Universal BPS structure of stationary supergravity solutions

Abstract

We study asymptotically flat stationary solutions of four-dimensional supergravity theories via the associated G/H ∗ pseudo-Riemannian non-linear sigma models in three spatial dimensions. The Noether charge C associated to G is shown to satisfy a characteristic equation that determines it as a function of the four-dimensional conserved charges. The matrix C is nilpotent for non-rotating extremal solutions. The nilpotency degree of C is directly related to the BPS degree of the corresponding solution when they are BPS. Equivalently, the charges can be described in terms of a Weyl spinor |C i of Spin ∗ (2N), and then the characteristic equation becomes equivalent to a generalisation of the Cartan pure spinor constraint on |C i . The invariance of a given solution with respect to supersymmetry is determined by an algebraic ‘Dirac equation’ on the Weyl spinor |C i . We explicitly solve this equation for all pure supergravity theories and we characterise the stratified structure of the moduli space of asymptotically Taub–NUT black holes with respect to their BPS degree. The analysis is valid for any asymptotically flat stationary solutions for which the singularities are protected by horizons. The H ∗ -orbits of extremal solutions are identified as Lagrangian submanifolds of nilpotent orbits of G, and so the moduli space of extremal spherically symmetric black holes is identified as a Lagrangian subvariety of the variety of nilpotent elements of g. We also generalise the notion of active duality transformations to an ‘almost action’ of the three-dimensional duality group G on asymptotically flat stationary solutions.