The spectrum of twisted Dirac operators on compact flat manifolds

MIATELLO R., PODESTA' R.

TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY

American Mathematical Society

Lugar: Providence; Año: 2006 vol. 358 p. 4569 - 4603

0002-9947

Let M be an orientable compact flat Riemannian manifold endowed with aspin structure. In this paper we determine the spectrum of Dirac operatorsacting on smooth sections of twisted spinor bundles of M, and we derivea formula for the corresponding eta series. In the case of manifolds withholonomy group Z_2^k, we give a very simple expression for themultiplicities of eigenvalues that allows us to compute explicitly theeta-series in terms of values Hurwitz zeta functions, and theeta-invariant. We give the dimension of the space of harmonic spinorsand characterize all Z_2^k-manifolds having asymmetric Dirac spectrum. Furthermore, we exhibit many examples of Dirac isospectral pairs ofZ_2^k-manifolds which do not satisfy other types of isospectrality. Inone of the main examples, we construct a large family of Dirac isospectralcompact flat n-manifolds, pairwise nonhomeomorphic to each other.