First-order flow equations for extremal black holes in very special geometry

TL;DR

This work develops a systematic method to construct single-center static extremal non-supersymmetric black holes in four-dimensional N=2 supergravity with cubic prepotentials by leveraging five-dimensional very special geometry and the Taub-NUT construction. By formulating first-order flow equations in five dimensions and exploiting the 5d/4d connection, the authors derive corresponding four-dimensional flows described by a four-dimensional superpotential $W_4$, obtaining explicit interpolating solutions and entropy expressions. The approach clarifies how rotating five-dimensional solutions map to static four-dimensional dyons with NUT charge, revealing both supersymmetric and non-supersymmetric branches and a non-standard 5d rotating solution. The results extend prior STU-model findings to general cubic prepotentials and highlight the non-uniqueness in rewriting the black hole potential in terms of a superpotential, with implications for hidden supersymmetry and fake-supergravity interpretations.

Abstract

We construct interpolating solutions describing single-center static extremal non-supersymmetric black holes in four-dimensional N=2 supergravity theories with cubic prepotentials. To this end, we derive and solve first-order flow equations for rotating electrically charged extremal black holes in a Taub-NUT geometry in five dimensions. We then use the connection between five- and four-dimensional extremal black holes to obtain four-dimensional flow equations and we give the corresponding solutions.