Renormalized Hennings Invariants and 2+1-TQFTs
Résumé
We extend Lyubashenko's non-semisimple modular functors to $2+1$-TQFTs. In order to do this, we first generalize Beliakova, Blanchet and Geer's logarithmic Hennings invariants based on quantum $\mathfrak{sl}_2$ to the setting of non-degenerate unimodular ribbon Hopf algebras. The tools used for this construction are a Hennings-augmented Reshetikhin-Turaev functor and modified traces. When the Hopf algebra is factorizable, we further show that the universal construction of Blanchet, Habegger, Masbaum and Vogel produces a $2+1$-TQFT on a not completely rigid monoidal subcategory of cobordisms. This provides a fully monoidal extension to non-connected surfaces of Kerler's non-semisimple TQFTs for connected surfaces.