A tale of two indices or how to count black hole microstates in AdS/CFT

In this thesis we investigate on the microscopic nature of the entropy of black holes in string theory in the context of AdS/CFT correspondence. We focus our attention on two particularly interesting cases in different dimensions. First, we consider static dyonic BPS black holes in AdS$_4$ in 4d $mathcal{N}=2$ gauged supergravities with vector and hyper multiplets. More precisely, we focus on the example of BPS black holes in AdS$_4 imes S^6$ in massive Type IIA, whose dual three-dimensional holographic description is known and simple. To provide a microscopic counting of the black hole entropy we employ a topologically twisted index in the dual field theory, which can be computed exactly and non-perturbatively with localization techniques. We find perfect match at leading order. Second, we turn to rotating electrically-charged BPS black holes in AdS$_5 imes S^5$ in Type IIB. Here, the microscopic entropy counting is done in terms of the superconformal index of the dual $cN=4$ Super-Yang-Mills theory. We proceed by deriving a new formula, which expresses the index as a finite sum over the solution set of certain transcendental equations, that we dub Bethe Ansatz Equations, of a function evaluated at those solutions. Then, using the latter formula, we reconsider the large $N$ limit of the superconformal index. We find an exponentially large contribution which exactly reproduces the Bekenstein-Hawking entropy of the black holes of Gutowski-Reall. Besides, the large $N$ limit exhibits a complicated structure, with many competing exponential contributions and Stokes lines, hinting at new physics.