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THE LOGARITHMIC RESIDUE DENSITY OF A GENERALIZED LAPLACIAN

Part of: Partial differential equations on manifolds; differential operators

Published online by Cambridge University Press:  18 May 2011

JOUKO MICKELSSON  and
SYLVIE PAYCHA
JOUKO MICKELSSON
Affiliation:
Department of Mathematics and Statistics, University of Helsinki, FI-00014 Helsinki, Finland Department of Theoretical Physics, Royal Institute of Technology, 10691 Stockholm, Sweden (email: jouko@kth.se)
SYLVIE PAYCHA*
Affiliation:
Laboratoire de Mathématiques, Complexe des Cézeaux, 63177 Aubière, France (email: Sylvie.Paycha@math.univ-bpclermont.fr)
*
✉For correspondence; e-mail: Sylvie.Paycha@math.univ-bpclermont.fr
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Abstract

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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.

Keywords

residueindexDirac operators

MSC classification

Secondary: 58J42: Noncommutative global analysis, noncommutative residues
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 90 , Issue 1 , February 2011 , pp. 53 - 80
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

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