Geometry of Killing spinors in neutral signature

TL;DR

This paper classifies supersymmetric solutions of minimal N=2 gauged supergravity in four dimensions with neutral signature, revealing new features such as spacelike bilinear vectors and detailed geometric structures depending on the cosmological constant.

Contribution

It provides a comprehensive classification of solutions, including new geometric structures and explicit forms, especially highlighting differences from Lorentzian signature cases.

Findings

Spacelike bilinear vectors are possible in neutral signature.

Solutions include fibrations over base spaces with U(1) holonomy and torsion.

No self-dual solutions exist in the null class for either sign of the cosmological constant.

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 $\Lambda<0$ non-null case, the canonical form of the metric is described by a fibration over a three-dimensional base space that has $\text{U}(1)$ 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-K\"ahler manifold, and admitting in addition a charged conformal Killing spinor. In the $\Lambda<0$ null case, the supersymmetric solutions define an integrable null K\"ahler structure. In the $\Lambda>0$ non-null case, the manifold is a fibration over a Lorentzian Gauduchon-Tod base space. Finally, in the $\Lambda>0$ null class, the metric is contained in the Kundt family, and it turns out that the holonomy is reduced to ${\rm Sim}(1)\times{\rm Sim}(1)$. There appear no self-dual solutions in the null class for either sign of the cosmological constant.