Geometry of Killing spinors in neutral signature

Abstract

We classify the supersymmetric solutions of minimal N = 2 gauged supergravity in four dimensions with neutral signature. They are distinguished according to the sign of the cosmological constant and whether the vector field constructed as a bilinear of the Killing spinor is null or non-null. In neutral signature the bilinear vector field can be spacelike, which is a new feature not arising in Lorentzian signature. In the non-null case, the canonical form of the metric is described by a fibration over a three-dimensional base space that has holonomy with torsion. We find that a generalized monopole equation determines the twist of the bilinear Killing field, which is reminiscent of an Einstein–Weyl structure. If, moreover, the electromagnetic field strength is self-dual, one gets the Kleinian signature analogue of the Przanowski–Tod class of metrics, namely a pseudo-hermitian spacetime determined by solutions of the continuous Toda equation, conformal to a scalar-flat pseudo-Kähler manifold, and admitting in addition a charged conformal Killing spinor. In the null case, the supersymmetric solutions define an integrable null Kähler structure. In the non-null case, the manifold is a fibration over a Lorentzian Gauduchon–Tod base space. Finally, in the null class, the metric is contained in the Kundt family, and it turns out that the holonomy is reduced to . There appear no self-dual solutions in the null class for either sign of the cosmological constant.