Universality of Sypersymmetric Attractors

TL;DR

The paper demonstrates that the macroscopic entropy-area relation for supersymmetric black holes in d=4 with N=2,4,8 is universal: at the attractor point the area satisfies $rac{A}{4}=\pi E^2$ with $E=M_{ADM}=\max_C |Z_C|$, and all other central charges vanish, a consequence of SUSY enhancement near the horizon. The ADM mass at the attractor is expressible in terms of duality-invariant invariants (quartic for N=8, cubic for d=5, etc.), yielding moduli-independent horizon areas such as $A=4\pi\sqrt{J}$ in N=8 or $A=2\pi\sqrt{q^2 p^2-(q\cdot p)^2}$ in simpler cases. The authors extend these results to d=5 theories and provide exact, duality-symmetric expressions for the ground-state energy, linking spontaneous SUSY breaking to the non-vanishing horizon area, and propose a universal, dimension-spanning formula $E=|Z_{\text{max}}|$ at the attractor. This framework connects macroscopic black-hole entropy to duality invariants and offers a coherent picture bridging classical attractor flows with microscopic string-state counting.

Abstract

The macroscopic entropy-area formula for supersymmetric black holes in N=2,4,8 theories is found to be universal: in d=4 it is always given by the square of the largest of the central charges extremized in the moduli space. The proof of universality is based on the fact that the doubling of unbroken supersymmetry near the black hole horizon requires that all central charges other than Z=M vanish at the attractor point for N=4,8. The ADM mass at the extremum can be computed in terms of duality symmetric quartic invariants which are moduli independent. The extension of these results for d=5, N=1,2,4 is also reported. A duality symmetric expression for the energy of the ground state with spontaneous breaking of supersymmetry is provided by the power 1/2 (2/3) of the black hole area of the horizon in d=4 (d=5). It is suggested that the universal duality symmetric formula for the energy of the ground state in supersymmetric gravity is given by the modulus of the maximal central charge at the attractor point in any supersymmetric theory in any dimension.

Figures (1)

• Figure 1: Evolution of the dilaton field $e^{-2\phi}$ and of the effective dilaton charge $\Sigma(\rho) = - {d\phi\over d\rho}$ for various initial conditions at $\rho = 0$ ($r = \infty$) to a common fixed point at $\rho =-\infty$ ($r = 0)$.