Black hole microstates and supersymmetric localization

This thesis focuses mainly on understanding the origin of the Bekenstein-Hawking entropy for a class of four- and five-dimensional BPS black holes in string/M-theory. To this aim, important ingredients are holography and supersymmetric localization. Using the method of supersymmetric localization, the Euclidean path integrals for supersymmetric field theories on $\Sigma_\mathfrak{g} \times T^{n}$ $(n=1,2)$, with at least four real supercharges, can be reduced to a matrix integral that depends on background magnetic fluxes and chemical potentials for the global symmetries of the theory. This defines the topologically twisted index which, upon extremization with respect to the chemical potentials, is conjectured to reproduce the entropy of magnetically charged static BPS AdS$_{4/5}$ black holes/strings. We solve a number of such matrix models both in three and four dimensions and provide general formulae in the large $N$ limit. We then use these results to provide the microscopic realization of the entropy of a class of BPS black holes in $\mathcal{N}=2$ gauged supergravity. Finally, inspired by our previous results, we put forward an extremization principle for reproducing the Bekenstein-Hawking entropy of a class of BPS electrically charged rotating black holes in AdS$_5\times S^5$.