Microstates of Four-Dimensional Rotating Black Holes from Near-Horizon Geometry

We show that a class of four-dimensional rotating black holes allow fivedimensional embeddings as black rotating strings. Their near-horizon geometry factorizes locally as a product of the three-dimensional anti-deSitter spacetime and a two-dimensional sphere (AdS3 × S2), with angular momentum encoded in the global space-time structure. Following the observation that the isometries on the AdS3 space induce a two-dimensional (super)conformal field theory on the boundary, we reproduce the microscopic entropy with the correct dependence on the black hole angular momentum. Typeset using REVTEX 1 Recent developments in nonperturbative string theory have provided a fruitful framework to consider quantum properties of black holes. In particular, extreme black holes with Ramond-Ramond (R-R) charges can be interpreted in higher dimensions as intersecting Dbranes (the nonperturbative objects in string theory that carry such charges [1]), and this has lead to a counting of black hole quantum states that agrees precisely with the BekensteinHawking (BH) entropy [2]. This counting is carried out in the weakly coupled regime where the D-brane constituents of the black hole experience flat space-time geometry; however, due to supersymmetry, it remains valid in the regime where the D-branes are strongly coupled, and the geometric space-time description of black holes emerges. Thus the microscopic derivation of the BH-entropy is justified, but it is difficult to explore the quantum black hole geometry in detail using D-branes. The success of the D-brane counting overshadowed prior attempts to shed light on the microscopics of black holes in string theory. In pioneering work, Sen attempted to identify the microstates of extreme electrically charged black holes with perturbative excitations of string theory [3]. However, it was not until the discovery of extreme dyonic black holes in string theory — with regular horizons and thus finite BH-entropy — that a quantitative agreement between the microscopic and macroscopic entropy became feasible . The microscopic features of these black holes are captured by string theory in the curved space-time geometry specified by their near-horizon region [5,6]. In particular, a SL(2,Z) × SU(2) Wess-Zumino-Witten (WZW) model [7,6] reproduces, at least qualitatively [6,8], the extreme black hole entropy directly from the near-horizon geometry. However, it is only very recently that a precise derivation of the BH-entropy from the near-horizon geometry was achieved by Strominger [9] (and also by Sachs et al. [10]). The 1Such black holes were originally specified by four Neveu-Schwarz Neveu-Schwarz (NS-NS) charges [4], two electric and two magnetic ones. Note, however, that these can be mapped onto solutions with R-R charges, by exploiting duality symmetry; their space-time is thus the same.