Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-26T13:32:42.667Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected])
SYLVIE PAYCHA*
Affiliation:
Laboratoire de Mathématiques, Complexe des Cézeaux, 63177 Aubière, France (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

References

[1]Berline, N., Getzler, E. and Vergne, M., Heat Kernels and Dirac Operators, Grundlehren der Mathematischen Wissenschaften, 298 (Springer, Berlin, 1992).CrossRefGoogle Scholar
[2]Gilkey, P., Invariance Theory, the Heat Equation and the Atiyah–Singer Index Theorem, 2nd edn, Studies in Advanced Mathematics (CRC Press, Boca Raton, FL, 1995).Google Scholar
[3]Kassel, Ch., ‘Le résidu non commutatif (d’après M. Wodzicki)’, Séminaire Bourbaki, Astérisque 177–178 (1989), 199–229.Google Scholar
[4]Lawson, H. B. and Michelson, M.-L., Spin Geometry (Princeton University Press, Princeton, NJ, 1989).Google Scholar
[5]Mc Kean, H. P. and Singer, I. M., ‘Curvature and the eigenvalues of the Laplacian’, J. Differential Geom. 1 (1967), 43–69.Google Scholar
[6]Okikiolu, K., ‘The Campbell–Hausdorff theorem for elliptic operators and a related trace formula’, Duke Math. J. 79 (1995), 687–722.CrossRefGoogle Scholar
[7]Okikiolu, K., ‘The multiplicative anomaly for determinants of elliptic operators’, Duke Math. J. 79 (1995), 722–749.CrossRefGoogle Scholar
[8]Paycha, S., ‘Noncommutative formal Taylor expansions and second quantised regularised traces’, in: Combinatorics and Physics, Clay Mathematics Institute Proceedings, to appear.Google Scholar
[9]Paycha, S. and Scott, S., ‘A Laurent expansion for regularised integrals of holomorphic symbols’, Geom. Funct. Anal. 17 (2007), 491–536.CrossRefGoogle Scholar
[10]Scott, S., ‘Logarithmic structures and TQFT’, Clay Math. Proc. 12 (2010), 309–331.Google Scholar
[11]Scott, S., Traces and Determinants of Pseudodifferential Operators, Math. Monographs (Oxford University Press, Oxford, 2009).Google Scholar
[12]Scott, S., ‘The residue determinant’, Comm. Partial Differential Equations 30 (2005), 483–507.CrossRefGoogle Scholar
[13]Seeley, R. T., ‘Complex powers of an elliptic operator, singular integrals’, Proc. Symp. Pure Math., Chicago (American Mathematical Society, Providence, RI, 1966), pp. 288–307.Google Scholar
[14]Wodzicki, M., ‘Spectral asymmetry and noncommutative residue’ (in Russian) Thesis, (former) Steklov Institute, Sov. Acad. Sci., Moscow, New York 1984.Google Scholar
[15]Wodzicki, M., Noncommutative Residue. Chapter I. Fundamentals, Lecture Notes in Mathematics, 1289 (Springer, Berlin, 1987), pp. 320–399.Google Scholar