Lessons from the Vacuum Structure of 4d N=2 Supergravity

This thesis is devoted to a study of supersymmetry preserving background solutions of N=2 supergravity in 4 dimensions. These models include arbitrary electric gaugings in the vector- and hypermultiplet sectors introduced in the beginning of the thesis. The main contents are divided into three major parts. In part 1 we consider vacua that can be fully analyzed by requiring preserved supersymmetry. We determine and analyze maximally supersymmetric configurations, preserving eight supercharges. We present several examples of such solutions and connect some of them to vacuum solutions of flux compactifications in string theory. We also provide a consistent truncation of the gauged theory based on integrating out massive supermultiplets of the gauged theory. The second part is devoted to the topic of supersymmetric black holes. These can be asymptotically flat or asymptotically AdS, and we analyze both cases in detail. We construct BPS black hole solutions in Minkowski space with charged hypermultiplets. We find solutions with vanishing scalar hair that resemble already known black holes, while the genuinely new solutions with hair that we find contain ghost modes. We also elaborate further on the static supersymmetric AdS black holes found in arXiv:0911.4926, investigating thoroughly the BPS constraints for spherical symmetry in N = 2 gauged supergravity. We find Killing spinors that preserve two of the original eight supercharges and investigate the conditions for genuine black holes free of naked singularities. The existence of a horizon is intimately related with the requirement that the scalars are not constant, but given in terms of harmonic functions in analogy to the attractor flow in ungauged supergravity. The third major topic of this thesis is BPS bounds, where we discuss asymptotically Minkowski and AdS solutions in full generality. We concentrate on asymptotically anti-de Sitter (AdS) spacetimes, and find that there exist two disconnected BPS ground states of the theory, depending on the presence of magnetic charge. Each of these ground states comes with a different superalgebra and a different BPS bound, which we derive. As a byproduct, we also demonstrate how the supersymmetry algebra has a built-in holographic renormalization method to define finite conserved charges. We derive the general form of the charges for all asymptotically flat, anti-de Sitter, and magnetic anti-de Sitter spacetimes. Some particular black hole examples from part 2 are considered to explicitly demonstrate how AdS and mAdS masses differ when solutions with non-trivial scalar profiles are considere