THE LOGARITHMIC RESIDUE DENSITY OF A GENERALIZED LAPLACIAN
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
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.
Other Physics Topics
IdentifiersURN: urn:nbn:se:kth:diva-36231DOI: 10.1017/S144678871100108XISI: 000292078900005ScopusID: 2-s2.0-79960280302OAI: oai:DiVA.org:kth-36231DiVA: diva2:430641
QC 201107112011-07-112011-07-112011-07-11Bibliographically approved