Entropy and Conformal Field Theories of AdS 3 Models

We present a derivation of the Bekenstein-Hawking entropy from conformal field theories. In particular we consider a six-dimensional string configuration with background metric AdS3 × S3 near the horizon. In addition we introduce momentum modes along the string, corresponding to a BanadosTeitelboim-Zanelli (BTZ) black hole in the anti-de Sitter (AdS) space-time, and a Taub-NUT soliton in the transverse Euclidean space, projecting out a discrete subgroub of S3. This spherical part is described by a SU(2) Z(m) WessZumino-Witten (WZW) model. The AdS3 space-time, on the other hand, is determined by two conformal field theories living at the two boundaries: One SL(2,R) U(1) WZW model on the horizon of the BTZ black hole and one Liouville model at infinity. The extremal BTZ black hole interpolates between these two conformal field theories. Moreover, we argue that the sum of the three conformal field theories yields the correct microscopic state-counting including all α′ corrections. PACS: 04.70, 11.25.H