An analytic Approach to Turaev's Shadow Invariant

TL;DR

This paper extends a gauge fixing approach for Chern-Simons models on manifolds of the form Sigma x S^1, deriving a path integral formula for Wilson loop observables that connects to Turaev's shadow invariant.

Contribution

It introduces a novel extension of the torus gauge fixing method, enabling explicit non-perturbative evaluation of Wilson loop observables linked to Turaev's shadow invariant.

Findings

Derived a heuristic path integral formula for Wilson loops.

Explicit evaluation connects to face models of Turaev's shadow invariant.

Provides a non-perturbative approach to Chern-Simons invariants.

Abstract

In the present paper we extend the "torus gauge fixing approach" by Blau and Thompson (Nucl. Phys. B408(1):345--390, 1993) for Chern-Simons models with base manifolds M of the form M= \Sigma x S^1 in a suitable way. We arrive at a heuristic path integral formula for the Wilson loop observables associated to general links in M. We then show that the right-hand side of this formula can be evaluated explicitly in a non-perturbative way and that this evaluation naturally leads to the face models in terms of which Turaev's shadow invariant is defined.