The gravitational index and its saddles with enhanced supersymmetry

The study of black hole physics plays a fundamental role in our understanding of the quantum nature of the gravitational interaction. In this context, string theory offers an elegant framework in which many open questions can be addressed, leading to important results such as the counting of black hole microstates, which reproduces the Bekenstein-Hawking entropy formula. A possible way of computing the statistical entropy of a black hole solution in string theory is through the definition of a supersymmetric index. Recently, it has been understood that the supergravity path integral, when supplemented with suitable boundary conditions, computes a supersymmetric index. The relevant saddle points of this gravitational index are supersymmetric, non-extremal and, in general, complex solutions of the classical equations of motion. In four-dimensional supergravity theories, these solutions can be studied and classified systematically. While solutions with minimal supersymmetry have already been studied within this formalism, a better understanding of the extended supergravity case could be interesting for the comprehension of the role of supersymmetry on the gravitational path integral. In this thesis we analyse a specific class of solutions of pure, ungauged N=4, d=4 supergravity, with particular emphasis on the higher-supersymmetric case. We focus on their Euclidean continuation, which describes supersymmetric, non-extremal black-hole configurations contributing to the gravitational index. These Euclidean black holes are studied both in the generic case where they preserve four supercharges, and in the special case where there are eight preserved supercharges. We show that, when the condition for enhanced supersymmetry is imposed, these configurations are associated with a class of small black holes. We discuss the supergravity path integral for which these configurations are saddles, focusing on the grand-canonical viewpoint.