Black hole attractors and the entropy function in four‐ and five‐dimensional N = 2 supergravity

Abstract

In this overview of selected aspects of the black hole attractor mechanism, after introducing the necessary foundations, we examine the relationship between two ways to describe the attractor phenomenon in four‐dimensional N = 2 supergravity: the entropy function and the black hole potential. We also exemplify their practical application to finding solutions to the attractor equations for a conifold prepotential. Next we describe an extension of the original definition of the entropy function to a class of rotating black holes in five‐dimensional N = 2 supergravity based on cubic polynomials, exploiting a connection between four‐ and five‐dimensional black holes. This link allows further the derivation of five‐dimensional first‐order differential flow equations governing the profile of the fields from infinity to the event horizon and construction of non‐supersymmetric interpolating solutions in four dimensions by dimensional reduction. Finally, since four‐dimensional extremal black holes in N = 2 supergravity can be viewed as certain two‐dimensional string compactifications with fluxes, we discuss implications of the conifold example in the context of the entropic principle, which postulates as a probability measure on the space of these string compactifications the exponentiated entropy of the corresponding black holes.