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.
Author details: | Jouko Mickelsson, Sylvie PaychaORCiDGND |
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DOI: | https://doi.org/10.1017/S144678871100108X |
ISSN: | 0263-6115 |
ISSN: | 1446-8107 |
Title of parent work (English): | Journal of the Australian Mathematical Society |
Publisher: | Cambridge Univ. Press |
Place of publishing: | Cambridge |
Publication type: | Article |
Language: | English |
Date of first publication: | 2011/02/01 |
Publication year: | 2010 |
Release date: | 2019/02/25 |
Tag: | Dirac operators; index; residue |
Volume: | 90 |
Issue: | 1 |
Number of pages: | 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): | Keine öffentliche Lizenz: Unter Urheberrechtsschutz |
External remark: | Zweitveröffentlichung in der Schriftenreihe Postprints der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe ; 649 |