Extremal multicenter black holes: nilpotent orbits and Tits Satake universality classes

A bstract Four dimensional supergravity theories whose scalar manifold is a symmetric coset manifold U D=4 /H c are arranged into a finite list of Tits Satake universality classes. Stationary solutions of these theories, spherically symmetric or not, are identified with those of an euclidian three-dimensional σ -model, whose target manifold is a Lorentzian coset U D=3 /H * and the extremal ones are associated with H * nilpotent orbits in the K * representation emerging from the orthogonal decomposition of the algebra D =3 with respect to H * . It is shown that the classification of such orbits can always be reduced to the Tits-Satake projection and it is a class property of the Tits Satake universality classes. The construction procedure of Bossard et al of extremal multicenter solutions by means of a triangular hierarchy of integrable equations is completed and converted into a closed algorithm by means of a general formula that provides the transition from the symmetric to the solvable gauge. The question of the relation between H * orbits and charge orbits W of the corresponding black holes is addressed and also reduced to the corresponding question within the Tits Satake projection. It is conjectured that on the vanishing locus of the Taub-NUT current the relation between H * -orbit and W -orbit is rigid and one-to-one. All black holes emerging from multicenter solutions associated with a given H * orbit have the same W -type. For the S 3 model we provide a complete survey of its multicenter solutions associated with all of the previously classified nilpotent orbits of within . We find a new intrinsic classification of the W -orbits of this model that might provide a paradigm for the analogous classification in all the other Tits Satake universality classes.