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

Abstract

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 $\mathbb{U}$ _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 $\mathfrak{s}\mathfrak{l}(2)\times \mathfrak{s}\mathfrak{l}(2)$ within ${{\mathfrak{g}}_{2,2 }}$. 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.