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 $\Lambda <0$ 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-Kähler manifold, and admitting in addition a charged conformal Killing spinor. In the $\Lambda <0$ null case, the supersymmetric solutions define an integrable null Kähler 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 $\mathrm{Sim}(1)\times \mathrm{Sim}(1)$. There appear no self-dual solutions in the null class for either sign of the cosmological constant.

40 pages, uses JHEP3.cls. v2: Appendix and ref. added. v3: Published version

References in corpus (14)

• The supersymmetric configurations of N=2, d=4 supergravity coupled to vector supermultiplets
• Characterization of all the supersymmetric solutions of gauged N=1,d=5 supergravity
• All null supersymmetric backgrounds of N=2, D=4 gauged supergravity coupled to abelian vector multiplets
• Cosmological Einstein-Maxwell Instantons and Euclidean Supersymmetry: Beyond Self-Duality
• Geometry of four-dimensional Killing spinors
• Gauduchon-Tod structures, Sim holonomy and De Sitter supergravity
• HKT Geometry and de Sitter Supergravity
• Time-Dependent Multi-Centre Solutions from New Metrics with Holonomy Sim(n-2)
• All the timelike supersymmetric solutions of all ungauged d=4 supergravities
• Cosmological rotating black holes in five-dimensional fake supergravity
• On the integrability of Einstein-Maxwell-(A)dS gravity in presence of Killing vectors
• Pseudo-Riemannian VSI spaces II
• A Note on Riemannian Anti-self-dual Einstein metrics with Symmetry
• Rotating black holes in an expanding universe from fake supergravity

Cited by in corpus (18)

• Supersymmetric black lenses in five dimensions
• Classification, geometry and applications of supersymmetric backgrounds
• Special Geometry and Space-time Signature
• Five-dimensional vector multiplets in arbitrary signature
• Killing-Yano tensor and supersymmetry of the self-dual Plebanski-Demianski solution
• Supersymmetric near-horizon geometry and Einstein-Cartan-Weyl spaces
• Real Killing Spinors in Neutral Signature
• Klein and Conformal Superspaces, Split Algebras and Spinor Orbits
• Euclidean supersymmetric solutions with the self-dual Weyl tensor
• Type-II Calabi-Yau compactifications, T-duality and special geometry in general spacetime signature
• Quantum Klein Space and Superspace
• A note on the Hyper--CR equation, and gauged $N=2$ supergravity
• Supersymmetry of the Robinson-Trautman solution
• Killing Spinors and Related Symmetries in Six Dimensions
• Flow Equations In Arbitrary Signature
• Cosmological Solutions, a New Wick-Rotation, and the First Law of Thermodynamics
• Supersymmetry algebras in arbitrary signature and their R-symmetry groups
• Neutral Signature Gauged Supergravity Solutions