Modular Categories and TQFTs Beyond Semisimplicity

Abstract

Vladimir Turaev discovered in the early years of quantum topology that the notion of modular category was an appropriate structure for building 3-dimensional Topological Quantum Field Theories (TQFTs for short) containing invariants of links in 3-manifolds such as Witten-Reshetikhin-Turaev ones. In recent years, generalized notions of modular categories, which relax the semisimplicity requirement, have been successfully used to extend Turaev's construction to various non-semisimple settings. We report on these recent developments in the domain, showing the richness of Vladimir's lineage.

arXiv comment26 pages, to appear in Topology and Geometry: A Collection of Essays Dedicated to Vladimir G. Turaev, ed. A. Papadopoulos, European Mathematical Society Publishing House, Berlin, 2021. Formulas for semisimple stabilization coefficients fixed in version 2

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Cited by in corpus (5)

• Extended TQFTs From Non-Semisimple Modular Categories
• The indecomposable objects in the center of Deligne's category $Rep(S_t)$
• Homological Construction of Quantum Representations of Mapping Class Groups
• Non-Semisimple 3-Manifold Invariants Derived From the Kauffman Bracket
• Kerler-Lyubashenko Functors on 4-Dimensional 2-Handlebodies