Supersymmetry and attractors

Abstract

We find a general principle which allows one to compute the area of the horizon of $N=2$ extremal black holes as an extremum of the central charge. One considers the ADM mass equal to the central charge as a function of electric and magnetic charges and moduli and extremizes this function in the moduli space (a minimum corresponds to a fixed point of attraction). The extremal value of the square of the central charge provides the area of the horizon, which depends only on electric and magnetic charges. The doubling of unbroken supersymmetry at the fixed point of attraction for $N=2$ black holes near the horizon is derived via conformal flatness of the Bertotti-Robinson-type geometry. These results provide an explicit model-independent expression for the macroscopic Bekenstein-Hawking entropy of $N=2$ black holes which is manifestly duality invariant. The presence of hypermultiplets in the solution does not affect the area formula. Various examples of the general formula are displayed. We outline the attractor mechanism in $N=4, 8$ super-symmetries and the relation to the $N=2$ case. The entropy-area formula in five dimensions, recently discussed in the literature, is also seen to be obtained by extremizing the $5d$ central charge.