Universal BPS structure of stationary supergravity solutions

TL;DR

This paper explores the algebraic and geometric structures underlying stationary supergravity solutions, revealing a universal BPS framework, characterizing solution moduli spaces, and extending duality transformations in four-dimensional supergravity theories.

Contribution

It introduces a universal BPS structure for stationary supergravity solutions, linking charge matrices to nilpotent orbits and pure spinor constraints, and generalizes duality actions.

Findings

Nilpotency degree of charge matrix C relates to BPS degree.

Explicit solutions for supersymmetry conditions in all pure supergravity theories.

Characterization of moduli space as Lagrangian subvarieties of nilpotent orbits.

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 > of Spin*(2N), and then the characteristic equation becomes equivalent to a generalisation of the Cartan pure spinor constraint on |C>. The invariance of a given solution with respect to supersymmetry is determined by an algebraic `Dirac equation' on the Weyl spinor |C>. 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 with 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 as a Lagrangian subvariety of the variety of nilpotent elements of Lie(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.