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Transactions of the American Mathematical Society

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The spectrum of twisted Dirac operators on compact flat manifolds


Authors: Roberto J. Miatello and Ricardo A. Podestá
Journal: Trans. Amer. Math. Soc. 358 (2006), 4569-4603
MSC (2000): Primary 58J53; Secondary 58C22, 20H15
DOI: https://doi.org/10.1090/S0002-9947-06-03873-6
Published electronically: May 9, 2006
MathSciNet review: 2231389
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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 $ \mathbb{Z}_2^k$, we give a very simple expression for the multiplicities of eigenvalues that allows us to compute explicitly the $ \eta$-series, in terms of values of Hurwitz zeta functions, and the $ \eta$-invariant. We give the dimension of the space of harmonic spinors and characterize all $ \mathbb{Z}_2^k$-manifolds having asymmetric Dirac spectrum.

Furthermore, we exhibit many examples of Dirac isospectral pairs of $ \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$-manifolds, pairwise nonhomeomorphic to each other of the order of $ a^n$.


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Additional Information

Roberto J. Miatello
Affiliation: FaMAF–CIEM, Universidad Nacional de Córdoba, Argentina
Email: miatello@mate.uncor.edu

Ricardo A. Podestá
Affiliation: FaMAF–CIEM, Universidad Nacional de Córdoba, Argentina
Email: podesta@mate.uncor.edu

DOI: https://doi.org/10.1090/S0002-9947-06-03873-6
Keywords: Dirac spectrum, flat manifolds, spinors, isospectrality
Received by editor(s): December 8, 2003
Received by editor(s) in revised form: May 12, 2004, and October 8, 2004
Published electronically: May 9, 2006
Additional Notes: This work was supported by Conicet and Secyt-UNC
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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