The phases of supersymmetric black holes in five dimensions

Abstract

In contrast to four dimensions, in five dimensions black hole solutions can have event horizons with nonspherical topology and violate the uniqueness theorems. The supersymmetric solutions of single black hole (also called BMPV black hole because it was first discovered by Breckenridge, Myers, Peet and Vafa) and single black ring (with horizon topology $S^1 \times S^2$) have been discovered in minimal $D=5$ supergravity. The first part of the thesis is devoted to the recent developments in five-dimensional supersymmetric black holes: first I briefly describe minimal $N=1,D=5$ supergravity theory, next I review the well-known supersymmetric black hole solutions in five dimensions and study the physical properties of the BMPV black hole and the supersymmetric black ring. However, the BPS (Bogomol'nyi-Prasad-Sommerfield) equations solved by the black ring appear to be nonlinear, hence this obscures the construction of multiple ring solutions via simple superpositions.\par

In order to construct solutions describing multiple supersymmetric black rings or superpositions of supersymmetric black rings with BMPV black holes, in the second part I review an alternative approach --- the harmonic function method. I first introduce four harmonic functions to characterize the single supersymmetric black ring solution. Via simple superpositions of harmonic functions for each ring, supersymmetric multiple concentric black rings are then constructed, in which the rings have a common center, and can lie either in the same plane or in orthogonal planes. In addition, this solution-generating method can also be applied to supersymmetric solutions with a black hole by taking the limit $R\rightarrow0$. As a result, I construct the most general supersymmetric solution --- black multiple bi-rings Saturn, which consists of multiple concentric black rings sitting in orthogonal planes with a BMPV black hole at the center. In the thesis I focus particularly on the bi-rings Saturn solution.\par

In the last part, I present the phase diagram of the established supersymmetric bi-ring Saturn in five dimensions and show that its structure is similar to those for extremal vacuum ones: a semi-infinite open strip, whose upper bound on the entropy is equal to the entropy of a static BMPV black hole of the same total mass for any value of the angular momentum. Following this, I provide a detailed analysis of the configurations that approach its three boundaries. Remarkably, I argue that for any $j\geq0$ the phase with highest entropy is a black bi-ring Saturn configuration with a central, close to static, $S^3$ BMPV black hole (accounting for the high entropy) surrounded by a pair of very large and thin orthogonal black rings (carrying the angular momentum). Moreover, I also study the outstanding feature of non-uniqueness arising from this exotic configuration. Possible generalizations to more supersymmetric black hole solutions including black Saturn with an off-center hole, non-orthogonal ring configurations and rings on Eguchi-Hanson space are discussed at the end of the thesis.