On some properties of the Attractor Equations

TL;DR

This paper explores the properties of Attractor Equations in N=2, d=4 supergravity, analyzing their solutions, stability, and geometric structure in the context of extremal black holes with various charges.

Contribution

It provides a detailed analysis of non-BPS Attractor solutions, their stability criteria, and the influence of Special K"ahler Geometry on these solutions.

Findings

Non-BPS solutions correspond to vanishing fermionic gaugino mass determinant.

Stability is governed by the Special K"ahler Geometry of the moduli space.

The 1-modulus case reveals insights into K"ahler gauge choices and entropic functionals.

Abstract

We discuss the Attractor Equations of N=2, $d=4$ supergravity in an extremal black hole background with arbitrary electric and magnetic fluxes (charges) for field-strength two-forms. The effective one-dimensional Lagrangian in the radial (evolution) variable exhibits features of a spontaneously broken supergravity theory. Indeed, non-BPS Attractor solutions correspond to the vanishing determinant of a (fermionic) gaugino mass matrix. The stability of these solutions is controlled by the data of the underlying Special K\"{a}hler Geometry of the vector multiplets' moduli space. Finally, after analyzing the 1-modulus case more in detail, we briefly comment on the choice of the K\"{a}hler gauge and its relevance for the recently discussed entropic functional.