Symplectic Killing spinors
Abstract
Let (M,ω) be a symplectic manifold admitting a metaplectic structure (a symplectic analogue of the Riemannian spin structure) and a torsion-free symplectic connection ∇. 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 topic of this paper. We derive a necessary condition satisfied by a symplectic Killing spinor field. The advantage of this condition consists in the fact that it is expressed by a zeroth order operator. This condition helps us substantionally to compute the symplectic Killing spinor fields for the standard symplectic vector spaces and the round sphere S2 equipped with the volume form of the round metric.