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Symplectic Killing spinors. (English) Zbl 1249.53093
Summary: Let $$(M,\omega )$$ be a symplectic manifold admitting a metaplectic structure (a symplectic analogue of the Riemannian spin structure) and a torsion-free symplectic connection $$\nabla$$. Symplectic Killing spinor fields for this structure are sections of the symplectic spinor bundle satisfying a certain first order partial differential equation and they are the main object of this paper. We derive a necessary condition which has to be satisfied by a symplectic Killing spinor field. Using this condition one may easily compute the symplectic Killing spinor fields for the standard symplectic vector spaces and the round sphere $$S^2$$ equipped with the volume form of the round metric.
##### MSC:
 53D05 Symplectic manifolds (general theory) 58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.) 53C07 Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills)
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