Quantum Black Hole Entropy and Localization in Supergravity

In this thesis, we examine in detail the notion of black hole entropy in Quantum Field Theories, with a specific focus on supersymmetric black holes and the perturbative and non-perturbative quantum corrections to the classical area-law of Bekenstein-Hawking, where the latter stipulates that the thermodynamic entropy of a black hole is proportional to the surface area of its event horizon. To examine such corrections, we employ the formalism of Sen's Quantum Entropy Function where the complete quantum entropy of a supersymmetric black hole in theories of Supergravity is defined as a path-integral in the near-horizon region of the black hole. Evaluation of this path-integral can then be conducted exactly using localization computation techniques. Such techniques reduce the evaluation of the infinite dimensional path-integral to the evaluation of a classical, finite-dimensional integral over the so-called localizing manifold, and a one-loop correction stemming from quantum field fluctuations orthogonal to this localizing manifold. Due to the exactness of the localization argument, the results obtained in this manner are therefore formally expected to re-sum all perturbative and non-perturbative corrections to the classical area-law, and thus connect to string-theoretic predictions, where supersymmetric black holes can be described as D-branes interacting quantum mechanically (following the original insight of Strominger and Vafa). We investigate such connections in detail for specific supersymmetric black holes in the hopes of strengthening a Boltzmann-type interpretation of their thermodynamic entropy as arising from the degeneracies of the microscopic gravitational constituents (the D-branes). We find that this picture holds very precisely for two types of black holes preserving four real supercharges in both four-dimensional N=8 (maximally supersymmetric) and N=4 (half-maximally supersymmetric) string theories and supergravities. From a broader point of view, such results can be interpreted as providing important examples where supergravity theories encode the complete low-energy dynamics of string theories and provide a consistent effective picture. Some interesting connections to the mathematical theory of modular forms and mock modular forms are also exhibited