The spectrum of twisted Dirac operators on compact flat manifolds

Abstract

Let M 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 M , and we derive a formula for the corresponding eta series. In the case of manifolds with holonomy group Z 2 k \mathbb {Z}_2^k , we give a very simple expression for the multiplicities of eigenvalues that allows us to compute explicitly the η \eta -series, in terms of values of Hurwitz zeta functions, and the η \eta -invariant. We give the dimension of the space of harmonic spinors and characterize all Z 2 k \mathbb {Z}_2^k -manifolds having asymmetric Dirac spectrum. Furthermore, we exhibit many examples of Dirac isospectral pairs of Z 2 k \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 n -manifolds, pairwise nonhomeomorphic to each other of the order of a n a^n .