Integrality and Gauge Dependence of Hennings TQFTs

TL;DR

This paper constructs integral Topological Quantum Field Theories (TQFTs) from finite Hopf algebras, demonstrating their specialization to Hennings invariants and establishing their integrality over cyclotomic integers without relying on skein or knot polynomial calculations.

Contribution

It introduces a general combinatorial method to build integral TQFTs from Frobenius, double balanced Hopf algebras, extending to quantum groups and relating gauge transformations to TQFT equivalences.

Findings

Hennings invariants for quantum sl_2 are integral over cyclotomic integers.

Constructs TQFTs from index 2 extensions of Lusztig's small quantum groups.

Shows invariance under gauge transformations via explicit natural isomorphisms.

Abstract

We provide a general construction of integral TQFTs over a general commutative ring, $\mathbf{k}$, starting from a finite Hopf algebra over $\mathbf{k}$ which is Frobenius and double balanced. These TQFTs specialize to the Hennings invariants of the respective doubles on closed 3-manifolds. We show the construction applies to index 2 extensions of the Borel parts of Lusztig's small quantum groups for all simple Lie types, yielding integral TQFTs over the cyclotoic integers for surfaces with boundary. We further establish and compute isomorphisms of TQFT functors constructed from Hopf algebras that are related by a strict gauge transformation in the sense of Drinfeld. Formulas for the natural isomorphisms are given in terms of the gauge twist element. These results are combined and applied to show that the Hennings invariant associated to quantum-$sl_2$ takes values in the cyclotomic integers. Using prior results of Chen et al we infer integrality also of the Witten-Reshetikhin-Turaev $SO(3)$ invariant for rational homology spheres. As opposed to most other approaches the methods described in this article do not invoke calculations of skeins, knots polynomials, or representation theory, but follow a combinatorial construction that uses only the elements and operations of the underlying Hopf algebras.