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The logarithmic residue density of a generalized Laplacian

  • We show that the residue density of the logarithm of a generalized Laplacian on a closed manifold definesan invariant polynomial-valued differential form. We express it in terms of a finite sum of residues ofclassical pseudodifferential symbols. In the case of the square of a Dirac operator, these formulas providea pedestrian proof of the Atiyah–Singer formula for a pure Dirac operator in four dimensions and for atwisted Dirac operator on a flat space of any dimension. These correspond to special cases of a moregeneral formula by Scott and Zagier. In our approach, which is of perturbative nature, we use either aCampbell–Hausdorff formula derived by Okikiolu or a noncommutative Taylor-type formula.

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Metadaten
Author details:Jouko Mickelsson, Sylvie PaychaORCiDGND
DOI:https://doi.org/10.1017/S144678871100108X
ISSN:0263-6115
ISSN:1446-8107
Parent title (English):Journal of the Australian Mathematical Society
Publisher:Cambridge Univ. Press
Place of publication:Cambridge
Document type:Article
Language:English
Date of first publication:2011/02/01
Year of completion:2010
Release date:2019/02/25
Tag:Dirac operators; index; residue
Volume:90
Issue:1
Page number:28
First page:53
Last Page:80
Organizational units:Mathematisch-Naturwissenschaftliche Fakultät
DDC classification:5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik
Peer review:Referiert
License (German):License LogoUrheberrechtsschutz
External remark:Zweitveröffentlichung in der Schriftenreihe Postprints der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe ; 649