Abstract
We discuss some of the basic features of extremal black holes in four-dimensional extended supergravities. Firstly, all regular solutions display an attractor behavior for the scalar field evolution towards the black hole horizon. Secondly, they can be obtained by solving first order flow equations even when they are not supersymmetric, provided one identifies a suitable superpotential W which also gives the black hole entropy at the horizon and its ADM mass at spatial infinity. We focus on N=8 supergravity and we review the basic role played by U-duality of the underlying supergravity in determining the attractors, their entropies, their masses and in classifying both regular and singular extremal black holes.
Highlights
Contribution to the Proceedings of the Conference in Honor of Murray Gell-Mann’s 80th Birthday, Singapore, 24th-26th February 2010 Lecture delivered by Sergio Ferrara
We focus on N=8 supergravity and we review the basic role played by U-duality of the underlying supergravity in determining the attractors, their entropies, their masses and in classifying both regular and singular extremal black holes
Once the background geometry of the d-dimensional spacetime and the number N of supersymmetry charges have been selected, all the important features of a given solution are encoded into the electric magnetic-duality group G acting on the vector fields AΛ, and in the geometric properties of the moduli space G/H parametrized by the scalar fields φi[6]
Summary
In 1976, at the dawn of the N=1 supersymmetric theory of gravity in four dimensions [1]( called “N=1 supergravity”), when its N=2 extension had just appeared, Murray Gell-Mann was the first to remark during a seminar at Caltech that, if higher N supergravity would have existed, there would be a bound such that Nmax=8 would be the end of the story. The scalars evolve to a common value at the horizon, φiH = φiH(Q), which they reach with zero velocity (φi → 0), and where they entirely depend on the electric-magnetic charge vector Q of the asymptotic configuration This attractive feature is called Attractor Mechanism [10, 11], and the attractor fixed points can be obtained as extrema of a suitable effective potential. This formula can be viewed as a differential equation defining W for a given black hole effective potential VBH , and it can lead to multiple choices: only one of those will corresponds to BPS solutions, while a different one will be associated to non BPS ones In both cases, W allows to rewrite the ordinary second order supergravity equations of motion d2U dτ 2.
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