CHERN-SIMONS QUANTIZATION OF (2+1)-ANTI-DE SITTER GRAVITY ON A TORUS

The Chern-Simons formulation of (2 + 1)-dimensional Einstein gravity with a negative cosmological constant is investigated when the spacetime has the topology R x T-2. The physical phase space is shown to be a direct product of two sub-phase spaces each of which is a non-Hausdorff manifold plus a set with non-zero co-dimensions. The spacetime geometrical interpretation of each point in the phase space is also given and we explain the 1 to 2 correspondence with the ADM formalism from the geometrical viewpoint. In quantizing this theory, we construct a 'modified phase space' which is a cotangent bundle on a torus. We also provide a modular invariant inner product and investigate the relation to the quantum theory which is directly related to the spinor representation of the ADM formalism.