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. 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. 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. 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 $\mathcal{N}=8$ and $\mathcal{N}=4$ 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.