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.

    Source link


    Leave A Reply