Microstate Counting of 3-charge Black Hole from Supergravity

Abstract

where A is the area of the horizon. Statistical mechanics then suggests that there should be e states of the black hole, but the geometry of the black hole is completely determined by its mass, charge and angular momentum : so called ‘Black holes have no hair’. This statement implies that the entropy of the black hole is S = ln 1 = 0. This is the ‘black hole entropy puzzle’, which is closely related to the ‘black hole information problems’. String theory has a rich structure and admits black-hole solutions in higher dimensions [2–4]. Then, the black holes in string theory may give an understanding of these puzzles. Black holes in string theory are widely studied by using a chiral null model [5–11]. This model provides a class of exact solutions of oscillating strings in various cases. From these solutions, Mathur proposed the ‘fuzzy ball’ conjecture for the black hole [12,13]. He constructed a geometry of the two-charge black hole by performing a chain of duality transformations from the metric of the oscillating fundamental string. The result gives a family of geometries for the D1-D5 system characterized by a single function ~ F , the vibration profile of the fundamental string. Moreover these geometries have no horizons and no singularity which corresponds to the so-called ‘black hole microstate geometries’. By quantizing the single function ~ F , we can obtain the number of geometries for the black hole [14]. This result agrees with the one from the calculation in the CFT side.