From simplicial Chern-Simons theory to the shadow invariant II

TL;DR

This paper develops a rigorous simplicial approach to non-Abelian Chern-Simons theory on specific 3-manifolds, providing explicit evaluations of Wilson loop observables that match known topological invariants.

Contribution

It introduces a new rigorous simplicial formulation of the Chern-Simons path integral and evaluates Wilson loop observables, connecting them to Turaev's shadow invariant.

Findings

Rigorous simplicial realization of Chern-Simons path integral.

Explicit non-perturbative evaluation of Wilson loop observables.

Agreement with Turaev's shadow invariant for certain links.

Abstract

This is the second 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|.