Communications in Mathematical Physics

, Volume 305, Issue 3, pp 641–656 | Cite as

The Dirac Operator on Generalized Taub-NUT Spaces

  • Andrei MoroianuEmail author
  • Sergiu Moroianu


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.


Manifold Line Bundle Dirac Operator Spin Structure Conformal Factor 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ammann B.: The Dirac operator on collapsing S 1-bundles. Sémin. Théor. Spec. Géom., Univ. Grenoble 16, 33–42 (1998)Google Scholar
  2. 2.
    Ammann B.: Spin-Strukturen und das Spektrum des Dirac-Operators. PhD Thesis, Freiburg, 1998Google Scholar
  3. 3.
    Ammann B., Bär C.: The Dirac Operator on Nilmanifolds and Collapsing Circle Bundles. Ann. Global Anal. Geom. 16, 221–253 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bär C., Gauduchon P., Moroianu A.: Generalized Cylinders in Semi-Riemannian and Spin Geometry. Math. Z. 249, 545–580 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bourguignon J.-P., Gauduchon P.: Spineurs, opérateurs de Dirac et variations de métriques. Commun. Math. Phys. 144, 581–599 (1992)ADSzbMATHCrossRefMathSciNetGoogle Scholar
  6. 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)ADSzbMATHCrossRefGoogle Scholar
  7. 7.
    Cotăescu I.I., Vişinescu M.: Runge-Lenz operator for Dirac field in Taub-NUT background. Phys. Lett. B 502, 229–234 (2001)ADSzbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Hitchin N.: Harmonic spinors. Adv. in Math. 14, 1–55 (1974)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Iwai T., Katayama N.: On extended Taub-NUT metrics. J. Geom. Phys. 12, 55–75 (1993)ADSzbMATHCrossRefMathSciNetGoogle Scholar
  10. 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)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Lott J.: The Dirac operator and conformal compactification. Internat. Math. Res. Notices 2001(4), 171–178 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Moroianu A.: La première valeur propre de l’opérateur de Dirac sur les variétés kähleriennes compactes. Commun. Math. Phys. 169, 373–384 (1995)ADSzbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Moroianu A.: Spinc Manifolds and Complex Contact Structures. Commun. Math. Phys. 193, 661–673 (1998)ADSzbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Moroianu, A.:Lectures on Kähler Geometry. LMS Student Texts 69, Cambridge, Cambridge Univ Press, 2007Google Scholar
  15. 15.
    Moroianu A., Moroianu S.: The Dirac spectrum on manifolds with gradient conformal vector fields. J. Funct. Analysis 253, 207–219 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 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)ADSCrossRefGoogle Scholar
  17. 17.
    O’Neill B.: Semi-Riemannian geometry. Acad. Press, New York (1983)zbMATHGoogle Scholar
  18. 18.
    Nistor V.: On the kernel of the equivariant Dirac operator. Ann. Global Anal. Geom. 17, 595–613 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Vaillant, B.: Index- and spectral theory for manifolds with generalized fibered cusps. Dissertation, Bonner Math. Schriften 344, Rheinische Friedrich-Wilhelms-Universität Bonn, 2001Google Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Centre de MathématiquesÉcole PolytechniquePalaiseau CedexFrance
  2. 2.Institutul de Matematică al Academiei RomâneBucharestRomania
  3. 3.Şcoala Normală Superioară BucharestBucharestRomania

Personalised recommendations