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Three-dimensional Black Holes and Liouville Field Theory

T. Nakatsu, H. Umetsu, N. Yokoi

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.

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 . 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 , 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 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.

Paper Structure

This paper contains 18 sections, 105 equations, 8 figures.

Table of Contents

  1. Introduction
  2. More On Geometry In Three Dimensions
  3. More on the black hole
  4. Generalization of flat connection
  5. Generalization of black hole
  6. 3-Geometries As The Virasoro Coadjoint Orbit
  7. Coadjoint orbits of the Virasoro group
  8. 3-geometries as the Virasoro coadjoint orbit
  9. Extremal black hole
  10. Anti de-Sitter space
  11. Quantization of the Virasoro coadjoint orbits
  12. Quantization Of Three-Dimensional Gravity
  13. Holonomy variable
  14. Realization by the Liouville field theory
  15. Quantization of $3d$ gravity based on the Liouville field theory
  16. ...and 3 more sections

Figures (8)

  • Figure 1: $SL_2({\bf R})$ is a solid torus. The upper and lower sides are identified.
  • Figure 2: The non-compact simply-connected region $Z_{0}$ of $SL_2({\bf R})$
  • Figure 3: The projection from $Z_{0}$ to the exterior of the outer horizon of the non-extremal black hole. $J = 0$ case is depicted as an example. The space between two surfaces $HRS$ and $H^{'}R^{'}S^{'}$ is the fundamental region.
  • Figure 4: The in-momentum allowed by the unitarity constraint.
  • Figure 5: The region $Z_{0}$ in $\widetilde{SL}_2({\bf R})$. It is the universal covering space of $Y_{(-\frac{\gamma^2l}{16G}, -\frac{\tilde{\gamma}^2l}{16G})}$.
  • ...and 3 more figures