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Article Dans Une Revue Memoirs of the American Mathematical Society Année : 2022

Non-semisimple extended topological quantum field theories

Résumé

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 s l 2 \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.

Dates et versions

hal-04230685 , version 1 (06-10-2023)

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Citer

Marco de Renzi. Non-semisimple extended topological quantum field theories. Memoirs of the American Mathematical Society, 2022, 277 (1364), ⟨10.1090/memo/1364⟩. ⟨hal-04230685⟩
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