Macroscopic Entropy of $N=2$ Extremal Black Holes

TL;DR

Strominger analyzes extremal BPS black holes in $N=2$, $d=4$ supergravity, showing horizon moduli are attracted to a charge-dependent fixed point and that the near-horizon geometry and horizon area depend only on the charges. He extends the attractor mechanism to configurations with both electric and magnetic charges and derives a universal entropy formula determined by the charges and the horizon moduli-space geometry at the fixed point, independent of asymptotic moduli. The work generalizes earlier magnetic cases and links the entropy to moduli-space data at the horizon, suggesting connections to Calabi–Yau moduli and future microscopic interpretations. It provides a covariant framework for computing the Bekenstein-Hawking entropy from low-energy supergravity data without detailed asymptotic boundary conditions.

Abstract

Extremal BPS-saturated black holes in $N=2$, $d=4$ supergravity can carry electric and magnetic charges $(q^Λ_{(m)},q_Λ^{(e)})$. It is shown that in smooth cases the moduli fields at the horizon take a fixed "rational" value $X^Λ(q_{(m)},q^{(e)})$ which is determined by the charges and is independent of the asymptotic values of the moduli fields. A universal formula for the Bekenstein-Hawking entropy is derived in terms of the charges and the moduli space geometry at $X^Λ(q_{(m)},q^{(e)})$. This work extends previous results of Ferrara, Kallosh and the author for the pure magnetic case.