Spectral theory of the Atiyah-Patodi-Singer operator on compact flat manifolds
Let \(M\) be an oriented Riemannian manifold of dimension \(n=4h-1=2m+1\), and let \(\Omega=\Omega(M)\) denote the de~Rham complex of \(M\). The operator \(\mathcal{D}:\Omega^{\text{ev}}\to\Omega^{\text{ev}}\), \(\mathcal{D}\phi=(-1)^{h+p+1}(\star d+d\star)\phi\) if \(\phi\in\Omega^{2p}\), was introduced by \textit{M. F. Atiyah, V. K. Patodi} and \textit{I. M. Singer} because its \(\eta\)-invariant is the defect of the signature operator on any compact Riemannian manifold \(N\) with \(\partial N=M\) [Math. Proc. Camb. Philos. Soc. 77, 43--69 (1975; Zbl 0297.58008)]. The operator \(\mathcal{D}\) is called the APS-boundary operator in this paper. It is elliptic, self-adjoint and of order one, and therefore it has a discrete spectrum \(\text{Spec}_{\mathcal{D}}\). The \(\eta\)-series of \(\mathcal{D}\) is \(\eta_{\mathcal{D}}(s)=\sum_{0\neq\lambda\in\text{Spec}_{\mathcal{D}}}\text{sign}(\lambda)|\lambda|^{-s}\). This series converges for \(\text{Re}(s)>n\) and defines a holomorphic function, which has a holomorphic continuation to \(\mathbb{C}\) with possible poles at \(s=n-k\) for \(k\in\mathbb{N}_0\). Moreover its residue at \(0\) vanishes, and \(\eta=\eta_{\mathcal{D}}(0)\) is its \(\eta\)-invariant, which gives a measure of the asymmetry of \(\text{Spec}_{\mathcal{D}}\). Assume from now on that \(M\) is flat. The paper is devoted to the study of \(\text{Spec}_{\mathcal{D}}\) in this case. To begin with, the authors give an explicit base of eigenfunctions of mixed degree, \((2p,n-2p-1)\), \((2p,n-2p+1)\), together with an expression of the multiplicities of the eigenvalues, in the case where \(M\) is a flat torus. This is illustrated for \(n=7,11\), with small eigenvalues. Next, the authors consider a general flat \(M\), and give a formula expressing the multiplicities of the eigenvalues in terms of the character of the exterior representations evaluated at elements of the maximal torus of \(SO(2m)\). They also give an expression for the characters of such elements. As a byproduct, they get large families of isospectral flat manifolds that are non-homeomorphic to each other. Then, the authors give an expression of \(\eta_{\mathcal{D}}(s)\) in terms of the Hurwitz zeta functions, which is used to compute \(\eta\) in many cases; in particular, when the holonomy group is \(\mathbb{Z}_p\) for odd primes \(p\).
scientific article; zbMATH DE number 6124337
scientific article