Structure of the curvature tensor on symplectic spinors

TL;DR

This paper analyzes the structure of the curvature tensor on symplectic spinors on symplectic manifolds with Fedosov connections, decomposing it into invariant parts and deriving related differential operators.

Contribution

It explicitly computes projections of the symplectic spinor curvature tensor onto invariant spaces and derives a complex of differential operators under certain conditions.

Findings

Decomposition of the curvature tensor into invariant spaces.

Explicit formulas for projections involving Ricci and Weyl tensors.

Derivation of a differential operator complex when the Weyl tensor vanishes.

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.