Extremal single-charge small black holes: Entropy function analysis

We study stretched horizons of the type AdS_2 x S^8 for certain spherically symmetric extremal small black holes in type IIA carrying only D0-brane charge making use of Sen's entropy function formalism for higher derivative gravity. A scaling argument is given to show that the entropy of this class of black holes for large charge behaves as \sqrt{|q|} where q is the electric charge. The leading order result arises from IIA string loop corrections. We find that for solutions to exist the force on a probe D0-brane has to vanish and we prove that this feature persists to all higher derivative orders. We comment on the nature of the extremum of these solutions and on the sub-leading corrections to the entropy. The entropy of other small black holes related by dualities to our case is also discussed.Comment: 19 pages, v2:typos corrected and references adde

Lectures on on Black Holes, Topological Strings and Quantum Attractors (2.0)

In these lecture notes, we review some recent developments on the relation between the macroscopic entropy of four-dimensional BPS black holes and the microscopic counting of states, beyond the thermodynamical, large charge limit. After a brief overview of charged black holes in supergravity and string theory, we give an extensive introduction to special and very special geometry, attractor flows and topological string theory, including holomorphic anomalies. We then expose the Ooguri-Strominger-Vafa (OSV) conjecture which relates microscopic degeneracies to the topological string amplitude, and review precision tests of this formula on ``small'' black holes. Finally, motivated by a holographic interpretation of the OSV conjecture, we give a systematic approach to the radial quantization of BPS black holes (i.e. quantum attractors). This suggests the existence of a one-parameter generalization of the topological string amplitude, and provides a general framework for constructing automorphic partition functions for black hole degeneracies in theories with sufficient degree of symmetry.Comment: 103 pages, 8 figures, 21 exercises, uses JHEP3.cls; v5: important upgrade, prepared for the proceedings of Frascati School on Attractor Mechanism; Sec 7 was largely rewritten to incorporate recent progress; more figures, more refs, and minor changes in abstract and introductio