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

Haïoun, Benjamin¹

¹Institut de Mathématiques de Toulouse, Université Toulouse 3 Paul Sabatier, Toulouse, France

Geometry & topology, Tome 29 (2025) no. 4, pp. 2175-2216

Résumé

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.

Détail

DOI : 10.2140/gt.2025.29.2175

Keywords: TQFT, nonsemisimple, dualizability, cobordism hypothesis, braided tensor category

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