Symplectic Killing spinors

TL;DR

This paper investigates symplectic Killing spinor fields on symplectic manifolds with metaplectic structures, deriving a necessary zeroth order condition and computing explicit solutions for standard symplectic spaces and the sphere.

Contribution

It introduces a zeroth order necessary condition for symplectic Killing spinors and applies it to compute solutions on standard symplectic spaces and the sphere.

Findings

Derived a zeroth order necessary condition for symplectic Killing spinors.

Computed explicit symplectic Killing spinors on standard symplectic vector spaces.

Determined symplectic Killing spinors on the round sphere S^2.

Abstract

Let $(M,\omega)$ be a symplectic manifold admitting a metaplectic structure (a symplectic analogue of the Riemannian spin structure) and a torsion-free symplectic connection $\nabla.$ Symplectic Killing spinor fields for this structure are sections of the symplectic spinor bundle satisfying a certain first order partial differential equation and they are the main topic of this paper. We derive a necessary condition satisfied by a symplectic Killing spinor field. The advantage of this condition consists in the fact that it is expressed by a zeroth order operator. This condition helps us substantionally to compute the symplectic Killing spinor fields for the standard symplectic vector spaces and the round sphere $S^2$ equipped with the volume form of the round metric.