First order flows for N=2 extremal black holes and duality invariants

We derive explicitly the superpotential W for the non-BPS branch of $N=2$ extremal black holes in terms of duality invariants of special geometry. Although this is done for a one-modulus case (the $t^3$ model), the example gives $Z\ne 0$ black holes and captures the basic distinction from previous attempts on the quadratic series (vanishing C tensor) and from the other $Z=0$ cases. The superpotential W turns out to be a non-polynomial expression (containing radicals) of the basic duality invariant quantities. These are the same which enter in the quartic invariant $I_4$ for $N=2$ theories based on symmetric spaces. Using the flow equations generated by W, we also provide the analytic general solution for the warp factor and for the scalar field supporting the non-BPS black holes.