@article{MickelssonPaycha2010, author = {Mickelsson, Jouko and Paycha, Sylvie}, title = {The logarithmic residue density of a generalized Laplacian}, series = {Journal of the Australian Mathematical Society}, volume = {90}, journal = {Journal of the Australian Mathematical Society}, number = {1}, publisher = {Cambridge Univ. Press}, address = {Cambridge}, issn = {0263-6115}, doi = {10.1017/S144678871100108X}, pages = {53 -- 80}, year = {2010}, abstract = {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.}, language = {en} } @misc{MickelssonPaycha2011, author = {Mickelsson, Jouko and Paycha, Sylvie}, title = {The logarithmic residue density of a generalized Laplacian}, series = {Postprints der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe}, journal = {Postprints der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe}, number = {649}, issn = {1866-8372}, doi = {10.25932/publishup-41368}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-413680}, pages = {28}, year = {2011}, abstract = {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.}, language = {en} }