The spectrum of twisted Dirac operators on compact flat manifolds

Details The spectrum of twisted Dirac operators on compact flat manifolds The spectrum of twisted Dirac operators on compact flat manifolds

2003-11-28

arxiv.org/abs/math/0312004v2

Mathematics

Differential Geometry

39 pages, revised and expanded version, to appear in TAMS

Scientific paper

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 $\Z_2^k$, we give a very simple expression for the multiplicities of eigenvalues that allows to compute explicitly the $\eta$-series in terms of values of Riemann-Hurwitz zeta functions, and the $\eta$-invariant. We give the dimension of the space of harmonic spinors and characterize all $\Z_2^k$-manifolds having asymmetric Dirac spectrum. Furthermore, we exhibit many examples of Dirac isospectral pairs of $\Z_2^k$-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 non-homeomorphic to each other.

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