Orateur: Marco De Renzi (Université Paris 7)
Titre: Renormalized Hennings Invariants and TQFTs.
Résumé: Non-semisimple constructions in quantum topology produce strong invariants and TQFTs with unprecedented properties. A first non-semisimple generalization of Witten--Reshetikhin--Turaev invariants was obtained by Hennings. His construction relied directly on finite-dimensional unimodular ribbon Hopf algebras, rather than their category of representations. These invariants enabled Lyubashenko to build a modular functor out of every finite-dimensional factorizable ribbon Hopf algebra. Further attempts at extending these functors to TQFTs only produced partial constructions, as the vanishing of Hennings invariants in many crucial situations made it impossible to treat non-connected surfaces. We will show how to overcome these problems. In order to do so, we will first renormalize Hennings invariants through the use of modified traces, as Beliakova, Blanchet and Geer did in the special case of restricted quantum sl2. When the Hopf algebra is factorizable, we further show that the universal construction of Blanchet, Habegger, Masbaum and Vogel produces a fully monoidal TQFT which extends Lyubashenko’s modular functor.
This is joint work with Nathan Geer and Bertrand Patureau.

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