Eigenspaces of the Spin Dirac operator over S^3

TL;DR

This paper computes the spectrum and eigenvectors of the Spin Dirac operator on the 3-sphere using quaternionic methods and Lie algebra representations, providing explicit spectral data and eigenbasis construction.

Contribution

It offers a detailed calculation of the spectrum and eigenbasis of the Spin Dirac operator on S^3, employing Hitchin's method and representation theory, which was not previously explicitly detailed.

Findings

Explicit spectrum of the Spin Dirac operator on S^3 derived.

Eigenbasis constructed using polynomial representations of sl(2,C).

Methodology can be applied to similar geometric operators on other manifolds.

Abstract

We calculate the spectrum and a basis of eigenvectors for the Spin Dirac operator over the standard 3-sphere. For the spectrum, we use the method of Hitchin which we transfer to quaternions and explain in more detail. The eigenbasis (in terms of polynomials) will be computed by means of representations of sl(2,C).