Toward a homological (re)construction of TQFTs that also unifies themMPIM

by Martel Jules (Université Cergy-Pontoise)

Monday Apr 27, 2026, 11:00 AM → 12:00 PM Europe/Berlin

MPIM, Vivatsgasse, 7 - Lecture Hall (Max Planck Institute for Mathematics)

Quantum topology seminar

Topological quantum field theories (TQFTs) originate from an idea of Witten that quantum field theories could be studied from the perspective of topological states and topological transitions.

It was mathematically formalized by Atiyah as a nice linearization of a category of cobordisms. It was concretely realized by Reshetikhin and Turaev (RT), who developed a philosophy for constructing TQFTs based on representation theory (of quantum groups with parameter q evaluated at roots of unity) and, more generally, on monoidal categories. This framework was later generalized to allow input categories to be non-semisimple; the Kerler–Lyubashenko (KL) TQFT is an important example extending RT.

In this talk, I will present a new way to (re)construct TQFTs from a different perspective: using twisted homologies of configuration spaces (it's an ongoing program). I will motivate and present the constructions and it will be the occasion to review some joint works with Bigelow, De Renzi, Detcherry, or Faes (depending on time).

An important feature is that, by working with homologies with local coefficients in the Heisenberg ring of a surface, we construct an overlying functor “at q generic". It recovers (parts of) KL TQFTs associated with quantum groups once we evaluate the ring at roots of unity, and we have established this correspondence for quantum representations of mapping class groups.

For the case of knot invariants, if time permits, I will also review earlier joint work with S. Willetts, where we define, from this setup, his unifying invariant living in a generalization of Habiro’s ring. This invariant encompasses both semisimple and non-semisimple knot invariants.