From simplicial Chern-Simons theory to the shadow invariant I

TL;DR

This paper develops a rigorous simplicial approach to non-Abelian Chern-Simons theory on manifolds of the form Sigma x S1, introducing a new Wilson loop observable and confirming its agreement with Turaev's shadow invariant for certain links.

Contribution

It introduces a novel simplicial realization of the Chern-Simons path integral and explicitly evaluates Wilson loop observables, aligning with known topological invariants.

Findings

Rigorous simplicial model for Chern-Simons path integral

Explicit non-perturbative evaluation of Wilson loop observables

Agreement with Turaev's shadow invariant for specific links

Abstract

This is the first of a series of papers in which we introduce and study a rigorous "simplicial" realization of the non-Abelian Chern-Simons path integral for manifolds M of the form M = Sigma x S1 and arbitrary simply-connected compact structure groups G. More precisely, we introduce, for general links L in M, a rigorous simplicial version WLO_{rig}(L) of the corresponding Wilson loop observable WLO(L) in the so-called "torus gauge" by Blau and Thompson (Nucl. Phys. B408(2):345-390, 1993). For a simple class of links L we then evaluate WLO_{rig}(L) explicitly in a non-perturbative way, finding agreement with Turaev's shadow invariant |L|.