(2+1)-Dimensional Quantum Gravity: Case of Torus Universe

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Abstract
The (2+1)-dimensional pure Einstein gravity is studied in the canonical ADM formalism, assuming that the spatial surface is closed and compact. In the torus case for the spatial manifold we can do an explicit analysis. Owing to the constraints, the dynamical variables are reduced to the moduli parameters of the 2-surface. Upon quantization, the system becomes a quantum mechanics of moduli parameters in a curved space endowed with the Weil-Peterson metric. The superspace, on which the wave function of universe is defined, turns out to be the fundamental region in the moduli space. The solution of the Wheeler-DeWitt equation is explicitly given as the Maass form which is perfectly regular in the superspace.

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