Renormalized Hennings Invariants and 2+1-TQFTs

TL;DR

This paper develops new non-semisimple 2+1-dimensional topological quantum field theories (TQFTs) using generalized Hennings invariants based on quantum groups, leading to novel mapping class group representations.

Contribution

It generalizes logarithmic Hennings invariants to non-semisimple Hopf algebras and constructs associated 2+1-TQFTs with mapping class group representations.

Findings

Constructed non-semisimple 2+1-TQFTs from generalized Hennings invariants.

Established TQFTs on a subcategory of cobordisms for factorizable Hopf algebras.

Connected the invariants to Lyubashenko's spaces and mapping class group representations.

Abstract

We construct non-semisimple $2+1$-TQFTs yielding mapping class group representations in Lyubashenko's spaces. 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 finite-dimensional 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.