Two-Center Black Holes Duality-Invariants for stu Model and its lower-rank Descendants

We classify 2-center extremal black hole charge configurations through duality-invariant homogeneous polynomials, which are the generalization of the unique invariant quartic polynomial for single-center black holes based on homogeneous symmetric cubic special Kaehler geometries. A crucial role is played by an horizontal SL(p,R) symmetry group, which classifies invariants for p-center black holes. For p = 2, a (spin 2) quintet of quartic invariants emerge. We provide the minimal set of independent invariants for the rank-3 N = 2, d = 4 stu model, and for its lower-rank descendants, namely the rank-2 st^2 and rank-1 t^3 models; these models respectively exhibit seven, six and five independent invariants. We also derive the polynomial relations among these and other duality invariants. In particular, the symplectic product of two charge vectors is not independent from the quartic quintet in the t^3 model, but rather it satisfies a degree-16 relation, corresponding to a quartic equation for the square of the symplectic product itself.