Black hole solutions to the image-model and their orbits (I) Original Research Article

In this paper we continue the program of the classification of nilpotent orbits using the approach developed in , within the study of black hole solutions in image supergravities. Our goal in this work is to classify static, single center black hole solutions to a specific image four-dimensional “magic” model, with special Kähler scalar manifold image, as orbits of geodesics on the pseudo-quaternionic manifold image with respect to the action of the isometry group image. Our analysis amounts to the classification of the orbits of the geodesic “velocity” vector with respect to the isotropy group image, which include a thorough classification of the nilpotent orbits associated with extremal solutions and reveals a richer structure than the one predicted by the image-labels alone, based on the Kostant–Sekiguchi approach. We provide a general proof of the conjecture made in which states that regular single center solutions belong to orbits with coinciding image-labels. We also prove that the reverse is not true by finding distinct orbits with the same image-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 image) and this allows us to identify the orbit corresponding to the regular four-dimensional metrics. image-orbits with degree of nilpotency greater than 3 are analyzed solely from a group theoretical point of view, leaving a systematic analysis of their possible interpretation in terms of static multicenter or stationary non-static solutions to a future work. We just limit ourselves to give (singular) single-center representatives of these orbits, to be possibly interpreted as singular limits of regular multicenter solutions. We provide the explicit transformations mapping the various image-orbits and in particular BPS into non-BPS regular solutions showing that they in general belong to the complexification of the global symmetry group in image.