Geometric Structures and Loop Variables in (2+1)-Dimensional Gravity

TL;DR

This paper reviews how geometric structures relate to loop variables in (2+1)-dimensional gravity, exploring their potential for reconstructing geometry and implications for quantum gravity quantization methods.

Contribution

It highlights the connection between metric formulation and loop observables, proposing a method to reconstruct geometry from loop variables in (2+1)D gravity.

Findings

Loop variables can encode geometric information.

Reconstruction of geometry from loop variables is feasible.

Implications for covariant canonical quantum gravity approaches.

Abstract

This paper is a review of the relationship between the metric formulation of (2+1)-dimensional gravity and the loop observables of Rovelli and Smolin. I emphasize the possibility of reconstructing the geometry, via the theory of geometric structures, from the values of the loop variables. I close with a brief discussion of implications for quantization, particularly for covariant canonical approaches to quantum gravity.