The spectrum of twisted Dirac operators on compact flat manifolds

ROBERTO J. MIATELLO; RICARDO A. PODESTÁ

TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY

American Mathematical Society

Año: 2006 vol. 358 p. 4569 - 4603

0002-9947

Let M be an orientable compact flat Riemannian manifold endowed with a spin structure. In this paper we determine the spectrum of Dirac operators acting on smooth sections of twisted spinor bundles of M, and we derive a formula for the corresponding eta series. In the case of manifolds with holonomy group (Z2)^k, we give a very simple expression for the multiplicities of eigenvalues that allows us to compute explicitly the eta-series in terms of values Hurwitz zeta functions, and the eta-invariant. We give the dimension of the space of harmonic spinors and characterize all (Z2)^k-manifolds having asymmetric Dirac spectrum.Furthermore, we exhibit many examples of Dirac isospectral pairs of (Z2)^k-manifolds which do not satisfy other types of isospectrality. In one of the main examples, we construct a large family of Dirac isospectral compact flat n-manifolds, pairwise nonhomeomorphic to each other.