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

Understand

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 $Z \neq 0$ 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.