In these lecture notes, we review some recent developments on the relationbetween the macroscopic entropy of four-dimensional BPS black holes and themicroscopic counting of states, beyond the thermodynamical, large charge limit.After a brief overview of charged black holes in supergravity and stringtheory, 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 relatesmicroscopic degeneracies to the topological string amplitude, and reviewprecision tests of this formula on ``small'' black holes. Finally, motivated bya holographic interpretation of the OSV conjecture, we give a systematicapproach to the radial quantization of BPS black holes (i.e. quantumattractors). This suggests the existence of a one-parameter generalization ofthe topological string amplitude, and provides a general framework forconstructing automorphic partition functions for black hole degeneracies intheories with sufficient degree of symmetry.
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