The non-Abelian Chern-Simons path integral on $M=\Sigma \times S^1$ in the torus gauge: a review

TL;DR

This paper reviews recent progress on evaluating the non-Abelian Chern-Simons path integral on 3-manifolds of the form  imes S^1 in the torus gauge, connecting it with Reshetikhin-Turaev invariants and proposing rigorous formulations.

Contribution

It provides explicit evaluation methods for the torus gauge fixed Chern-Simons path integral and compares results with Reshetikhin-Turaev invariants, advancing understanding in quantum topology.

Findings

Explicit evaluation of the path integral in certain cases

Agreement with Reshetikhin-Turaev invariants in special cases

Proposals for rigorous mathematical realization of the gauge-fixed path integral

Abstract

In the present paper we review the main results of a series of recent papers on the non-Abelian Chern-Simons path integral on $M=\Sigma \times S^1$ in the so-called "torus gauge". More precisely, we study the torus gauge fixed version of the Chern-Simons path integral expressions $Z(\Sigma \times S^1,L)$ associated to $G$ and $k \in N$ where $\Sigma$ is a compact, connected, oriented surface, $L$ is a framed, colored link in $\Sigma \times S^1$, and $G$ is a simple, simply-connected, compact Lie group. We demonstrate that the torus gauge approach allows a rather quick explicit evaluation of $Z(\Sigma \times S^1,L)$. Moreover, we verify in several special cases that the explicit values obtained for $Z(\Sigma \times S^1,L)$ agree with the values of the corresponding Reshetikhin-Turaev invariant. Finally, we sketch three different approaches for obtaining a rigorous realization of the torus gauge fixed CS path integral. It remains to be seen whether also for general $L$ the explicit values obtained for $Z(\Sigma \times S^1,L)$ agree with those of the corresponding Reshetikhin-Turaev invariant. If this is indeed the case then this could lead to progress towards the solution of several open questions in Quantum Topology.