Hennings TQFTs for Cobordisms Decorated With Cohomology Classes

TL;DR

The paper constructs a braided monoidal TQFT $J_H$ on the category $3\mathrm{Cob}^G$ of connected framed cobordisms decorated with $G$-cohomology data, derived from a factorizable ribbon Hopf $G$-bialgebra $H$. It shows $J_H$ extends Kerler–Lyubashenko TQFTs in the trivial decoration case and connects with non-semisimple CGP-type invariants in decorated settings, including a detailed quantum $\mathfrak{sl}_2$ example with $G=\mathbb{C}/2\mathbb{Z}$. The construction uses a diagrammatic model via $G$-labeled top tangles in labeled handlebodies and a bead-insertion calculus to define morphisms, ensuring invariance under framed Kirby moves and functoriality. A homological perspective is proposed, seeking a model that encodes cohomology-dependent representations of mapping class groups, with a concrete isomorphism in the quantum $sl_2$ case linking to twisted homology representations. This work thus provides a first step toward a homological, decorated, non-semisimple TQFT framework and offers a concrete, integrality-friendly instance in the $sl_2$ setting.

Abstract

Starting from an abelian group $G$ and a factorizable ribbon Hopf $G$-bialgebra $H$, we construct a TQFT $J_H$ for connected framed cobordisms between connected surfaces with connected boundary decorated with cohomology classes with coefficients in $G$. When restricted to the subcategory of cobordisms with trivial decorations, our functor recovers a special case of Kerler-Lyubashenko TQFTs, namely those associated with factorizable ribbon Hopf algebras. Our result is inspired by the work of Blanchet-Costantino-Geer-Patureau, who constructed non-semisimple TQFTs for admissible decorated cobordisms using the unrolled quantum group of $\mathfrak{sl}_2$, and by that of Geer-Ha-Patureau, who reformulated the underlying invariants of admissible decorated $3$-manifolds using ribbon Hopf $G$-coalgebras. Our work represents the first step towards a homological model for non-semisimple TQFTs decorated with cohomology classes that appears in a conjecture by the first two authors.

Figures (1)

• Figure 1: Generators of $\mathrm{Tan}^G$, with $\alpha, \beta, \gamma \in G$.

Theorems & Definitions (36)

• Theorem 1.1
• Corollary 1.2
• Remark 2.1
• Remark 2.2
• Remark 2.3
• Proposition 2.4
• proof
• Remark 2.5
• Remark 3.1
• Proposition 3.2