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

Downes, R. J., Levitin, M. ORCID: https://orcid.org/0000-0003-0020-3265 and Vassiliev, D. (2013) Spectral asymmetry of the massless Dirac operator on a 3-torus. Journal of Mathematical Physics, 54 (11). 111503. ISSN 0022-2488 doi: 10.1063/1.4828858

Abstract/Summary

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.

Item Type	Article
URI	https://reading-clone.eprints-hosting.org/id/eprint/34885
Identification Number/DOI	10.1063/1.4828858
Refereed	Yes
Divisions	Science > School of Mathematical, Physical and Computational Sciences > Department of Mathematics and Statistics
Publisher	American Institute of Physics
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