Three-dimensional black holes and Liouville field theory

Toshio Nakatsu, Hiroshi Umetsu, Naoto Yokoi

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17 Citations (Scopus)


A quantization of (2 + 1)-dimensional gravity with negative cosmological constant is presented, and it is used to study quantum aspects of (2 + 1)-dimensional black holes. The quantization consists of two procedures. One is related with quantization of the asymptotic Virasoro symmetry. The concept 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 the appropriate coadjoint orbits of the Virasoro groups diffS1±. Its quantization provides unitary irreducible representations of the Virasoro algebra, in which the 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 are shown, as long as c ≫ 1, to be 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 spectra, can recognize the outer horizon only as a one-dimensional object in SL2(R) and realize it as insertions of the corresponding vertex operator. Therefore it cannot be a conformal field theory on the horizon. Two possible descriptions of the horizon conformal field theory are proposed.

Original languageEnglish
Pages (from-to)867-896
Number of pages30
JournalProgress of Theoretical Physics
Issue number4
Publication statusPublished - 1999 Oct
Externally publishedYes

ASJC Scopus subject areas

  • Physics and Astronomy (miscellaneous)


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