Critical points of the Black-Hole potential for homogeneous special geometries

TL;DR

The paper extends the N=2, D=4 black-hole attractor analysis to homogeneous special Kähler geometries by constructing a rigid-index formulation of the coset spaces and deriving the relevant connections and curvature. It shows that non-BPS attractors with nonzero central charge satisfy the same entropy relation as symmetric spaces, with $S_{BH}=4\pi\,|Z|^2_{|extr.}$, while deviations arise for non-homogeneous geometries via geometric data with background symplectic charges. The authors classify Z=0 and Z≠0 non-BPS attractors across examples such as $L(0,P,\dot P)$, $L(1,2)$, and $L(2,2)$, detailing the possible charge configurations and residual symmetry groups. These results suggest robustness of the non-BPS entropy relation in a broad class of homogeneous geometries and have potential implications for brane dynamics and higher-derivative corrections.

Abstract

We extend the analysis of N=2 extremal Black-Hole attractor equations to the case of special geometries based on homogeneous coset spaces. For non-BPS critical points (with non vanishing central charge) the (Bekenstein-Hawking) entropy formula is the same as for symmetric spaces, namely four times the square of the central charge evaluated at the critical point. For non homogeneous geometries the deviation from this formula is given in terms of geometrical data of special geometry in presence of a background symplectic charge vector.