About fifty years ago, Gibbons and Hawking argued that the Euclidean gravitational path integral with suitable boundary conditions can be interpreted as a grand-canonical partition function. Classical gravitational solutions, including black holes, arise as saddles of this path integral, and from the saddle-point action one can extract the black hole entropy. In the talk, I will discuss some recent developments of these ideas. Working in five dimensions, we will see how imposing supersymmetric boundary conditions converts the partition function into an index. Then we will construct a class of saddles of this index which interpolates between supersymmetric black holes and horizonless microstate geometries. I will discuss how the saddle point action can be computed via equivariant localization. Finally, I will comment on the relevance of these findings for black hole microstate counting and holography.

The fundamental theory of quantum gravity is expected to manifest itself at low energies via a series of higher-derivative corrections to Einstein’s theory. Holography and supersymmetry are of great help to characterize such corrections in controlled scenarios: through holography, quantum gravity in Anti de Sitter (AdS) space has a rigorous definition in terms of a conformal field theory (CFT), while supersymmetry makes it possible to compute exact observables and make quantitative predictions. In this context, we will illustrate how a CFT generating function known as the superconformal index provides a microscopic explanation of the entropy of five-dimensional supersymmetric black holes in AdS. We will show how this match goes beyond the leading Bekenstein-Hawking term and includes higher-derivative corrections.

Supersymmetry on curved spaces has recently attracted much attention, mainly as a tool towards the exact computation of quantum field theory observables via localization. Taking a holographic perspective, I will discuss how the conditions for rigid supersymmetry to be preserved on a curved boundary arise from the bulk supergravity Killing spinor equations. In particular, I will show that a four-dimensional superconformal field theory can be put on a curved, Lorentzian spacetime if and only if this admits a null conformal Killing vector. For a supersymmetric field theory with an R-symmetry, not necessarily conformal, the vector is further restricted to be Killing. After having presented some illustrative examples, I will conclude comparing with the Euclidean case.

Consistent truncations have proved to be powerful tools in the construction of new string theory solutions. Recently, they have been employed in the holographic description of condensed matter systems. In the talk, I will present a rich class of supersymmetric consistent truncations of higher-dimensional supergravity which are based on geometric structures, focusing on the tri-Sasakian case. Then I will discuss some applications, including a general result relating AdS backgrounds to solutions with non-relativistic Lifshitz symmetry.

In the context of gauge-gravity duality, consistent truncations have proved to be powerful solution-generating tools, their latest application being to the holographic description of condensed matter systems. In the talk, I will discuss a rich class of consistent truncations of type IIB supergravity on squashed Sasaki-Einstein manifolds, leading to N=4 or N=2 gauged supergravity in five dimensions. As examples of the several possible applications, I will present an approach to domain wall gradient flows, as well as a new class of AdS5 backgrounds on the T(1,1) coset, with a comment on some related Lifshitz solutions.