Lectures on black holes, topological strings and quantum attractors

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 discuss the radial quantization of BPS black holes (i.e. quantum attractors) and present a recent conjecture relating exact black hole degeneracies to Fourier coefficients of certain automorphic forms.