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The Dehn Twist Action for Quantum Representations of Mapping Class Groups

Lukas Müller, Lukas Woike

TL;DR

The paper analyzes Dehn twist actions on spaces of conformal blocks arising from modular categories that need not be semisimple, using a robust framework of modular and ansular functors. It proves that for Dehn twists about non-separating curves, the order of the induced automorphism equals the ribbon twist order $|\theta|$, while for separating curves the order is $\min\{ |\theta_{\mathbb{A}^{\otimes g'}}|, |\theta_{\mathbb{A}^{\otimes g''}}| \}$ with $\mathbb{A}=\int_{X\in\mathcal{A}} X\otimes X^{\vee}$, linking topological data to intrinsic categorical invariants. The work further translates these orders into consequences for the Johnson kernel and Torelli group visibility, providing criteria in terms of the canonical end $\mathbb{A}$ (and its ribbon twist) for when these subgroups act trivially. By developing and exploiting the generalized ribbon element and the ansular extension, the authors connect the algebraic structure of the underlying category with mapping class group representations, shedding light on faithfulness, unitarity, and kernel phenomena in non-semisimple settings.

Abstract

We calculate the Dehn twist action on the spaces of conformal blocks of a not necessarily semisimple modular category. In particular, we give the order of the Dehn twists under the mapping class group representations of closed surfaces. For Dehn twists about non-separating simple closed curves, we prove that this order is the order of the ribbon twist, thereby generalizing a result that De Renzi-Gainutdinov-Geer-Patureau-Mirand-Runkel obtained for the small quantum group. In the separating case, we express the order using the order of the ribbon twist on monoidal powers of the canonical end. As an application, we prove that the Johnson kernels of the mapping class groups act trivially if and only if for the canonical end the ribbon twist and double braiding with itself are trivial. We give a similar result for the visibility of the Torelli groups.

The Dehn Twist Action for Quantum Representations of Mapping Class Groups

TL;DR

The paper analyzes Dehn twist actions on spaces of conformal blocks arising from modular categories that need not be semisimple, using a robust framework of modular and ansular functors. It proves that for Dehn twists about non-separating curves, the order of the induced automorphism equals the ribbon twist order , while for separating curves the order is with , linking topological data to intrinsic categorical invariants. The work further translates these orders into consequences for the Johnson kernel and Torelli group visibility, providing criteria in terms of the canonical end (and its ribbon twist) for when these subgroups act trivially. By developing and exploiting the generalized ribbon element and the ansular extension, the authors connect the algebraic structure of the underlying category with mapping class group representations, shedding light on faithfulness, unitarity, and kernel phenomena in non-semisimple settings.

Abstract

We calculate the Dehn twist action on the spaces of conformal blocks of a not necessarily semisimple modular category. In particular, we give the order of the Dehn twists under the mapping class group representations of closed surfaces. For Dehn twists about non-separating simple closed curves, we prove that this order is the order of the ribbon twist, thereby generalizing a result that De Renzi-Gainutdinov-Geer-Patureau-Mirand-Runkel obtained for the small quantum group. In the separating case, we express the order using the order of the ribbon twist on monoidal powers of the canonical end. As an application, we prove that the Johnson kernels of the mapping class groups act trivially if and only if for the canonical end the ribbon twist and double braiding with itself are trivial. We give a similar result for the visibility of the Torelli groups.
Paper Structure (10 sections, 20 theorems, 34 equations, 2 figures)

This paper contains 10 sections, 20 theorems, 34 equations, 2 figures.

Table of Contents

  1. Introduction and summary
  2. Building modular and ansular functors
  3. The generalized ribbon element
  4. The action of a Dehn twist about a non-separating simple closed curve
  5. The action of a Dehn twist about a separating simple closed curve
  6. Applications, special cases and examples
  7. Unitary structures
  8. Extending faithfulness beyond single Dehn twists
  9. Visibility of the Johnson kernel
  10. Visibility of the Torelli group

Key Result

Proposition 3.1

For a ribbon Grothendieck-Verdier category $\mathcal{A}$ in ${\mathsf{Rex}}^{\mathsf{f}}$, we have $\nu \in Z(B)$ and This number is a uniform bound for the images of all Dehn twists under the handlebody group representations provided by the ansular functor $\widehat{\mathcal{A}}$.

Figures (2)

  • Figure 1: The Dehn twists generating the subgroup $\mathbb{Z}^g \subset {\mathsf{Map}}(\Sigmait_g)$ in Proposition \ref{['propzg']}
  • Figure 2: The Dehn twists producing the bounding pair map for a genus two surface with two boundary components (one incoming, one outgoing) labeled by $X$. The simple closed curves for the Dehn twists are indicated in blue. The arrow indicates a gluing.

Theorems & Definitions (40)

  • Proposition 3.1
  • proof
  • Theorem 4.1
  • Remark 4.2
  • Lemma 4.3
  • proof : Proof of Lemma \ref{['lemmablockforH']}
  • Corollary 4.4
  • Lemma 4.5
  • proof
  • Lemma 4.6
  • ...and 30 more