Spectral asymmetry of the massless Dirac operator on a 3-torus

TL;DR

This paper develops a perturbation theory for the smallest eigenvalue of the massless Dirac operator on a 3-torus, revealing spectral asymmetry and relating it to the eta invariant, with explicit calculations for specific metrics.

Contribution

It introduces a novel perturbation approach for eigenvalues with even multiplicity in the Dirac operator on a 3-torus, including explicit eigenvalue evaluations for certain metrics.

Findings

Derived an asymptotic formula for the smallest eigenvalue under metric perturbations

Identified specific metric families with explicitly computable eigenvalues

Connected the eigenvalue asymptotics to the eta invariant

Abstract

Consider the massless Dirac operator on a 3-torus equipped with Euclidean metric and standard spin structure. It is known that the eigenvalues can be calculated explicitly: the spectrum is symmetric about zero and zero itself is a double eigenvalue. The aim of the paper is to develop a perturbation theory for the eigenvalue with smallest modulus with respect to perturbations of the metric. Here the application of perturbation techniques is hindered by the fact that eigenvalues of the massless Dirac operator have even multiplicity, which is a consequence of this operator commuting with the antilinear operator of charge conjugation (a peculiar feature of dimension 3). We derive an asymptotic formula for the eigenvalue with smallest modulus for arbitrary perturbations of the metric and present two particular families of Riemannian metrics for which the eigenvalue with smallest modulus can be evaluated explicitly. We also establish a relation between our asymptotic formula and the eta invariant.