Three-dimensional black holes and Liouville field theory

A quantization of (2 + 1)-dimensional gravity with negative cosmological co nstant is presented, and it is used to study quantum aspects of (2 + 1)-dim ensional black holes. The quantization consists of two procedures. One is r elated with quantization of the asymptotic Virasoro symmetry. The concept o f the Virasoro deformation of 3-geometry is introduced. For a given black h ole, the deformation of the exterior of the outer horizon is identified wit h a product of the appropriate coadjoint orbits of the Virasoro groups <(di ffS(1))over cap>(+/-). Its quantization provides unitary irreducible repres entations of the Virasoro algebra, in which the state of the black hole bec omes primary. To make the quantization complete, holonomies, the global deg rees of freedom, are taken into account. By an identification of these topo logical operators with zero modes of the Liouville field, the aforementione d unitary representations are shown, as long as c much greater than 1, to b e the Hilbert space of this two-dimensional conformal field theory. This co nformal field theory, living on the cylinder at infinity of the black hole and having continuous spectra, can recognize the outer horizon only as a on e-dimensional object in SL2(R) and realize it as insertions of the correspo nding vertex operator. Therefore it cannot be a conformal field theory on t he horizon. Two possible descriptions of the horizon conformal field theory are proposed.