Structure of the curvature tensor on symplectic spinors

Abstract

We study symplectic manifolds ( M 2 l , 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 over M and let R S be the curvature field of the symplectic spinor covariant derivative ∇ S 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 subspaces with respect to the structure group, which is the metaplectic group M p ( 2 l , R ) in this case. For a symplectic spinor field ϕ ∈ Γ ( M , S ) , we compute explicitly the projections of R S ϕ ∈ Γ ( M , ⋀ 2 T ∗ 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.