Spectral Theory of the Atiyah–Patodi–Singer Operator on Compact Flat Manifolds

Abstract

We study the spectral theory of the Dirac-type boundary operator \(\mathcal{D}\) defined by Atiyah, Patodi, and Singer, acting on smooth even forms of a compact flat Riemannian manifold M. We give an explicit formula for the multiplicities of the eigenvalues of \(\mathcal{D}\) in terms of values of characters of exterior representations of SO(n), where n=dim M. As a consequence, we give large families of \(\mathcal{D}\)-isospectral flat manifolds that are non-homeomorphic to each other. Furthermore, we derive expressions for the eta series in terms of special values of Hurwitz zeta functions and, as a result, we obtain a simple explicit expression of the eta invariant.