Loop Representations for 2+1 Gravity on a Torus

TL;DR

This paper investigates the loop representation of 2+1 dimensional quantum gravity on a torus, analyzing the loop transform's properties and constructing isomorphic representations despite certain ambiguities.

Contribution

It provides a detailed comparison between loop and connection representations for 2+1 gravity on a torus, highlighting the construction of isomorphic loop representations and their limitations.

Findings

Loop transform is dense and not continuous in the connection representation.

Isomorphic loop representations can be constructed with care.

Ambiguities remain in associating functions of loops with functions of connections.

Abstract

We study the loop representation of the quantum theory for 2+1 dimensional general relativity on a manifold, $M = {\cal T}^2 \times {\cal R}$, where ${\cal T}^2$ is the torus, and compare it with the connection representation for this system. In particular, we look at the loop transform in the part of the phase space where the holonomies are boosts and study its kernel. This kernel is dense in the connection representation and the transform is not continuous with respect to the natural topologies, even in its domain of definition. Nonetheless, loop representations isomorphic to the connection representation corresponding to this part of the phase space can still be constructed if due care is taken. We present this construction but note that certain ambiguities remain; in particular, functions of loops cannot be uniquely associated with functions of connections.