The spectrum of twisted Dirac operators on compact flat manifolds

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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 ₂ᵏ, we give a very simple expression for the multiplicities of eigenvalues that allows to compute explicitly the η-series in terms of values of Riemann-Hurwitz zeta functions, and the η-invariant. We give the dimension of the space of harmonic spinors and characterize all ₂ᵏ-manifolds having asymmetric Dirac spectrum. Furthermore, we exhibit many examples of Dirac isospectral pairs of ₂ᵏ-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.