Abstract
Hamiltonian path integral quantization of the SO(2, 1)/SO(2) and SO(2, 1)/SO(1, 1) gauged WZW models is considered. The SO(2, 1)/SO(2) model yields the Liouville theory coupled to a free scalar field theory after eliminating the group coordinates in favor of their conjugate momenta. The SO(2, 1)/SO(1, 1) model, recently argued to be a two-dimensional black-hole solution to string theory, can also be studied in momentum space. The resulting negative-metric Liouville-like theory has indefinite cosmological constant, is coupled to a positive-metric free scalar field, and lives on a Minkowski world sheet. The scaling variable |μ| of Liouville theory is related to the mass of the black hole. The connection to Liouville theory should provide the exact solution for correlation functions in the euclidean black-hole geometry using matrix models and Liouville theory; for the Lorentz-signature Schwarzschild geometry it gives us a tool to investigate the properties of singularities in string theory.
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