On some properties of the Attractor Equations

TL;DR

The paper analyzes attractor equations in $N=2$, $d=4$ supergravity for extremal black holes with general electric/magnetic charges. It derives the black-hole potential $V_{BH}$, characterizes BPS ($D_i Z=0$) and non-BPS attractors, and links stability to Special Kähler Geometry data via the gaugino mass matrix, $M_{ij}$. In the one-modulus case, it provides explicit non-BPS results, including $V_{BH}|_{non-BPS}$ and a multiplicative entropy renormalization $S_{BH,non-BPS} = \pi \gamma |Z|^2$ with $\gamma-1 = 4 ( g^{3} / |C|^{2} )_{non-BPS}$, and highlights the role of $D_z C$ in stability. The Hessian analysis connects complex and real bases and shows how SKG structure constrains stability, while the Kähler gauge-invariant formulation $V_{BH} = e^{G}[1 + g^{i\bar i}(\partial_i G)(\bar{\partial}_{\bar i} G)]$ clarifies the BPS case, whose entropic interpretation aligns with the GSV functional, $\mathcal{S}_{GSV} = (\pi/4) e^{-K}$, up to gauge choices. Overall, the work clarifies how SKG data governs attractor stability and links attractor dynamics to entropic functionals in black-hole thermodynamics.

Abstract

We discuss the Attractor Equations of N=2, $d=4$ supergravity in an extremal black hole background with arbitrary electric and magnetic fluxes (charges) for field-strength two-forms. The effective one-dimensional Lagrangian in the radial (evolution) variable exhibits features of a spontaneously broken supergravity theory. Indeed, non-BPS Attractor solutions correspond to the vanishing determinant of a (fermionic) gaugino mass matrix. The stability of these solutions is controlled by the data of the underlying Special Kähler Geometry of the vector multiplets' moduli space. Finally, after analyzing the 1-modulus case more in detail, we briefly comment on the choice of the Kähler gauge and its relevance for the recently discussed entropic functional.