There has been some recent progress in understanding the microscopic derivation of the entropy of supersymmetric AdS black holes using holography and a localisation computation in the dual quantum field theory. In this talk, after a general discussion of the overall picture, I discuss the large N factorization properties of supersymmetric partition functions for CFT with a holographic dual in various dimensions. I show how this factorization is related to a universal gluing formula for the entropy functionals of known AdS black holes in terms of elementary objects called gravitational blocks.

Abstract: We define indices for topologically twisted supersymmetric theories from two to five dimensions. We apply them to the holographic study of AdS black holes, black strings and domain-walls across dimensions.

One of the great success of string theory is the microscopical explanation of the entropy of a class of asymptotically flat black holes. Much less is known about asymptotically AdS black holes in four dimensions or higher. In this talk I explain how to derive the entropy of a class of asymptotically AdS supersymmetric black holes in four dimensions using holography. The counting of black hole micro-states is related to a counting of states in the dual 3d gauge theory which can be explicitly performed using localization.

I discuss some aspects of the full moduli space of N=1 supersymmetric gauge theories, focusing on those arising on D-branes at Calabi-Yau singularities. Of particola importance is the full moduli space for one brane, called the master space. I discuss its geometrical properties and its relation with the uncovering of hidden symmetries and the counting of BPS states of the gauge theory.

I discuss some aspects of the conformal field theories obtained by placing D3 branes at conical singularities focusing on the rich interplay between the properties of the gauge theories and the geometry of the internal manifold. I discuss in particular the role of baryons, a character for counting all BPS baryonic operators and its geometrical interpretation.