Attractor Horizon Geometries of Extremal Black Holes

TL;DR

This work surveys attractor horizon geometries for extremal black holes in $oldsymbol{N}=2$, $d=4$ supergravity with vector multiplets, showing that horizon data are fixed by critical points of the black-hole potential $V_{BH}$. It develops the framework of Special Kähler geometry to express the central charge $Z$, matter charges $Z_i$, and the holomorphic prepotential, and derives attractor equations and a Hessian-based stability analysis. The paper distinguishes $rac{1}{2}$-BPS attractors, which are always stable, from non-BPS attractors with $Z eq0$, where stability depends on SKG data such as the tensor $C_{ijk}$ and derivatives like $D_i C_{jkl}$; in the one-modulus case, stability reduces to explicit inequalities involving $oldsymbol{ ext N}$ and $oldsymbol{ ext M}$. It also reviews group-theoretic classifications of attractor solutions for homogeneous symmetric SKGs, and outlines extensions to non-cubic SKGs, Fermat CY$_3$ compactifications, and multi-centre scenarios, highlighting connections to $U$-duality orbits and potential physical implications for black hole entropy and moduli stabilization.

Abstract

We report on recent advances in the study of critical points of the ``black hole effective potential'' V_{BH} (usually named \textit{attractors}) of N=2, d=4 supergravity coupled to n_{V} Abelian vector multiplets, in an asymptotically flat extremal black hole background described by 2n_{V}+2 dyonic charges and (complex) scalar fields which are coordinates of an n_{V}-dimensional Special Kahler manifold.