Stationary D = 4 black holes in supergravity: The issue of real nilpotent orbits

The complete classification of the nilpotent orbits of SO(2,2)2 in the representation (2,2,2,2), achieved in [14], is applied to the study of multi-center, asymptotically flat, extremal black hole solutions to the STU model. These real orbits provide an intrinsic characterization of regular single-center solutions, which is invariant with respect to the action of the global symmetry group SO(4,4), underlying the stationary solutions of the model, and provide stringent regularity constraints on multi-centered solutions. The known almost-BPS and composite non-BPS solutions are revisited in this setting. We systematically provide, for the relevant SO(2,2)2-nilpotent orbits of the global Noether charge matrix, regular representatives thereof. This analysis unveils a composition law of the orbits according to which those containing regular multi-centered solutions can be obtained as combinations of specific single-center orbits defining the constituent black holes. Some of the SO(2,2)2-orbits of the total Noether charge matrix are characterized as “intrinsically singular” in that they cannot contain any regular solution.