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Rigidity of $\mathbf{SU(2)}$ and $\mathbf{SO(3)}$ quantum representations of mapping class groups at prime levels

Pierre Godfard

TL;DR

This work proves infinitesimal rigidity for Witten–Reshetikhin–Turaev quantum representations of mapping class groups at prime levels on genus $g\ge7$ surfaces, using a multi-faceted approach that blends modular-category Ocneanu rigidity, partial homological stability in degree 1, and non-Abelian Hodge theory to lift and control deformations. The strategy reduces the cohomological rigidity problem to vanishing results on twisted moduli spaces $\overline{\mathcal{M}}_{g}^{n}(r)$, then lifts local deformation classes to genuine modular-functor deformations and shows they must be trivial. A key innovation is constructing low-genus deformation data from high-genus classes via gluing and embeddings, together with harmonic representatives to realize deformations, enabling Ocneanu rigidity to force triviality. The results link rigidity phenomena for mapping class groups with rigidity conjectures (Ivanov, fT, fD) and yield corollaries such as a streamlined proof of Fibonacci representations’ rigidity, highlighting the broader impact on understanding representation stability and rigidity in quantum topology.

Abstract

We prove the rigidity of Witten-Reshetikhin-Turaev $\mathrm{SU}(2)$ and $\mathrm{SO}(3)$ quantum representations of mapping class groups at all prime levels for closed surfaces of genus at least $7$. The proof relies on Ocneanu rigidity of modular categories and harmonic representatives in Hodge theory.

Rigidity of $\mathbf{SU(2)}$ and $\mathbf{SO(3)}$ quantum representations of mapping class groups at prime levels

TL;DR

This work proves infinitesimal rigidity for Witten–Reshetikhin–Turaev quantum representations of mapping class groups at prime levels on genus surfaces, using a multi-faceted approach that blends modular-category Ocneanu rigidity, partial homological stability in degree 1, and non-Abelian Hodge theory to lift and control deformations. The strategy reduces the cohomological rigidity problem to vanishing results on twisted moduli spaces , then lifts local deformation classes to genuine modular-functor deformations and shows they must be trivial. A key innovation is constructing low-genus deformation data from high-genus classes via gluing and embeddings, together with harmonic representatives to realize deformations, enabling Ocneanu rigidity to force triviality. The results link rigidity phenomena for mapping class groups with rigidity conjectures (Ivanov, fT, fD) and yield corollaries such as a streamlined proof of Fibonacci representations’ rigidity, highlighting the broader impact on understanding representation stability and rigidity in quantum topology.

Abstract

We prove the rigidity of Witten-Reshetikhin-Turaev and quantum representations of mapping class groups at all prime levels for closed surfaces of genus at least . The proof relies on Ocneanu rigidity of modular categories and harmonic representatives in Hodge theory.

Paper Structure

This paper contains 24 sections, 35 theorems, 42 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The main result on rigidity of SU(2) and SO(3) quantum representations at prime levels
  3. Outline of the proof
  4. Motivation: rigidity conjectures on mapping class groups
  5. Organization of the paper
  6. Acknowledgements
  7. Modular functors
  8. Topological definition of modular functor
  9. Levels for modular functors
  10. Twisted moduli spaces of curves
  11. Geometric definition of modular functors
  12. Comments on the relation between the 2 definitions of modular functor
  13. SO(3) and SU(2) modular functors at prime levels
  14. Main result
  15. Proof of Theorem \ref{['mainresult']}
  16. ...and 9 more sections

Key Result

Theorem 1

Let $\ell\geq 3$ be a prime number. Denote by $(\rho_g(\ul))_{g,\ul}$ the collection of $\mathrm{SU}(2)$ quantum representations at level $2\ell$ or $\mathrm{SO}(3)$ at level $\ell$. Then, for $g\geq 7$, $\rho_g$ is infinitesimally rigid, that is, $H^1(\mathrm{Mod}(S_g),\mathrm{ad}\, \rho_g)=0$.

Figures (1)

  • Figure 2.1: Gluing $S_{\pm}$ of $S$ along $\partial_{\pm}S$. The $\bullet$ represents the image of $1$ in the boundary parametrization.

Theorems & Definitions (99)

  • Theorem : \ref{['mainresult']}
  • Proposition : \ref{['propositioninjectivity']}
  • Proposition : \ref{['corollaryinjectivity']}
  • Proposition : \ref{['theoremuniversalsectionofH1']}
  • Remark 1.1
  • Remark 1.2
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • ...and 89 more