Relationship of the Hennings and Chern-Simons Invariants For Higher Rank Quantum Groups

TL;DR

This paper demonstrates that the Hennings invariant for small quantum groups associated with simple Lie algebras at roots of unity coincides with the Chern-Simons invariant derived from quantum field theory on integer homology three-spheres, extending previous results.

Contribution

It establishes the equivalence of Hennings and Chern-Simons invariants for higher rank quantum groups at roots of unity on all integer homology three-spheres.

Findings

Hennings invariant matches the Chern-Simons invariant for simple Lie algebras.

The result generalizes previous work from SL(2) and SO(3) cases.

The equivalence holds for all integer homology three-spheres at roots of unity.

Abstract

The Hennings invariant for the small quantum group associated to an arbitrary simple Lie algebra at a root of unity is shown to agree with Jones- Witten-Reshetikhin-Turaev invariant arising from Chern-Simons filed theory for the same Lie algebra and the same root of unity on all integer homol- ogy three-spheres, at roots of unity where both are defined. This partially generalizes the work of Chen, et al. ([CYZ12, CKS09]) which relates the Hennings and Chern-Simons invariants for SL(2) and SO(3) for arbitrary rational homology three-spheres.