Einstein-Hilbert Path Integrals and Chern-Simons Integrals

TL;DR

This paper develops a method to compute Wilson Loop observables in quantum gravity using path integrals, linking Einstein-Hilbert and Chern-Simons theories, and demonstrating invariance under hyperlink equivalence.

Contribution

It introduces a novel approach to evaluate Wilson Loop observables via axial-gauge fixing and link diagram projections, connecting Einstein-Hilbert and Chern-Simons path integrals.

Findings

Wilson Loop observable computed from link diagrams.

Path integral limit relates Einstein-Hilbert and Chern-Simons theories.

Invariance under hyperlink equivalence established.

Abstract

A hyperlink is a finite set of non-intersecting simple closed curves in $\mathbb{R} \times \mathbb{R}^3$. We compute the Wilson Loop observable using a path integral with an Einstein-Hilbert action. Using axial-gauge fixing, we can write this path integral as the limit of a sequence of Chern-Simons integrals, studied earlier in our previous work on the Chern-Simons path integrals in $\mathbb{R}^3$. We will show that the Wilson Loop observable can be computed from a link diagram of a hyperlink, projected on a plane. Only crossings in the diagram will contribute to the path integral. Furthermore, we will show that it is invariant under an equivalence relation defined on the set of hyperlinks.