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

Renaud Detcherry, Ramanujan Santharoubane

TL;DR

This work analyzes the kernels of the SO$$(3)$$-WRT quantum representations $$\rho_p$$ of mapping class groups, focusing on whether the kernel is generated by $$p$$-th powers of Dehn twists. Employing the $$h$$-adic expansion, Johnson filtration, and skein-theoretic computations, the authors derive explicit kernel descriptions at low levels and containment results for the full representations: for $$g\ge 3, p\ge 5$$, $$\ker \rho_p\subseteq\langle p\text{th powers of Dehn twists},\text{ separating twists}\rangle$$, and for $$g\ge 6$$ with large $$p$$, $$\ker \rho_p\subseteq [J_2(\Sigma),J_2(\Sigma)]\,T_p$$. They prove $$\ker(\rho_{p,1})=[J_1(\Sigma),J_1(\Sigma)]\,T_p$$ and, under suitable conditions, $$\ker(\rho_{p,2})=[J_2(\Sigma),J_2(\Sigma)]\,T_p$$, while providing detailed module-structure results for the mod $$p$$ abelianizations of the Torelli and Johnson subgroups. The analysis connects quantum invariants, Casson theory, and Johnson filtrations, clarifying how low-level kernels reflect the algebraic structure of surface mapping class groups and their $$p$$-adic reductions. These findings advance understanding of the kernel landscape of quantum representations and illuminate the interplay between topological and quantum invariants in high-genus regimes.

Abstract

In this paper, we study the kernels of the $\mathrm{SO}(3)$-Witten-Reshetikhin-Turaev quantum representations $ρ_p$ of mapping class groups of closed orientable surfaces $Σ_g$ of genus $g.$ We investigate the question whether the kernel of $ρ_p$ 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 $\mathrm{Ker} \, ρ_p$ 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 $\mathrm{Ker} \, ρ_p$ is contained in the subgroup generated by the commutator subgroup of the Johnson subgroup and by $p$-th powers of Dehn twists.

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

TL;DR

This work analyzes the kernels of the SO-WRT quantum representations of mapping class groups, focusing on whether the kernel is generated by -th powers of Dehn twists. Employing the -adic expansion, Johnson filtration, and skein-theoretic computations, the authors derive explicit kernel descriptions at low levels and containment results for the full representations: for , , and for with large , . They prove and, under suitable conditions, , while providing detailed module-structure results for the mod abelianizations of the Torelli and Johnson subgroups. The analysis connects quantum invariants, Casson theory, and Johnson filtrations, clarifying how low-level kernels reflect the algebraic structure of surface mapping class groups and their -adic reductions. These findings advance understanding of the kernel landscape of quantum representations and illuminate the interplay between topological and quantum invariants in high-genus regimes.

Abstract

In this paper, we study the kernels of the -Witten-Reshetikhin-Turaev quantum representations of mapping class groups of closed orientable surfaces of genus We investigate the question whether the kernel of for prime is exactly the subgroup generated by -th powers of Dehn twists. We show that if and then is contained in the subgroup generated by -th powers of Dehn twists and separating twists, and if and is a large enough prime then is contained in the subgroup generated by the commutator subgroup of the Johnson subgroup and by -th powers of Dehn twists.
Paper Structure (14 sections, 27 theorems, 50 equations)

This paper contains 14 sections, 27 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic properties of $\mathrm{SO}(3)$-WRT representations
  4. Computations of $\rho_p$ on bounding pairs and separating twists
  5. Proof of Theorem \ref{['thm:main-thm1.2']}
  6. Structure of the abelianization of the Torelli and Johnson subgroups
  7. Irreducibility of some modular representations of $\mathrm{Sp}_{2g}(\mathbb{Z})$
  8. Simply intersecting pairs and the contraction map
  9. Casson invariant and quantum invariant
  10. $h$-adic expansion of quantum representations and Johnson filtration
  11. $h$-adic expansion of quantum representations
  12. Analysis of $\mathrm{Ker}(\rho_{p,1})$
  13. Analysis of $\mathrm{Ker}(\rho_{p,2})$
  14. Further comments

Key Result

Theorem 1.2

Let $p\geq 5$ be a prime, $\Sigma$ a closed surface of genus $g\geq 3,$ and $\rho_p$ be the $\mathrm{SO}(3)$-WRT quantum representation at level $p.$ Then where $J_1(\Sigma)$ is the Torelli subgroup of $\mathop{\mathrm{\mathrm{Mod}}}\nolimits(\Sigma)$ and $T_p$ is the subgroup generated by $p$-th powers of Dehn twists. In particular the kernel of $\rho_p$ is contained in $[J_1(\Sigma),J_1(\Sigma)

Theorems & Definitions (48)

  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Corollary 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 38 more