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

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