The spectrum of twisted Dirac operators on compact flat manifolds

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 $\mathbb{Z}_2^k$, we give a very simple expression for the multiplicities of eigenvalues that allows us to compute explicitly the $\eta$-series, in terms of values of Hurwitz zeta functions, and the $\eta$-invariant. We give the dimension of the space of harmonic spinors and characterize all $\mathbb{Z}_2^k$-manifolds having asymmetric Dirac spectrum.

Furthermore, we exhibit many examples of Dirac isospectral pairs of $\mathbb{Z}_2^k$-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 nonhomeomorphic to each other of the order of $a^n$.