Towards multi-center AdS black holes

In the last decades interest in black holes with an anti-de Sitter asymptotic behaviour has been renewed, mainly thanks to the possibilities opened by the gauge/gravity correspondence. Progress has been made in reproducing the Bekenstein--Hawking entropy through the counting of degenerate microstates by making use of the dual conformal field theories. Despite this, we still lack a proper understanding of these solutions from the gravity side, mainly with respect to rotating solutions and multi-center configurations. This work is motivated by recent developments: the finding of a first order description for rotating BPS black holes in gauged supergravity and of stable bound states from probe analysis in an AdS black hole background. The aim of this work is to recover the first order description of rotating BPS black holes in gauged supergravity in a duality invariant formulation and to work out possible developments with the ambition of proving the existence of multi-center configurations. The flow equations are formulated in an explicit way by making use of the symplectic invariance of the vector sector of the theory, starting from an appropriate ansatz for the metric and vector fields. We provide an analysis of the solutions in a simplified setting and this gives hints as to which is the role of the various quantities entering these equations. Lastly we show that, in this simplified case, a superpotential drives the flow of the scalar fields and warp factors.