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 x = 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 . We argue that these generalized transformations leave invariant not only the quadratic preotential theories but also the general stringy extremal quartic form Δ 4 , Δ 4 ( x ) = Δ 4 (cos θx + sin θx ) and therefore its entropy at lowest order.