Extremal black holes and nilpotent orbits

TL;DR

This paper explores the mathematical structure of extremal black holes in supergravity theories, showing that their charges lie in specific nilpotent orbits and deriving a superpotential governing scalar field behavior.

Contribution

It introduces a novel parameterization of nilpotent orbits to analyze extremal black hole solutions and derives the fake superpotential from this framework.

Findings

No naked singularities require specific charge conditions.

Extremal black hole charges lie in a Lagrangian submanifold of a nilpotent orbit.

Derived the fake superpotential governing scalar field evolution.

Abstract

The stationary solutions of a large variety of (super)gravity theories can be described within a non-linear sigma model G / H* coupled to Euclidean gravity in three-dimensions, for which G is a simple group and H* a non-compact real form of its maximal compact subgroup. The absence of naked singularities in four dimensions requires the G Noether charge in 3D to satisfy a characteristic equation that determines it in function of the mass, the NUT charge and the electro-magnetic charges of the solution. It follows that the Noether charge associated to extremal black holes must lie in a certain Lagrangian submanifold of a nilpotent orbit of G. Constructing a suitable parameterisation of this Lagrangian, we are able to determine the so-called `fake superpotential' that governs the radial dependency of the scalar fields.