Symplectic Killing spinors. (English) Zbl 1249.53093

Krýsl, Svatopluk

Commentat. Math. Univ. Carol. 53, No. 1, 19-35 (2012).

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)

Keywords:

Fedosov manifold; symplectic spinor; symplectic Killing spinor; symplectic Dirac operator; Segal-Shale-Weil representation