Analyzing the spectral (a)symmetry of the massless Dirac operator on the 3-torus

TL;DR

This paper investigates the spectral symmetry of the massless Dirac operator on the 3-torus, showing that metric perturbations can induce spectral asymmetry, contrasting with the symmetric spectrum in the flat case.

Contribution

It demonstrates how perturbing the Euclidean metric on the 3-torus can produce spectral asymmetry in the massless Dirac operator, a phenomenon not present in the flat case.

Findings

Spectral asymmetry can be achieved through metric perturbations.

Eigenvalue asymptotics are derived using perturbation theory.

Symmetry of the spectrum is not guaranteed on general 3-manifolds.

Abstract

We analyze the spectrum of the massless Dirac operator on the 3-torus $\mathbb{T}^3$. It is known that it is possible to calculate this spectrum explicitly, that it is symmetric about zero and that each eigenvalue has even multiplicity. However, for a general oriented closed Riemannian 3-manifold $(M,g)$ there is no reason for the spectrum of the massless Dirac operator to be symmetric. Using perturbation theory, we derive the asymptotic formulae for its eigenvalues and prove that by the perturbation of the Euclidean metric on the 3-torus, it is possible to obtain spectral asymmetry of the massless Dirac operator in the axisymmetric case.