Quantum black holes: supersymmetry and exact results

The explanation of black hole entropy as statistical entropy is one of the big successes of string theory. In this article we review recent progress in this subject, focussing on understanding quantum effects on black hole entropy. Supersymmetry plays a key role in these developments and leads to prototype systems where we can discuss quantum effects to great precision. Our discussion has two strands, both of which involve the gravitational path integral that calculates the supersymmetric index. In the first strand we discuss supersymmetric black holes in the microcanonical ensemble, which are decoupled from the environment and can be treated as independent quantum systems. Using methods of supersymmetric localization one can arrive at the integer quantum degeneracies of such systems purely in terms of the gravitational variables. In the second strand we consider grand-canonical ensembles in gravity in which black holes arise as a finite-action excitation in asymptotically flat or Anti de Sitter space. In this context we discuss the saddle-points of the gravitational index, and how they reproduce the black hole action and entropy that agree with the index of the holographic superconformal field theory even beyond the leading order in the semiclassical approximation. Finally we discuss how the gravitational index informs us about the detailed structure of the non-perturbative sum over saddle points and the resulting phases of the theory. Throughout, we try to highlight the central role played by the methods as well as foundational concepts of supergravity in driving these developments.