N=2 SUGRA BPS Multi-center solutions, quadratic prepotentials and Freudenthal transformations

We present a detailed description of N=2 stationary BPS multicenter black hole solutions for quadratic prepotentials with an arbitrary number of centers and scalar fields making a systematic use of the algebraic properties of the matrix of second derivatives of the prepotential, $\mathcal{S}$, which in this case is a scalar-independent matrix. In particular we obtain bounds on the physical parameter of the multicenter solution such as horizon areas and ADM mass. We discuss the possibility and convenience of setting up a basis of the symplectic vector space built from charge eigenvectors of the $\ssigma$, the set of vectors $(\Ppm q_a)$ with $\Ppm$ $\ssigma$-eigenspace proyectors. The anti-involution matrix $\mathcal{S}$ can be understood as a Freudenthal duality $\tilde{x}=\ssigma x$. We show that this duality can be generalized to "Freudenthal transformations" $$x\to \lambda\exp(\theta \ssigma) x= a x+b\tilde{x}$$ under which the horizon area, ADM mass and intercenter distances scale up leaving constant the fix point scalars. In the special case $\lambda=1$, "$\ssigma$-rotations", the transformations leave invariant the solution. The standard Freudental duality can be written as $\tilde x= \exp(\pi/2 \ssigma) x .$ We argue that these generalized transformations leave also invariant the general stringy extremal quartic form $\Delta_4$, $\Delta_4(x)= \Delta_4(\cos\theta x+\sin\theta\tilde{x})$.