On the kernel of $\mathrm{SO}(3)$-Witten-Reshetikhin-Turaev quantum representations

TL;DR

This paper investigates the kernels of $ ext{SO}(3)$-Witten-Reshetikhin-Turaev quantum representations of surface mapping class groups, providing bounds and containment results related to Dehn twists and Johnson subgroups for various genus and prime conditions.

Contribution

It offers new insights into the structure of kernels of quantum representations, showing their containment within specific subgroups under certain genus and prime constraints.

Findings

For genus g ≥ 3 and prime p ≥ 5, kernels are contained in subgroups generated by p-th powers of Dehn and separating twists.

For genus g ≥ 6 and sufficiently large primes p, kernels are contained in subgroups generated by the Johnson subgroup's commutator and p-th powers of Dehn twists.

Abstract

In this paper, we study the kernels of the $\mathrm{SO}(3)$-Witten-Reshetikhin-Turaev quantum representations $\rho_p$ of mapping class groups of closed orientable surfaces $\Sigma_g$ of genus $g.$ We investigate the question whether the kernel of $\rho_p$ for $p$ prime is exactly the subgroup generated by $p$-th powers of Dehn twists. We show that if $g\geq 3$ and $p\geq 5$ then $\mathrm{Ker} \, \rho_p$ is contained in the subgroup generated by $p$-th powers of Dehn twists and separating twists, and if $g\geq 6$ and $p$ is a large enough prime then $\mathrm{Ker} \, \rho_p$ is contained in the subgroup generated by the commutator subgroup of the Johnson subgroup and by $p$-th powers of Dehn twists.