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THE LOGARITHMIC RESIDUE DENSITY OF A GENERALIZED LAPLACIAN
KTH, School of Engineering Sciences (SCI), Theoretical Physics, Mathematical Physics.
2011 (English)In: Journal of the Australian Mathematical Society, ISSN 1446-7887, E-ISSN 1446-8107, Vol. 90, no 1, 53-80 p.Article in journal (Refereed) Published
Abstract [en]

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.

Place, publisher, year, edition, pages
2011. Vol. 90, no 1, 53-80 p.
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URN: urn:nbn:se:kth:diva-36231DOI: 10.1017/S144678871100108XISI: 000292078900005Scopus ID: 2-s2.0-79960280302OAI: oai:DiVA.org:kth-36231DiVA: diva2:430641
Note
QC 20110711Available from: 2011-07-11 Created: 2011-07-11 Last updated: 2017-12-11Bibliographically approved

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