Renormalized Hennings Invariants and 2 + 1-TQFTs

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 sl 2 to the setting of finite-dimensional nondegenerate 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.

On dualizability of braided tensor categories

“…Moving beyond the modular case, non-semisimple versions of the Witten–Reshetikhin–Turaev invariants have recently been constructed in a series of papers [BBG18, BCGP16, DGP18]. It is natural to expect that the TFT determined by a finite and factorizable but non-semisimple braided tensor category, regarded as an object of , can be related to those in a similar way.…”

DIAGRAMMATIC CONSTRUCTION OF REPRESENTATIONS OF SMALL QUANTUM $$ \mathfrak{sl} $$2

“…First introduced by Lusztig in [Lu90], Ū arises from a quantum deformation of the universal enveloping algebra of sl 2 with deformation parameter q = e 2πi/r , where 3 r ∈ Z is an odd integer called the level. The motivation behind this work is topological: indeed, Ū is a factorizable ribbon Hopf algebra, which means it can be used as an algebraic tool for building nonsemisimple TQFTs [DGP17]. At present, constructions rely either on the structure of Hopf algebras, or on some general categorical machinery [DGGPR19].…”

“…However, for even values of r, the Hopf algebra Ū is not ribbon, and although it admits a ribbon extension, the latter is not factorizable. In particular, the machinery of [DGP17] only produces quantum invariants of closed 3-manifolds, not TQFTs. This is why we turn our attention to the less-discussed odd level case.…”

Extended TQFTs From Non-Semisimple Modular Categories

“…The use of modified traces was subsequently integrated into Hennings' construction first [DGP17], and into Lyubashenko's construction later [DGGPR19], yielding TQFTs which extend Lyubashenko's mapping class group representations to symmetric monoidal functors defined over non-rigid categories of cobordisms [DGGPR20]. A discussion of the relation between this approach and the CGP one, in the case of quantum groups, can be found in [DGP18].…”