Extremal black holes and nilpotent orbits

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.