Skip to main content Accessibility help
×
×
Home

THE LOGARITHMIC RESIDUE DENSITY OF A GENERALIZED LAPLACIAN

  • JOUKO MICKELSSON (a1) (a2) and SYLVIE PAYCHA (a3)

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.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      THE LOGARITHMIC RESIDUE DENSITY OF A GENERALIZED LAPLACIAN
      Available formats
      ×

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      THE LOGARITHMIC RESIDUE DENSITY OF A GENERALIZED LAPLACIAN
      Available formats
      ×

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      THE LOGARITHMIC RESIDUE DENSITY OF A GENERALIZED LAPLACIAN
      Available formats
      ×

Copyright

Corresponding author

For correspondence; e-mail: Sylvie.Paycha@math.univ-bpclermont.fr

References

Hide All
[1]Berline, N., Getzler, E. and Vergne, M., Heat Kernels and Dirac Operators, Grundlehren der Mathematischen Wissenschaften, 298 (Springer, Berlin, 1992).
[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).
[3]Kassel, Ch., ‘Le résidu non commutatif (d’après M. Wodzicki)’, Séminaire Bourbaki, Astérisque 177178 (1989), 199229.
[4]Lawson, H. B. and Michelson, M.-L., Spin Geometry (Princeton University Press, Princeton, NJ, 1989).
[5]Mc Kean, H. P. and Singer, I. M., ‘Curvature and the eigenvalues of the Laplacian’, J. Differential Geom. 1 (1967), 4369.
[6]Okikiolu, K., ‘The Campbell–Hausdorff theorem for elliptic operators and a related trace formula’, Duke Math. J. 79 (1995), 687722.
[7]Okikiolu, K., ‘The multiplicative anomaly for determinants of elliptic operators’, Duke Math. J. 79 (1995), 722749.
[8]Paycha, S., ‘Noncommutative formal Taylor expansions and second quantised regularised traces’, in: Combinatorics and Physics, Clay Mathematics Institute Proceedings, to appear.
[9]Paycha, S. and Scott, S., ‘A Laurent expansion for regularised integrals of holomorphic symbols’, Geom. Funct. Anal. 17 (2007), 491536.
[10]Scott, S., ‘Logarithmic structures and TQFT’, Clay Math. Proc. 12 (2010), 309331.
[11]Scott, S., Traces and Determinants of Pseudodifferential Operators, Math. Monographs (Oxford University Press, Oxford, 2009).
[12]Scott, S., ‘The residue determinant’, Comm. Partial Differential Equations 30 (2005), 483507.
[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.
[14]Wodzicki, M., ‘Spectral asymmetry and noncommutative residue’ (in Russian) Thesis, (former) Steklov Institute, Sov. Acad. Sci., Moscow, New York 1984.
[15]Wodzicki, M., Noncommutative Residue. Chapter I. Fundamentals, Lecture Notes in Mathematics, 1289 (Springer, Berlin, 1987), pp. 320399.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of the Australian Mathematical Society
  • ISSN: 1446-7887
  • EISSN: 1446-8107
  • URL: /core/journals/journal-of-the-australian-mathematical-society
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

Keywords

MSC classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed