Invariant Spinors on Homogeneous Spheres

TL;DR

This paper classifies invariant spinors on all nine homogeneous spheres, providing explicit descriptions, dimensions, and geometric insights, including new examples of generalized Killing spinors with four eigenvalues.

Contribution

It offers a complete classification of invariant spinors on homogeneous spheres and introduces new examples of generalized Killing spinors with four eigenvalues.

Findings

Explicit classification of invariant spinors on all homogeneous spheres

Determination of the dimension and description of these spinors

Discovery of new generalized Killing spinors with four eigenvalues

Abstract

Using the characterization of the spin representation in terms of exterior forms, we give a complete classification of invariant spinors on the nine homogeneous realizations of the sphere $S^n$. In each of the cases we determine the dimension of the space of such spinors, give their explicit description, and study the underlying related geometric structures depending on the metric. We recover some known results in the Sasaki and 3-Sasaki cases and find several new examples: in particular we give the first known examples of generalized Killing spinors with four distinct eigenvalues.