Quantum Black Hole Entropy and Localization in Supergravity

TL;DR

This thesis explores quantum corrections to black hole entropy in supergravity using localization techniques, confirming microscopic degeneracy interpretations and connecting to string theory and modular forms.

Contribution

It applies localization to compute the exact quantum entropy of supersymmetric black holes, linking supergravity results with string theory predictions and mathematical structures.

Findings

Localization computes exact quantum entropy including all corrections.

Results support the microscopic degeneracy interpretation of black hole entropy.

Connections to modular and mock modular forms are established.

Abstract

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.