Reduced Phase Space of the first order Einstein Gravity on $\mathbf{R}\times T^2$

Abstract

Chern-Simons formulation of the 2+1 dimensional Einstein gravity with negative cosmological constant is investigated when the spacetime has the topology $\mathbf{R}\times T^2$. The physical phase space is shown to be a direct product of two sub-phase spaces having symplectic structures with opposite signs. Each sub-phase space is found to be a non-Hausdorff manifold plus a set of measure zero. Geometrical interpretation of each point in the phase space is also given. A prescription to quantize this phase space is proposed.

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