Solution space of 2+1 gravity on ${\bf R} \times T^2$ in Witten's connection formulation

TL;DR

This paper explores the structure of the solution space for 2+1 gravity on a spacetime with topology R times T^2, revealing its unique topological features and implications for quantization.

Contribution

It characterizes the solution space's topology, shows it forms a cotangent bundle over a non-Hausdorff base, and discusses novel quantization approaches.

Findings

Solution space is connected but not a manifold.

Removing measure-zero sets yields a cotangent bundle structure.

Quantization can map states supported on spacetime metrics to non-metric states.

Abstract

We investigate the space ${\cal M}$ of classical solutions to Witten's formulation of 2+1 gravity on the manifold ${\bf R} \times T^2$. ${\cal M}$ is connected, unlike the spaces of classical solutions in the cases where $T^2$ is replaced by a higher genus surface. Although ${\cal M}$ is neither Hausdorff nor a manifold, removing from ${\cal M}$ a set of measure zero yields a manifold which is naturally viewed as the cotangent bundle over a non-Hausdorff base space~${\cal B}$. We discuss the relation of the various parts of ${\cal M}$ to spacetime metrics, and various possibilities of quantizing~${\cal M}$. There exist quantizations in which the exponentials of certain momentum operators, when operating on states whose support is entirely on the part of ${\cal B}$ corresponding to conventional spacetime metrics, give states whose support is entirely outside this part of~${\cal B}$. Similar results hold when the gauge group ${\rm SO}_0(2,1)$ is replaced by ${\rm SU}(1,1)$.