Supersymmetric Kundt four manifolds and their spinorial evolution flows

We investigate the differential geometry and topology of four-dimensional Lorentzian manifolds ( M , g ) equipped with a real Killing spinor ε , where ε is defined as a section of a bundle of irreducible real Clifford modules satisfying the Killing spinor equation with nonzero real constant. Such triples (M,g,ε ) are precisely the supersymmetric configurations of minimal AdS four-dimensional supergravity and necessarily belong to the class Kundt of space-times, hence we refer to them as supersymmetric Kundt configurations. We characterize a class of Lorentzian metrics on ℝ^2× X , where X is a two-dimensional oriented manifold, to which every supersymmetric Kundt configuration is locally isometric, proving that X must be an elementary hyperbolic Riemann surface when equipped with the natural induced metric. This yields a class of space-times that vastly generalize the Siklos class of space-times describing gravitational waves in AdS _4 . Furthermore, we study the Cauchy problem posed by a real Killing spinor and we prove that the corresponding evolution problem is equivalent to a system of differential flow equations, the real Killing spinorial flow equations, for a family of functions and coframes on any Cauchy hypersurface Σ⊂ M . Using this formulation, we prove that the evolution flow defined by a real Killing spinor preserves the Hamiltonian and momentum constraints of the Einstein equation with negative curvature and is therefore compatible with the latter. Moreover, we explicitly construct all left-invariant evolution flows defined by a Killing spinor on a simply connected three-dimensional Lie group, classifying along the way all solutions to the corresponding constraint equations, some of which also satisfy the constraint equations associated to the Einstein condition.