Three-dimensional Black Holes and Liouville Field Theory

Abstract

A quantization of (2+1)-dimensional gravity with negative cosmological constant is presented and quantum aspects of the (2+1)-dimensional black holes are studied thereby. The quantization consists of two procedures. One is related with quantization of the asymptotic Virasoro symmetry. A notion of the Virasoro deformation of 3-geometry is introduced. For a given black hole, the deformation of the exterior of the outer horizon is identified with a product of appropriate coadjoint orbits of the Virasoro groups $\hat{diff S^1}_{\pm}$. Its quantization provides unitary irreducible representations of the Virasoro algebra, in which state of the black hole becomes primary. To make the quantization complete, holonomies, the global degrees of freedom, are taken into account. By an identification of these topological operators with zero modes of the Liouville field, the aforementioned unitary representations reveal, as far as $c \gg 1$, as the Hilbert space of this two-dimensional conformal field theory. This conformal field theory, living on the cylinder at infinity of the black hole and having continuous spectrums, can recognize the outer horizon only as a it one-dimensional object in $SL_2({\bf R})$ and realize it as insertions of the corresponding vertex operator. Therefore it can not be a conformal field theory on the horizon. Two possible descriptions of the horizon conformal field theory are proposed.