Non-Semisimple Extended Topological Quantum Field Theories

Abstract

We develop the general theory for the construction of Extended Topological Quantum Field Theories (ETQFTs) associated with the Costantino-Geer-Patureau quantum invariants of closed 3-manifolds. In order to do so, we introduce relative modular categories, a class of ribbon categories which are modeled on representations of unrolled quantum groups, and which can be thought of as a non-semisimple analogue to modular categories. Our approach exploits a 2-categorical version of the universal construction introduced by Blanchet, Habegger, Masbaum, and Vogel. The 1+1+1-EQFTs thus obtained are realized by symmetric monoidal 2-functors which are defined over non-rigid 2-categories of admissible cobordisms decorated with colored ribbon graphs and cohomology classes, and which take values in 2-categories of complete graded linear categories. In particular, our construction extends the family of graded 2+1-TQFTs defined for the unrolled version of quantum ${\mathfrak{sl}}_2$ by Blanchet, Costantino, Geer, and Patureau to a new family of graded ETQFTs. The non-semisimplicity of the theory is witnessed by the presence of non-semisimple graded linear categories associated with critical 1-manifolds.

172 pages, 46 figures, entirely rewritten, several appendices added

References in corpus (6)

• Non-Semisimple Quantum Invariants and TQFTs from Small and Unrolled Quantum Goups
• Logarithmic Hennings invariants for restricted quantum sl(2)
• Renormalized Hennings Invariants and 2+1-TQFTs
• Projective objects and the modified trace in factorisable finite tensor categories
• Kuperberg and Turaev-Viro Invariants in Unimodular Categories
• Quantum Invariants of 3-Manifolds Arising from Non-Semisimple Categories

Cited by in corpus (10)

• 3-Dimensional TQFTs From Non-Semisimple Modular Categories
• Modular Categories and TQFTs Beyond Semisimplicity
• Modified graded Hennings invariants from unrolled quantum groups and modified integral
• Non-semisimple Levin-Wen Models and Hermitian TQFTs from quantum (super)groups
• Categories of Weight Modules for Unrolled Restricted Quantum Groups at Roots of Unity
• Homological Construction of Quantum Representations of Mapping Class Groups
• Kerler-Lyubashenko Functors on 4-Dimensional 2-Handlebodies
• Hennings TQFTs for Cobordisms Decorated With Cohomology Classes
• Stated skein algebras and their representations
• Anomaly-free TQFTs from the super Lie algebra $\mathfrak{sl}(2|1)$