The spectrum of twisted Dirac operators on compact flat manifolds

2003/11/28 by Roberto J. Miatello, Roberto Miatello, Miatello, Roberto +3 · 1 citation
Mathematics · #58J53 #Advanced Algebra and Geometry #Advanced Operator Algebra Research #Advanced Topics in Algebra #Differential Geometry (math.DG) #FOS: Mathematics #math.DG #msc:58J53

paper · pdf · doi:10.48550/arxiv.math/0312004

39 pages, revised and expanded version, to appear in TAMS

openalex publication_date 2003/11/28 · arxiv created 2004/11/05 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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 \Z₂^k, we give a very simple expression for the multiplicities of eigenvalues that allows to compute explicitly the η-series in terms of values of Riemann-Hurwitz zeta functions, and the η-invariant. We give the dimension of the space of harmonic spinors and characterize all \Z₂^k-manifolds having asymmetric Dirac spectrum. Furthermore, we exhibit many examples of Dirac isospectral pairs of \Z₂^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 non-homeomorphic to each other.

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• Supersymmetry-breaking compactifications on Riemann-flat manifolds

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