Killing superalgebras for Lorentzian six-manifolds

Abstract

We calculate the Spencer cohomology of the $(1,0)$ Poincaré superalgebras in six dimensions: with and without R-symmetry. As the cases of four and eleven dimensions taught us, we may read off from this calculation a Killing spinor equation which allows the determination of which geometries admit rigidly supersymmetric theories in this dimension. We prove that the resulting Killing spinors generate a Lie superalgebra and determine the geometries admitting the maximal number of such Killing spinors. They are divided in two branches. One branch consists of the lorentzian Lie groups with bi-invariant metrics and, as a special case, it includes the lorentzian Lie groups with a self-dual Cartan three-form which define the maximally supersymmetric backgrounds of $(1,0)$ Poincaré supergravity in six dimensions. The notion of Killing spinor on the other branch does not depend on the choice of a three-form but rather on a one-form valued in the R-symmetry algebra. In this case, we obtain three different (up to local isometry) maximally supersymmetric backgrounds, which are distinguished by the causal type of the one-form.

arXiv comment38 pages (final version to appear in Journal of Geometry and Physics)

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Cited by in corpus (8)

• Contact metric three manifolds and Lorentzian geometry with torsion in six-dimensional supergravity
• Kinematical superspaces
• Symmetries of $\mathcal{N}=(1,0)$ supergravity backgrounds in six dimensions
• Non-Relativistic Supersymmetry on Curved Three-Manifolds
• Supergravities and Branes from Hilbert-Poincaré Series
• Killing superalgebras for lorentzian five-manifolds
• Kaluza-Klein reductions of maximally supersymmetric five-dimensional lorentzian spacetimes
• Decomposable $(5,6)$-solutions in eleven-dimensional supergravity