Skip to main content
SpringerLink
Log in

The Dirac Operator on Generalized Taub-NUT Spaces

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We find sufficient conditions for the absence of harmonic L 2 spinors on spin manifolds constructed as cone bundles over a compact Kähler base. These conditions are fulfilled for certain perturbations of the Euclidean metric, and also for the generalized Taub-NUT metrics of Iwai-Katayama, thus proving a conjecture of Vişinescu and the second author.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ammann B.: The Dirac operator on collapsing S 1-bundles. Sémin. Théor. Spec. Géom., Univ. Grenoble 16, 33–42 (1998)

    Google Scholar 

  2. Ammann B.: Spin-Strukturen und das Spektrum des Dirac-Operators. PhD Thesis, Freiburg, 1998

  3. Ammann B., Bär C.: The Dirac Operator on Nilmanifolds and Collapsing Circle Bundles. Ann. Global Anal. Geom. 16, 221–253 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bär C., Gauduchon P., Moroianu A.: Generalized Cylinders in Semi-Riemannian and Spin Geometry. Math. Z. 249, 545–580 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bourguignon J.-P., Gauduchon P.: Spineurs, opérateurs de Dirac et variations de métriques. Commun. Math. Phys. 144, 581–599 (1992)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  6. Cotăescu I.I., Moroianu S., Vişinescu M.: Gravitational and axial anomalies for generalized Euclidean Taub-NUT metrics. J. Phys. A – Math. Gen. 38, 7005–7019 (2005)

    Article  ADS  MATH  Google Scholar 

  7. Cotăescu I.I., Vişinescu M.: Runge-Lenz operator for Dirac field in Taub-NUT background. Phys. Lett. B 502, 229–234 (2001)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  8. Hitchin N.: Harmonic spinors. Adv. in Math. 14, 1–55 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  9. Iwai T., Katayama N.: On extended Taub-NUT metrics. J. Geom. Phys. 12, 55–75 (1993)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  10. Kirchberg K.-D.: An estimation for the first eigenvalue of the Dirac operator on closed Kähler manifolds of positive scalar curvature. Ann. Global Anal. Geom. 3, 291–325 (1986)

    Article  MathSciNet  Google Scholar 

  11. Lott J.: The Dirac operator and conformal compactification. Internat. Math. Res. Notices 2001(4), 171–178 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  12. Moroianu A.: La première valeur propre de l’opérateur de Dirac sur les variétés kähleriennes compactes. Commun. Math. Phys. 169, 373–384 (1995)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  13. Moroianu A.: Spinc Manifolds and Complex Contact Structures. Commun. Math. Phys. 193, 661–673 (1998)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  14. Moroianu, A.:Lectures on Kähler Geometry. LMS Student Texts 69, Cambridge, Cambridge Univ Press, 2007

  15. Moroianu A., Moroianu S.: The Dirac spectrum on manifolds with gradient conformal vector fields. J. Funct. Analysis 253, 207–219 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  16. Moroianu S., Vişinescu M.: L 2-index of the Dirac operator of generalized Euclidean Taub-NUT metrics. J. Phys. A - Math. Gen. 39, 6575–6581 (2006)

    Article  ADS  Google Scholar 

  17. O’Neill B.: Semi-Riemannian geometry. Acad. Press, New York (1983)

    MATH  Google Scholar 

  18. Nistor V.: On the kernel of the equivariant Dirac operator. Ann. Global Anal. Geom. 17, 595–613 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  19. Vaillant, B.: Index- and spectral theory for manifolds with generalized fibered cusps. Dissertation, Bonner Math. Schriften 344, Rheinische Friedrich-Wilhelms-Universität Bonn, 2001

Download references

Author information

Authors and Affiliations

  1. Centre de Mathématiques, École Polytechnique, 91128, Palaiseau Cedex, France

    Andrei Moroianu

  2. Institutul de Matematică al Academiei Române, P. O. Box 1-764, RO-014700, Bucharest, Romania

    Sergiu Moroianu

  3. Şcoala Normală Superioară Bucharest, Calea Griviţei 21, Bucharest, Romania

    Sergiu Moroianu

Authors
  1. Andrei Moroianu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sergiu Moroianu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Andrei Moroianu.

Additional information

Communicated by A. Connes

Rights and permissions

Reprints and permissions

About this article

Cite this article

Moroianu, A., Moroianu, S. The Dirac Operator on Generalized Taub-NUT Spaces. Commun. Math. Phys. 305, 641–656 (2011). https://doi.org/10.1007/s00220-011-1263-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-011-1263-4

Keywords

Search

Navigation