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

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 Kahler 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. We also prove that the reverse is not true by finding distinct orbits with the same $beta-gamma$ labels, which are distinguished by suitably devised tensor classifiers. Only one of these is generated by regular solutions. Since regular static solutions only occur with nilpotent degree not exceeding 3, we only discuss representatives of these orbits in terms of black hole solutions. We prove that these representatives can be found in the form of a purely dilatonic four-charge solution (the generating solution in D=3) and this allows us to identify the orbit corresponding to the regular four-dimensional metrics.