Supersymmetry and Attractors

TL;DR

This work establishes a universal attractor mechanism for supersymmetric black holes by showing that the horizon area (or entropy via $S=A/4$) is fixed by extremizing the central charge $|Z|$ over the moduli space. In four-dimensional N=2 supergravity, the fixed point characterized by $D_i Z=0$ yields $|Z_{ m fix}|^2$, and the horizon area is $A/4 = \\pi |Z_{ m fix}|^2$, independent of asymptotic moduli and governed solely by electric and magnetic charges. The authors extend the analysis to N=4 and N=8 via consistent truncations to N=2, demonstrating that the attractor behavior reduces to simple charge products in fixed points and providing explicit examples (dilatonic dyons, BR near-horizon geometry, and N=8 truncations). They also discuss five-dimensional analogs where $S \\sim |Z_{ m fix}|^{3/2}$, suggesting a potentially universal principle across dimensions and supersymmetries for macroscopic entropy and its microscopic interpretation.

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 supersymmetries 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.

Figures (2)

• Figure 1: Evolution of the dilaton from various initial conditions at infinity to a common fixed point at $r=0$.
• Figure 2: Extremum of the central charge in the moduli space.