Noncompact 3d TQFTs from non-semisimple modular categories

Abstract: Perhaps the most (mathematically) well understood 3d TQFT is that of Reshetikhin–Turaev. Famously, the input data for their construction is that of a semisimple modular tensor category (MTC). Attempts at generalizing this construction to the non-semisimple case date back to the 90's with work of Hennings, Lyubashenko and Kerler–Lyubashenko. However, only partial results were achieved. This was until De Renzi et al. defined a 3d TQFT from such non-semisimple modular categories. Importantly, they had to impose an admissibility condition on the cobordism categories they use. My work has been in the direction of defining a once-extended 3d TQFT from this data. However, Bartlett et al. proved that such TQFTs are classified by semisimple modular categories. We will investigate the most natural method of circumventing this. This will lead to the notion of noncompact TQFT. I will then proceed to talk about my work on constructing such a TQFT from the data of a (potentially) non-semisimple MTC, with an emphasis on the key ingredients of this construction. Time permitting, I will also discuss how to extract 3-manifold invariants and a modified trace from such a noncompact TQFT.

mathematical physicsalgebraic topologycategory theoryquantum algebra

Audience: researchers in the topic

Topological Quantum Field Theory Club (IST, Lisbon)

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