First order flows for image extremal black holes and duality invariants Original Research Article

We derive explicitly the superpotential W for the non-BPS branch of image extremal black holes in terms of duality invariants of special geometry. Although this is done for a one-modulus case (the image model), the example gives image black holes and captures the basic distinction from previous attempts on the quadratic series (vanishing C tensor) and from the other image 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 image for image 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.