Structure of the curvature tensor on symplectic spinors

We study symplectic manifolds (M2l,ω)(M2l,ω) equipped with a symplectic torsion-free affine (also called Fedosov) connection ∇∇ and admitting a metaplectic structure. Let SS be the so-called symplectic spinor bundle over MM and let RSRS be the curvature field of the symplectic spinor covariant derivative ∇S∇S associated to the Fedosov connection ∇∇. It is known that the space of symplectic spinor valued exterior differential 22-forms, Γ(M,⋀2T∗M⊗S)Γ(M,⋀2T∗M⊗S), decomposes into three invariant subspaces with respect to the structure group, which is the metaplectic group Mp(2l,R)Mp(2l,R) in this case. For a symplectic spinor field ϕ∈Γ(M,S)ϕ∈Γ(M,S), we compute explicitly the projections of RSϕ∈Γ(M,⋀2T∗M⊗S)RSϕ∈Γ(M,⋀2T∗M⊗S) onto the three mentioned invariant subspaces 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 curvature tensor field of the Fedosov connection is trivial.