Black hole solutions to the $F_4$-model and their orbits (I)

Understand

In this paper we continue the program of the classification of nilpotent orbits using the approach developed in arXiv:1107.5986, within the study of black hole solutions in D=4 supergravities.

• Our goal in this work is to classify static, single center black hole solutions to a specific N=2 four dimensional "magic" model, with special K\"ahler scalar manifold ${\rm Sp}(6,\mathbb{R})/{\rm U}(3)$, as orbits of geodesics on the pseudo-quaternionic manifold ${\rm F}_{4(4)}/[{\rm SL}(2,\mathbb{R})\times {\rm Sp}(6,\mathbb{R})]$ with respect to the action of the isometry group ${\rm F}_{4(4)}$.
• Our analysis amounts to the classification of the orbits of the geodesic "velocity" vector with respect to the isotropy group $H^*={\rm SL}(2,\mathbb{R})\times {\rm Sp}(6,\mathbb{R})$, which include a thorough classification of the \emph{nilpotent orbits} associated with extremal solutions and reveals a richer structure than the one predicted by the $\beta-\gamma$ labels alone, based on the Kostant Sekiguchi approach.
• We provide a general proof of the conjecture made in arXiv:0908.1742 which states that regular single center solutions belong to orbits with coinciding $\beta-\gamma$ labels.