Structure of the curvature tensor on symplectic spinors

Abstract

We study symplectic manifolds $(M^{2l},\omega)$ equipped with a symplectic torsion-free affine (also called Fedosov) connection $\nabla$ and admitting a metaplectic structure. Let $\mathcal{S}$ be the so called symplectic spinor bundle and let $R^S$ be the curvature tensor field of the symplectic spinor covariant derivative $\nabla^S$ associated to the Fedosov connection $\nabla.$ It is known that the space of symplectic spinor valued exterior differential 2-forms, $\Gamma(M,\bigwedge^2T^*M\otimes {\mathcal{S}}),$ decomposes into three invariant spaces with respect to the structure group, which is the metaplectic group $Mp(2l,\mathbb{R})$ in this case. For a symplectic spinor field $\phi \in \Gamma(M,\mathcal{S}),$ we compute explicitly the projections of $R^S\phi \in \Gamma(M,\bigwedge^2T^*M \otimes \mathcal{S})$ onto the three mentioned invariant spaces in terms of the symplectic Ricci and symplectic Weyl curvature tensor fields of the connection $\nabla.$ Using this decomposition, we derive a complex of first order differential operators provided the Weyl tensor of the Fedosov connection is trivial.