Thursday, June 27, 2013

1306.5689 (Robert J. Downes et al.)

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

Robert J. Downes, Michael Levitin, Dmitri Vassiliev
We consider the massless Dirac operator on a 3-torus. It is known that if the metric is Euclidean then the eigenvalues can be calculated explicitly: the spectrum is symmetric about zero and zero itself is an eigenvalue. The aim of the paper is to show that by perturbing the metric one can shift the zero eigenvalue, thus achieving spectral asymmetry. 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 develop a perturbation theory for the massless Dirac operator which accounts for this charge conjugation symmetry and derive an asymptotic formula for the eigenvalue with smallest modulus. We also present two families of Riemannian metrics for which the eigenvalue with smallest modulus can be evaluated explicitly.
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