Precision microstate counting of black hole entropy from N=1 toric quiver gauge theories

The AdS/CFT correspondence, which conjectures a mathematical equivalence between string theories and field theories, has proven to be extremely successful in probing the microscopic structure of black holes. In this thesis we apply the AdS/CFT correspondence to investigate the entropy of five-dimensional, electrically charged rotating black holes. We do so by computing a field theoretical observable called the superconformal index. Specifically, we first study the superconfomal index of a large class of four-dimensional toric quiver gauge theories using a Bethe-Ansatz approach. Relying on a particular set of solutions to the corresponding Bethe-Ansatz equations we evaluate the superconformal index in the limit where the rank $N$ of the gauge group of the field theory is large. We present explicit results for field theories arising from a stack of N D3 branes at the tip of toric Calabi-Yau cones whose gravity dual are IIB string theories on AdS$_5 imes ext{SE}_5$, where $ ext{SE}_5 = Y^{p,q}, X^{p,q}, L^{a,b,c}$ are Sasaki-Einstein manifolds. For a suitable choice of the chemical potentials of the theory we find agreement of the superconformal index and the function whose Legendre transform yields the black hole entropy, even in those situations where the explicit black hole supergravity solution is not known. Furthermore, we systematically study various sub-leading structures in the superconformal index of ${cal N}=4$ supersymmetric Yang-Mills theory with $SU(N)$ gauge group. We concentrate in the two descriptions of the superconformal index, one as a matrix model of elliptic gamma functions, another in the Bethe-Ansatz presentation. Our saddle-point approximation goes beyond the Cardy-like limit and we uncover various saddles governed by a matrix model corresponding to $SU(N)$ Chern-Simons theory. The dominant saddle, however, leads to perfect agreement with the Bethe-Ansatz approach. We also determine the logarithmic correction to the superconformal index to be $log N$, finding precise agreement between the saddle-point and Bethe-Ansatz approaches in their respective approximations. We generalize the two approaches to cover a large class of 4d ${cal N}=1$ toric quiver gauge theories. We find that also in this case both approximations agree all the way down to a universal contribution of the form $log N$. The universality of this last result constitutes a robust signature of this ultraviolet description of asymptotically AdS$_5$ black holes and could be tested by low-energy IIB one-loop supergravity.