A Simple Introduction to Complex Manifolds

  • Luis Alvarez-Gaume
  • Daniel Z. Freedman
Part of the Ettore Majorana International Science Series book series (EMISS, volume 7)


The material discussed at the conference consisted of two applications of Kahler differential geometry nonlinear σ-models [1,2]. Since this material will be published in the standard literature, there is little point in interesting it again here. However, in the course of our work we found that Kahler geometry is a beautiful subject not well-known to most theoretical physicists and the essential part not as accessible as one might hope from the many mathematical textbooks. Kahler geometry not only has applications in supersymmetric σ-models, as was shown first by Zumino [3] and then in our work, but also in quantum gravity where every self-dual gravitational instantation is a Kahler manifold [4].


Complex Manifold Parallel Transport Holonomy Group Mathematics Text Real Manifold 
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.
    L. Alvarez-Gaumé and D.Z. Freedman, Stony Brook preprint ITP-SB-80-13 (1980, to be published in Physical Review D.Google Scholar
  2. 2.
    L. Alvarez-Gaumé and D.Z. Freedman, Stony Brook preprint ITP-SB-80-14 (1980),to be published in Physics Letters B.Google Scholar
  3. 3.
    B. Zumino, Phys. Letters 87B (1979) 203.ADSCrossRefGoogle Scholar
  4. 4.
    M.F. Atiyah, N. Hitchin and I.M. Singer, Proc.Roy. Soc. A362 (1978) 425.MathSciNetADSMATHCrossRefGoogle Scholar
  5. 5.
    S. Kobayashi and K. Nomizu, “Foundations of Differential Geometry”, Vol. II ( Wiley Interscience, New York, 1963 ).MATHGoogle Scholar
  6. 6.
    K. Yano, “Differential Geometry on Complex and Almost Complex Manifolds” ( Macmillan Ed. Co., 1965 ).Google Scholar
  7. 7.
    A. Lichnerowicz, “Théorie Globale des connections et des groupes d’holonomie”, Consiglio Nazionale delle Ricerche, Roma (1962).Google Scholar
  8. 8.
    S. Goldberg, “Curvature and Homology” ( Academic Press, New York, 1962 ).MATHGoogle Scholar
  9. 9.
    G.W. Gibbons and C.N. Pope, Comm. Math. Phys. 66 (1979) 267.MathSciNetADSCrossRefGoogle Scholar
  10. 10.
    H. Eichenherr, Nuclear Phys. B146 (1978) 215.ADSCrossRefGoogle Scholar

Copyright information

© Plenum Press, New York 1980

Authors and Affiliations

  • Luis Alvarez-Gaume
    • 1
  • Daniel Z. Freedman
    • 1
  1. 1.Institute for Theoretical PhysicsState University of New YorkStony Brook, L.I.USA

Personalised recommendations