THE SOLUTION SPACE OF 2+1 GRAVITY ON R-X-T2 IN WITTEN CONNECTION FORMULATION

We investigate the space M of classical solutions of Witten's formulation of 2 + 1 gravity on the manifold R x T2. M is connected, unlike the spaces of classical solutions in the cases where T2 is replaced by a higher genus surface. Although M is neither Hausdorff nor a manifold. removing from M a set of measure zero yields a manifold which is naturally viewed as the cotangent bundle over a non-Hausdorff base space B. We discuss the relation of the various parts of M with spacetime metrics, and various possibilities of quantizing M. There exist quantizations in which the exponentials of certain momentum operators, when operating on states whose support is entirely on the part of B corresponding to conventional spacetime metrics, give states whose support is entirely outside this part of B. Similar results hold when the gauge group SO0(2, 1) is replaced by SU(1, 1).