42517
2010
2011
eng
53
80
28
1
90
article
Cambridge Univ. Press
Cambridge
1
2011-05-18
2011-02-01
--
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 definesan invariant polynomial-valued differential form. We express it in terms of a finite sum of residues ofclassical pseudodifferential symbols. In the case of the square of a Dirac operator, these formulas providea pedestrian proof of the Atiyah–Singer formula for a pure Dirac operator in four dimensions and for atwisted Dirac operator on a flat space of any dimension. These correspond to special cases of a moregeneral formula by Scott and Zagier. In our approach, which is of perturbative nature, we use either aCampbell–Hausdorff formula derived by Okikiolu or a noncommutative Taylor-type formula.
Journal of the Australian Mathematical Society
10.1017/S144678871100108X
0263-6115
1446-8107
<a href="http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-413680">Zweitveröffentlichung in der Schriftenreihe Postprints der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe ; 649</a>
Keine öffentliche Lizenz: Unter Urheberrechtsschutz
Jouko Mickelsson
Sylvie Paycha
eng
uncontrolled
residue
eng
uncontrolled
index
eng
uncontrolled
Dirac operators
Mathematik
Mathematisch-Naturwissenschaftliche Fakultät
Referiert
41368
2011
2019
eng
28
649
postprint
1
2019-02-25
2019-02-25
--
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.
Postprints der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe
10.25932/publishup-41368
urn:nbn:de:kobv:517-opus4-413680
1866-8372
online registration
Journal of the Australian Mathematical Society 90 (2011), pp. 53–80 DOI 10.1017/S144678871100108X
<a href="http://publishup.uni-potsdam.de/opus4-ubp/frontdoor/index/index/docId/42517">Bibliographieeintrag der Originalveröffentlichung/Quelle</a>
Keine öffentliche Lizenz: Unter Urheberrechtsschutz
Jouko Mickelsson
Sylvie Paycha
Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe
649
eng
uncontrolled
residue
eng
uncontrolled
index
eng
uncontrolled
Dirac operators
Mathematik
open_access
Mathematisch-Naturwissenschaftliche Fakultät
Referiert
Open Access
Cambridge University Press (CUP)
Universität Potsdam
https://publishup.uni-potsdam.de/files/41368/pmnr649.pdf