Spectral analysis of the Dirac operator on a 3-sphere

TL;DR

This paper investigates how the eigenvalues of the massless Dirac operator on a 3-sphere change under smooth metric perturbations, deriving explicit formulas and analyzing spectral symmetry and asymmetry.

Contribution

It provides explicit perturbation formulas for the eigenvalues near zero of the Dirac operator on a 3-sphere under arbitrary smooth metric changes, highlighting symmetry preservation and volume effects.

Findings

Eigenvalues near zero remain double eigenvalues under perturbations.

Spectral asymmetry appears only in second-order perturbations.

Explicit eigenvalue formulas are derived for generalized Berger spheres.

Abstract

We study the (massless) Dirac operator on a 3-sphere equipped with Riemannian metric. For the standard metric the spectrum is known. In particular, the eigenvalues closest to zero are the two double eigenvalues +3/2 and -3/2. Our aim is to analyse the behaviour of eigenvalues when the metric is perturbed in an arbitrary smooth fashion from the standard one. We derive explicit perturbation formulae for the two eigenvalues closest to zero, taking account of the second variations. Note that these eigenvalues remain double eigenvalues under perturbations of the metric: they cannot split because of a particular symmetry of the Dirac operator in dimension three (it commutes with the antilinear operator of charge conjugation). Our perturbation formulae show that in the first approximation our two eigenvalues maintain symmetry about zero and are completely determined by the increment of Riemannian volume. Spectral asymmetry is observed only in the second approximation of the perturbation process. As an example we consider a special family of metrics, the so-called generalized Berger spheres, for which the eigenvalues can be evaluated explicitly.