Structure of the curvature tensor on symplectic spinors

We study symplectic manifolds (M, ω) equipped with a symplectic torsion-free affine (also called Fedosov) connection ∇ and admitting a metaplectic structure. Let S be the so called symplectic spinor bundle and let R be the curvature tensor field of the symplectic spinor covariant derivative ∇ associated to the Fedosov connection ∇. It is known that the space of symplectic spinor valued exterior differential 2-forms, Γ(M, ∧2 T M ⊗ S), decomposes into three invariant spaces with respect to the structure group, which is the metaplectic group Mp(2l,R) in this case. For a symplectic spinor field φ ∈ Γ(M,S), we compute explicitly the projections of Rφ ∈ Γ(M, ∧2 T M⊗S) onto the three mentioned invariant spaces in terms of the symplectic Ricci and symplectic Weyl curvature tensor fields of the connection ∇. Using this decomposition, we derive a complex of first order differential operators provided the Weyl tensor of the Fedosov connection is trivial. Math. Subj. Class.: 53C07, 53D05, 58J10.