On the ellipticity of symplectic twistor complexes

TL;DR

This paper investigates symplectic twistor complexes on Fedosov manifolds with metaplectic structures, proving ellipticity of certain truncated complexes and setting the stage for future analytical research.

Contribution

It introduces symplectic twistor complexes analogous to Riemannian spin geometry and proves their ellipticity under specific conditions.

Findings

Truncated parts of symplectic twistor complexes are elliptic.

The complexes are well-defined when the connection is of Ricci type.

Provides a foundation for future analytical studies on these complexes.

Abstract

For a Fedosov manifold (symplectic manifold equipped with a symplectic torsion-free affine connection $\nabla$) admitting a metaplectic structure, we shall investigate two sequences of first order differential operators acting on sections of certain bundles over this manifold. The operators are symplectic analogues of the twistor operators known from Riemannian spin geometry. Therefore we call the mentioned sequences symplectic twistor sequences. These sequences are complexes if the connection $\nabla$ is of Ricci type. We shall prove that the so called truncated parts of these complexes are elliptic. This establishes a background for a future analytic study.