On Invariant Structures of Black Hole Charges
  We study "minimal degree" complete bases of duality- and "horizontal"- invariant homogeneous polynomials in the flux representation of two-centered black hole solutions in two classes of D=4 Einstein supergravity models with symmetric vector multiplets' scalar manifolds. Both classes exhibit an SL(2,R) "horizontal" symmetry. The first class encompasses N=2 and N=4 matter-coupled theories, with semi-simple U-duality given by SL(2,R) x SO(m,n); the analysis is carried out in the so-called Calabi-Vesentini symplectic frame (exhibiting maximal manifest covariance) and until order six in the fluxes included. The second class, exhibiting a non-trivial "horizontal" stabilizer SO(2), includes N=2 minimally coupled and N=3 matter coupled theories, with U-duality given by the pseudo-unitary group U(r,s) (related to complex flux representations). Finally, we comment on the formulation of special Kaehler geometry in terms of "generalized" groups of type E7. 
Sergio Ferrara, Alessio Marrani, Armen Yeranyan
hep-th