Complex of twistor operators in symplectic spin geometry

TL;DR

This paper constructs a sequence of differential operators on symplectic manifolds with metaplectic structures, forming a complex under certain curvature conditions, thus extending twistor theory into symplectic geometry.

Contribution

It introduces a new complex of differential operators in symplectic spin geometry, generalizing twistor operators from Riemannian geometry and establishing conditions for the complex's validity.

Findings

Sequence forms a complex when the symplectic Weyl curvature tensor vanishes.

Operators include symplectic analogues of Riemannian twistor operators.

Applicable to Ricci type symplectic manifolds with metaplectic structures.

Abstract

For a symplectic manifold admitting a metaplectic structure (a symplectic analogue of the Riemannian spin structure), we construct a sequence consisting of differential operators using a symplectic torsion-free affine connection. All but one of these operators are of first order. The first order ones are symplectic analogues of the twistor operators known from Riemannian spin geometry. We prove that under the condition the symplectic Weyl curvature tensor field of the symplectic connection vanishes, the mentioned sequence forms a complex. This gives rise to a new complex for the so called Ricci type symplectic manifolds, which admit a metaplectic structure.