First order flows for [formula omitted] extremal black holes and duality invariants

Anna Ceresole,Gianguido Dall'Agata,Sergio Ferrara,Armen Yeranyan

doi:10.1016/j.nuclphysb.2009.09.003

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 ≠ 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.

Highlights

It has long been known ([1]-[5]) that the properties of the N = 2 extremal, static, spherically symmetric black holes of Einstein-Maxwell theories coupled to the special Kahler geometry of n complex scalar fields zi are encoded in the effective potentialVBH = ZZ + gi ̄DiZD ̄Z, (1.1)where Z(z, z; q, p) is the central charge of the N = 2 supersymmetry algebra, gi ̄ =∂i∂ ̄K(z, z) is the metric of the scalar σ-model and Di ≡ ∂i +∂iK is the Kahler covariant derivative
Using the flow equations generated by W, we provide the analytic general solution for the warp factor and for the scalar field supporting the non-BPS black holes
Z plays the role of a superpotential that drives the first order radial flows for the warp factor and scalar fields towards the black hole horizon: U = −eU |Z|, z i = −2eU gi∂ ̄|Z|

Summary

Introduction

Based on the above considerations, the claim of this paper is that the “fake” superpotential W for the class of configurations corresponding to the non-BPS flows is given in terms of a non-polynomial expression of the purely duality invariant quantities i1–i5 (at least for symmetric special geometries). We will demonstrate this fact by first considering the quadratic and the cubic series and illustrating the path towards the general case

The quadratic series

The cubic series

Full Non-BPS solution

General case and outlook