The kernel of the Rarita-Schwinger operator on Riemannian spin manifolds

TL;DR

This paper investigates the Rarita-Schwinger operator on compact Riemannian spin manifolds, identifying conditions for non-trivial kernels and classifying manifolds with special geometric structures that admit Rarita-Schwinger fields.

Contribution

It provides a classification of manifolds with Rarita-Schwinger fields, including Einstein, quaternion Kähler, symmetric, Calabi-Yau, hyperkähler, G2, and Spin(7) manifolds, and relates kernels to harmonic forms.

Findings

Existence of non-trivial kernels on certain Einstein manifolds.

Complete classification for positive quaternion Kähler and symmetric spaces.

Identification of kernels with harmonic forms on special holonomy manifolds.

Abstract

We study the Rarita-Schwinger operator on compact Riemannian spin manifolds. In particular, we find examples of compact Einstein manifolds with positive scalar curvature where the Rarita-Schwinger operator has a non-trivial kernel. For positive quaternion K\"ahler manifolds and symmetric spaces with spin structure we give a complete classification of manifolds admitting Rarita-Schwinger fields. In the case of Calabi-Yau, hyperk\"ahler, $G_2$ and Spin(7) manifolds we find an identification of the kernel of the Rarita-Schwinger operator with certain spaces of harmonic forms. We also give a classification of compact irreducible spin manifolds admitting parallel Rarita-Schwinger fields.