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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 defines an invariant polynomial-valued differential form. We express it in terms of a finite sum of residues of classical pseudodifferential symbols. In the case of the square of a Dirac operator, these formulas provide a pedestrian proof of the Atiyah–Singer formula for a pure Dirac operator in four dimensions and for a twisted Dirac operator on a flat space of any dimension. These correspond to special cases of a more general formula by Scott and Zagier. In our approach, which is of perturbative nature, we use either a Campbell–Hausdorff formula derived by Okikiolu or a noncommutative Taylor-type formula.
Author details: | Jouko Mickelsson, Sylvie PaychaORCiDGND |
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URN: | urn:nbn:de:kobv:517-opus4-413680 |
DOI: | https://doi.org/10.25932/publishup-41368 |
ISSN: | 1866-8372 |
Title of parent work (English): | Postprints der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe |
Publication series (Volume number): | Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe (649) |
Publication type: | Postprint |
Language: | English |
Date of first publication: | 2019/02/25 |
Publication year: | 2011 |
Publishing institution: | Universität Potsdam |
Release date: | 2019/02/25 |
Tag: | Dirac operators; index; residue |
Issue: | 649 |
Number of pages: | 28 |
Source: | Journal of the Australian Mathematical Society 90 (2011), pp. 53–80 DOI 10.1017/S144678871100108X |
Organizational units: | Mathematisch-Naturwissenschaftliche Fakultät |
DDC classification: | 5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik |
Peer review: | Referiert |
Publishing method: | Open Access |
Grantor: | Cambridge University Press (CUP) |
License (German): | Keine öffentliche Lizenz: Unter Urheberrechtsschutz |
External remark: | Bibliographieeintrag der Originalveröffentlichung/Quelle |