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

TL;DR

This paper explicitly derives the superpotential for non-BPS N=2 extremal black holes in a one-modulus model, revealing its complex structure and providing analytic solutions for the warp factor and scalar fields.

Contribution

It introduces a new explicit form of the superpotential W for non-BPS black holes using duality invariants, including radicals, and solves the flow equations analytically.

Findings

Derived superpotential W in terms of duality invariants.

Provided analytic solutions for warp factor and scalar fields.

Highlighted differences from previous models with quadratic series and Z=0 cases.

Abstract

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. 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.