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

Haïoun, Benjamin

doi:10.2140/gt.2025.29.2175

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

Benjamin Haïoun

Geometry & Topology 29 (2025) 2175–2216

DOI: 10.2140/gt.2025.29.2175

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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Volume 29, issue 4 (2025)