Structure of the curvature tensor on symplectic spinors

We study symplectic manifolds (M(21), omega) equipped with a symplectic torsion-free affine (also called Fedosov) connection del 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 del(S) associated to the Fedosov connection del. It is known that the space of symplectic spinor valued exterior differential 2-forms, Gamma(M, Lambda(2) T*M circle times S), decomposes into three invariant subspaces with respect to the structure group, which is the metaplectic group Mp(2l, R) in this case. For a symplectic spinor field phi Gamma(M, s), we compute explicitly the projections of R(S)phi is an element of Gamma(M, Lambda(2) T*M circle times S) onto the three mentioned invariant subspaces in terms of the symplectic Ricci and symplectic Weyl curvature tensor fields of the connection del. Using this decomposition, we derive a complex of first order differential operators provided the Weyl curvature tensor field of the Fedosov connection is trivial. (C) 2010 Elsevier B.V. All rights reserved.