A quantum Goldman bracket in 2+1 quantum gravity

TL;DR

This paper advances the understanding of quantum holonomies and Wilson loops in 2+1 quantum gravity by constructing a quantum Goldman bracket for intersecting loops, revealing how rerouted paths relate to integer points within parallelograms.

Contribution

It introduces a quantum Goldman bracket for intersecting loops in 2+1 quantum gravity, incorporating noncommutative holonomies and the role of integer points in path rerouting.

Findings

Holonomies and Wilson loops relate via phases depending on signed area.

Rerouted paths at intersections pass through integer points.

The quantum Goldman bracket captures loop interactions in quantum gravity.

Abstract

In the context of quantum gravity for spacetimes of dimension 2+1, we describe progress in the construction of a quantum Goldman bracket for intersecting loops on surfaces. Using piecewise linear paths in R^2 (representing loops on the spatial manifold, i.e. the torus) and a quantum connection with noncommuting components, we review how holonomies and Wilson loops for two homotopic paths are related by phases in terms of the signed area between them. Paths rerouted at intersection points with other paths occur on the r.h.s. of the Goldman bracket. To better understand their nature we introduce the concept of integer points inside the parallelogram spanned by two intersecting paths, and show that the rerouted paths must necessarily pass through these integer points.