Communications in Mathematical Physics

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

The Dirac Operator on Generalized Taub-NUT Spaces

  • Andrei Moroianu
  • 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 
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  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)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bär C., Gauduchon P., Moroianu A.: Generalized Cylinders in Semi-Riemannian and Spin Geometry. Math. Z. 249, 545–580 (2005)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bourguignon J.-P., Gauduchon P.: Spineurs, opérateurs de Dirac et variations de métriques. Commun. Math. Phys. 144, 581–599 (1992)ADSMATHCrossRefMathSciNetGoogle 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)ADSMATHCrossRefGoogle 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)ADSMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Hitchin N.: Harmonic spinors. Adv. in Math. 14, 1–55 (1974)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Iwai T., Katayama N.: On extended Taub-NUT metrics. J. Geom. Phys. 12, 55–75 (1993)ADSMATHCrossRefMathSciNetGoogle 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)MATHCrossRefMathSciNetGoogle 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)ADSMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Moroianu A.: Spinc Manifolds and Complex Contact Structures. Commun. Math. Phys. 193, 661–673 (1998)ADSMATHCrossRefMathSciNetGoogle 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)MATHCrossRefMathSciNetGoogle 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)MATHGoogle Scholar
  18. 18.
    Nistor V.: On the kernel of the equivariant Dirac operator. Ann. Global Anal. Geom. 17, 595–613 (1999)MATHCrossRefMathSciNetGoogle 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

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© 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

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