Statistical entropy and AdS/CFT correspondence in BTZ black holes

We study AdS/CFT correspondence in the case of AdS3. We obtain the statistical entropy of the BTZ black hole in terms of the correct central charge and the conformal dimensions for the states corresponding to the BTZ black hole. We point out the difference between our method and the old fashioned approaches based on SL(2, R) Wess-Zumino-Witten model or Liouville theory. email: [email protected] email: [email protected] email: [email protected] 1 Recently there have been great interests in three-dimensional locally anti-de Sitter (AdS3) black hole solutions of Banãdos-Teitelboim-Zanelli (BTZ) [1]. It is relevant for the study of higher-dimensional black-hole solutions of string/M theory whose near horizon geometries are those of BTZ solutions [2]. As is well-known, the asymptotic symmetry of AdS3 is generated by two copies of Virasoro algebra [3], which indicates the structure of conformal field theory (CFT) at the two-dimensional boundary at infinity. The central charge of the algebra is given by c = 3l 2G , (1) where G is the three-dimensional Newton’s constant and Λ = − 1 l is the cosmological constant. By using this observation and identifying the BTZ black hole with mass M and angular momentum J as the boundary conformal field with dimension (h, h̄), h = 1 2 (lM + J), (2) h̄ = 1 2 (lM − J), (3) Strominger has obtained [4] the microscopic counting of Bekenstein-Hawking black hole entropy of BTZ solutions (and higher dimensional solutions) using Cardy’s formula [7]: