Extremal black holes in N=2 supergravity

Abstract

An explanation for the entropy of black holes has been an outstanding problem in recent decades. A special case where this is possible is that of extremal black holes in N=2 supergravity in four and five dimensions. The best developed case is for black holes preserving some supersymmetry (BPS), which are well understood in the context of the lowest order two derivative action. For ungauged theories, one can use the experience gained from this example to study the effect of higher order corrections on BPS solutions or to construct nonsupersymmetric black holes. We first present a detailed discussion of the four derivative terms allowed by N=2 supersymmetry in four and five dimensions respectively. Each of these invariants is parametrised by an arbitrary function, which can be fixed when one considers the embedding in a microscopic theory, such as string theory. We present a detailed investigation of the possible N=2 backgrounds, described in a unified way for both black holes and rings. Restricting to the class of theories obtained by string theoretic models, we connect to known microscopic constructions of black holes. The end result is a successful match of the entropy to the string theory counting for black rings and for large black holes. Issues remain with small black holes in the five-dimensional case. Beyond supersymmetric solutions, we discuss some recent results on the construction of non-BPS solutions in N=2 theories, emphasising the description in terms of first order equations and the attractor mechanism. We introduce a special class of five-dimensional solutions which are supersymmetric on local patches of spacetime, but nevertheless completely break supersymmetry globally. The new solutions obtained turn out to allow for a number of features that are present in generic extremal solutions, but forbidden in the supersymmetric case. Based on this fact, we then go on to propose a framework for constructing more general extremal solutions, including rotating black holes.