Supersymmetric Kundt four manifolds and their spinorial evolution flows

Abstract

We investigate the differential geometry and topology of four-dimensional Lorentzian manifolds $(M,g)$ equipped with a real Killing spinor $\varepsilon$, where $\varepsilon$ is defined as a section of a bundle of irreducible real Clifford modules satisfying the Killing spinor equation with non-zero real constant. Such triples $(M,g,\varepsilon)$ are precisely the supersymmetric configurations of minimal 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 $\mathbb{R}^2\times X$, 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 $\Sigma\subset M$. 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.