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Unit inclusion in a (nonsemisimple) braided tensor category and (noncompact) relative TQFTs

Benjamin Haïoun

Geometry & Topology 29 (2025) 2175–2216
Abstract

The inclusion of the unit in a braided tensor category 𝒱 induces a 1-morphism in the Morita 4-category of braided tensor categories  BrTens. We give criteria for the dualizability of this morphism.

When 𝒱 is a semisimple (resp. nonsemisimple) modular category, we show that the unit inclusion induces, under the cobordism hypothesis, a (resp. noncompact) relative 3-dimensional topological quantum field theory. Following Jordan, Reutter and Safronov, we conjecture that these relative field theories together with their bulk theories recover Witten–Reshetikhin–Turaev (resp. De Renzi–Gainutdinov–Geer–Patureau-Mirand–Runkel) theories, in a fully extended setting. In particular, we argue that these theories can be obtained by the cobordism hypothesis.

Keywords
TQFT, nonsemisimple, dualizability, cobordism hypothesis, braided tensor category
Mathematical Subject Classification
Primary: 18M20, 57R56
References
Publication
Received: 24 May 2023
Revised: 5 July 2024
Accepted: 9 August 2024
Published: 27 June 2025
Proposed: Nathalie Wahl
Seconded: Ciprian Manolescu, Ulrike Tillmann
Authors
Benjamin Haïoun
Institut de Mathématiques de Toulouse
Université Toulouse 3 Paul Sabatier
Toulouse
France

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