Global structure of Witten's 2+1 gravity on ${\bf R}\times T^2$

TL;DR

This paper analyzes the classical solution space of Witten's 2+1 gravity on a manifold with topology R times T^2, revealing its complex structure and discussing implications for quantization.

Contribution

It characterizes the global structure of the solution space, showing it is connected but non-Hausdorff, and explores how this impacts approaches to quantizing the theory.

Findings

Solution space is connected but not a manifold.

Removing measure-zero sets yields a cotangent bundle structure.

Discusses potential pathways for quantization.

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, but neither Hausdorff nor a manifold. However, removing from ${\cal M}$ a set of measure zero yields a connected manifold which is naturally viewed as the cotangent bundle over a non-Hausdorff base space. Avenues towards quantizing the theory are discussed in view of the relation between spacetime metrics and the various parts of~${\cal M}$. (Contribution to the proceedings of the Lanczos Centenary Conference, Raleigh, NC, December 12--17, 1993.)