For supersymmetric setups the counting makes use of holography and known techniques such as localization. The presence of supersymmetry usually makes computations more tractable, for instance removing some unwanted divergencies. In the first part of the project we studied the thermodynamics in the BPS limit of AdS black holes, including the definition of a correct limiting procedure to reach extremality. We computed the density of near-BPS states by making use of an effective 2d theory that takes into accounts fluctuations close to extremality: we showed the presence of a gap between the BPS black holes and the lightest near-BPS ones. In addition, we have found a remarkable behaviour displayed by black holes in a particular M-theory truncation. Their spectrum is altered due to the presence of anomalies, and both the gap and index are reduced. Their density of states can be qualitatively reproduced by an SYK model with an odd number of fermions.
For more realistic Kerr black holes (such as the ones seen at the Event Horizon Telescope) we instead developed a strategy for regulating divergencies (due to so-called "zero-modes") in the gravitational path integral in the four-dimensional geometry near the black hole horizon. Our techniques make use of a near-extremal geometry, and render the path integral finite. We find that the quantum-corrected near-extremal entropy exhibits 3/2logT behavior, predicting a lifting of the ground state degeneracy for the extremal Kerr black hole, and the absence of a mass gap. We have corroborated these results via a computation of the quantum corrections that makes use of the full black hole geometry, by making use black hole quasinormal modes. This simple 3d computation confirmed the 3/2logT behaviour.
Further studies concern the physics of black holes in presence of positive cosmological constant (de Sitter black holes). These spacetimes are characterized by the presence of a cosmological horizon that bounds the black hole in size. Different extremal (zero-temperature) limits exist, and one of our goals was to study how these branches react to deformations from extremality, to investigate whether (and to which extent) the black hole response to a small temperature increase is universal. In this framework, we have computed response coefficients and mass gap, and we have identified a sector in the near-horizon theory that is responsible for the deviation from extremality and studied its dynamics. Subsequently, we focused on the quantum corrections to black hole entropy in presence of a positive cosmological constant, finding for instance that one of the limiting configurations (Nariai solution) is plagued with pathologies such as negative norm modes, and negative eigenvalue corrections. Further studies investigated the consequences of the addition of angular momentum on the horizon dynamics at small temperatures.
Lastly, we performed studies on the deformability properties of black hole horizons, constructing for instance new solutions of composite black holes (Black Saturns) that are useful playground for the study of the equilibrium conditions between black objects in spaces with cosmological constant and studied possible contributions to the index, also in light of newly found Grey Galaxies solutions.
The results of the research were disseminated through seminars and conferences in research institutions around the world. I have presented these results in workshops in Asia, USA and Europe and I have been the lecturer in PhD summer schools in the UK and in Brazil. Finally, I have organized two workshops in my field and I have taken part in a number of outreach events (i.e. a podcast for RadioAspen, and the INFN school laboratory initiative Lab2go) and initiatives aimed at raising awareness of gender issues in physics.
