Skip to main content Accessibility help
×
Home
Hostname: page-component-dc8c957cd-qr7d4 Total loading time: 0.173 Render date: 2022-01-27T17:11:54.449Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": true, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true, "newUsageEvents": true }

THE LOGARITHMIC RESIDUE DENSITY OF A GENERALIZED LAPLACIAN

Published online by Cambridge University Press:  18 May 2011

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: jouko@kth.se)
SYLVIE PAYCHA*
Affiliation:
Laboratoire de Mathématiques, Complexe des Cézeaux, 63177 Aubière, France (email: Sylvie.Paycha@math.univ-bpclermont.fr)
*
For correspondence; e-mail: Sylvie.Paycha@math.univ-bpclermont.fr
Rights & Permissions[Opens in a new window]

Abstract

HTML view is 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.

Type
Research Article
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 177178 (1989), 199229.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), 4369.CrossRefGoogle Scholar
[6]Okikiolu, K., ‘The Campbell–Hausdorff theorem for elliptic operators and a related trace formula’, Duke Math. J. 79 (1995), 687722.CrossRefGoogle Scholar
[7]Okikiolu, K., ‘The multiplicative anomaly for determinants of elliptic operators’, Duke Math. J. 79 (1995), 722749.Google 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), 491536.CrossRefGoogle Scholar
[10]Scott, S., ‘Logarithmic structures and TQFT’, Clay Math. Proc. 12 (2010), 309331.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), 483507.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. 320399.Google Scholar
You have Access
2
Cited by

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
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *