TY - JOUR
A1 - Mickelsson, Jouko
A1 - Paycha, Sylvie
T1 - The logarithmic residue density of a generalized Laplacian
JF - Journal of the Australian Mathematical Society
N2 - 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.
KW - residue
KW - index
KW - Dirac operators
Y1 - 2011
U6 - https://doi.org/10.1017/S144678871100108X
SN - 0263-6115
SN - 1446-8107
VL - 90
IS - 1
SP - 53
EP - 80
PB - Cambridge Univ. Press
CY - Cambridge
ER -
TY - GEN
A1 - Mickelsson, Jouko
A1 - Paycha, Sylvie
T1 - The logarithmic residue density of a generalized Laplacian
T2 - Postprints der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe
N2 - 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.
T3 - Postprints der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 649
KW - residue
KW - index
KW - Dirac operators
Y1 - 2019
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-413680
SN - 1866-8372
IS - 649
ER -