Some Notes on Dirac operators on the ${\rm S}^3$ and ${\rm S}^2$ spheres

TL;DR

This paper analyzes the Dirac and Hodge-de Rham operators on the spheres S^3 and S^2, providing explicit spectral resolutions using globally defined eigenspinors based on Kähler's formalism.

Contribution

It offers a detailed description and spectral resolution of Dirac operators on S^3 and S^2, extending Kähler's formalism to these spheres.

Findings

Explicit spectral resolutions for Dirac operators on S^3 and S^2

Construction of globally defined eigenspinors

Application of Kähler's formalism to these spheres

Abstract

We describe both the Hodge - de Rham and the spin manifold Dirac operator on the spheres ${\rm S}^3$ and ${\rm S}^2$, following the formalism introduced by K\"ahler, and exhibit a complete spectral resolution for them in terms of suitably globally defined eigenspinors.