Orateur
Description
TQFTs are the idea of Witten that we can study quantum field theories from the point of view of the topology of state spaces and of topological transitions. Mathematically, it was formalised by Atiyah as a linearization of a cobordism category. It was concretely realized by Reshetikhin--Turaev (RT) coupled with Blanchet--Habbeger--Masbaum--Vogel universal construction. The RT philosophy relies on a diagrammatic model for cobordisms (based on knot diagrams) and uses modules on quantum groups (or more generally monoidal categories) to model linearly these diagrams. This has more recently been extended so to permit the utilization of non semisimple monoidal categories as input of the construction giving rise to a new generation of TQFTs, for which a nice example is the Kerler--Lyubashenko (KL) construction. These constructions need abstract algebra tools applied on diagrammatic representations of manifolds, and we will try to avoid this by using homology theories, allowing a more global definition of TQFTs.
In this talk I'll introduce a new philosophy to build TQFTs based on homology of configuration spaces. Representations of mapping class groups constitute an important
byproduct of TQFTs, while more natural ones can be built out of twisted homologies of configuration spaces of surfaces. We will show that from this latter new framework we can recover
step by step the properties of KL TQFTs associated with quantum groups and even a unifying framework. I'll stay introductory most of the talk but the constructions and their motivations will be the occasion to review joint works with: S. Bigelow, M. De Renzi, R. Detcherry or Q. Faes (depending on the time).