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We classify 2-center extremal black hole charge configurations throughduality-invariant homogeneous polynomials, which are the generalization of theunique invariant quartic polynomial for single-center black holes based onhomogeneous symmetric cubic special Kaehler geometries. A crucial role isplayed by an horizontal SL(p,R) symmetry group, which classifies invariants forp-center black holes. For p = 2, a (spin 2) quintet of quartic invariantsemerge. 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^2and rank-1 t^3 models; these models respectively exhibit seven, six and fiveindependent invariants. We also derive the polynomial relations among these andother duality invariants. In particular, the symplectic product of two chargevectors is not independent from the quartic quintet in the t^3 model, butrather it satisfies a degree-16 relation, corresponding to a quartic equationfor the square of the symplectic product itself.

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