳ}\left(Y\right)$ has non-trivial center only if $Y$ is a sphere with $0,1,$ or $2$ punctures, a torus with $0,1,$ or $2$ punctures, or the closed surface of genus $=2$.) Specifically, for a non-central $h\in \mathcal{ℳ}\left(Y\right)$ there is an $r_0\left(h\right)$ such that if $r\ge r_0\left(h\right)$ and $A$ is a primitive $4r$th root of unity then $h$ acts projectively nontrivially on $V_A\left(Y\right)$. Jones’ original representation ${\rho }_n$ of the braid groups $B_n$, sometimes called the generic $q$–analog–$SU\left(2\right)$–representation, is not known to be faithful. However, we show that any braid $h\ne id\in B_n$ admits a cabling $c=c_1,\dots ,c_n$ so that ${\rho }_N\left(c\left(h\right)\right)\ne id$, $N=c_1+\dots +c_n$.