Black hole entropy functions and attractor equations

TL;DR

This work unifies two entropy-function formalisms for extremal black holes in N=2 supergravity: Sen's near-horizon reduced-action entropy function and the BPS entropy function based on supersymmetry at the horizon. It shows that, for BPS black holes, both approaches yield identical attractor equations and semiclassical OSV-type entropy results, and extends the analysis to non-BPS configurations with higher-derivative (R^2) couplings. The authors derive the variational equations without and with R^2 terms, discuss duality covariance, and illustrate non-BPS corrections with a perturbative example. They further address non-holomorphic corrections and outline how to incorporate them consistently via a function Ω, clarifying implications for duality invariance and microscopic-macroscopic entropy matching. Overall, the paper deepens the understanding of how entropy functionals encode horizon data and how higher-derivative and non-holomorphic effects modify black hole entropy in four-dimensional N=2 theories.

Abstract

The entropy and the attractor equations for static extremal black hole solutions follow from a variational principle based on an entropy function. In the general case such an entropy function can be derived from the reduced action evaluated in a near-horizon geometry. BPS black holes constitute special solutions of this variational principle, but they can also be derived directly from a different entropy function based on supersymmetry enhancement at the horizon. Both functions are consistent with electric/magnetic duality and for BPS black holes their corresponding OSV-type integrals give identical results at the semi-classical level. We clarify the relation between the two entropy functions and the corresponding attractor equations for N=2 supergravity theories with higher-derivative couplings in four space-time dimensions. We discuss how non-holomorphic corrections will modify these entropy functions.