Loop Variables in Topological Gravity

TL;DR

This paper explores the connection between covariant and canonical loop variables in topological gravity models, performing canonical quantization in different topologies and gauge groups to deepen understanding of quantum gravity formulations.

Contribution

It provides a detailed analysis of loop variables in topological gravity, including canonical quantization in 2+1 and 3+1 dimensions with specific gauge groups and topologies.

Findings

Canonical quantization achieved for models with SO(2,1) and SO(3,1) gauge groups.

Analysis of loop variables in different topologies $T^3$ and $S^2\times S^1$.

Clarification of the relationship between covariant and canonical formulations.

Abstract

We examine the relationship between covariant and canonical (Ashtekar/Rovelli/Smolin) loop variables in the context of BF type topological field theories in 2+1 and 3+1 dimensions, with respective gauge groups SO(2,1) and SO(3,1). The latter model can be considered as the simplest topological gravity theory in 3+1 dimensions. We carry out the canonical quantization of this model in both the connection and loop representations, for the two spatial topologies $T^3$ and $S^2\times S^1$.