Mapping Class Group Representations From Non-Semisimple TQFTs

Abstract

In [arXiv:1912.02063], we constructed 3-dimensional Topological Quantum Field Theories (TQFTs) using not necessarily semisimple modular categories. Here, we study projective representations of mapping class groups of surfaces defined by these TQFTs, and we express the action of a set of generators through the algebraic data of the underlying modular category $\mathcal{C}$. This allows us to prove that the projective representations induced from the non-semisimple TQFTs of [arXiv:1912.02063] are equivalent to those obtained by Lyubashenko via generators and relations in [arXiv:hep-th/9405167]. Finally, we show that, when $\mathcal{C}$ is the category of finite-dimensional representations of the small quantum group of ${\mathfrak{sl}}_2$, the action of all Dehn twists for surfaces without marked points has infinite order.

arXiv comment41 pages, minor corrections, Section 2.4 and Appendix C added

References in corpus (7)

• Modular group representations and fusion in logarithmic conformal field theories and in the quantum group center
• Invariants of 3-manifolds and projective representations of mapping class groups via quantum groups at roots of unity
• Quantum invariants of 3-manifolds via link surgery presentations and non-semi-simple categories
• Genealogy of Nonperturbative Quantum-Invariants of 3-Manifolds: The Surgical Family
• 3-Dimensional TQFTs From Non-Semisimple Modular Categories
• SL(2,Z)-action for ribbon quasi-Hopf algebras
• Projective objects and the modified trace in factorisable finite tensor categories

Cited by in corpus (11)

• 3-Dimensional TQFTs From Non-Semisimple Modular Categories
• Modular Categories and TQFTs Beyond Semisimplicity
• Pseudo-Hermitian Levin-Wen models from non-semisimple TQFTs
• Non-semisimple Levin-Wen Models and Hermitian TQFTs from quantum (super)groups
• Entanglement in Typical States of Chern-Simons Theory
• Diagrammatic Construction of Representations of Small Quantum ${\mathfrak{sl}}_2$
• A non-semisimple non-invertible symmetry
• Homological Construction of Quantum Representations of Mapping Class Groups
• Non-Semisimple 3-Manifold Invariants Derived From the Kauffman Bracket
• Mapping Class Group Representations Derived from Stated Skein Algebras
• Non-semisimple link and manifold invariants for symplectic fermions