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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.

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Author details:Jouko Mickelsson, Sylvie PaychaORCiDGND
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
Date of first publication:2019/02/25
Completion year:2011
Publishing institution:Universität Potsdam
Release date:2019/02/25
Tag:Dirac operators; index; residue
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):License LogoUrheberrechtsschutz
External remark:Bibliographieeintrag der Originalveröffentlichung/Quelle