N=2 SUGRA BPS multi-center black holes and freudenthal triple systems

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, S, which in this case is a scalar-independent matrix. The anti-involution matrix S can be understood as a Freudenthal duality x̃ = Sx. We show that this duality can be generalized to “Freudenthal transformations” x→λexp(θS)x=ax+bx˜$x \to \lambda \exp \left( {\theta S} \right)x = ax + b\tilde x$ under which the horizon area, ADM mass and intercenter distances scale up leaving constant the scalars at the fixed points. In the special case λ = 1, “S-rotations”, the transformations leave invariant the solution. The standard Freudenthal duality can be written as x˜=exp (π2S) x$\tilde x = {\rm{exp }}\left( {{\pi \over 2}S} \right){\rm{ }}x$. We argue that these generalized transformations leave invariant not only the quadratic preotential theories but also the general stringy extremal quartic form Δ4, Δ4(x) = Δ4(cos θx + sin θx̃) and therefore its entropy at lowest order