Communications in Mathematical Physics

, Volume 80, Issue 3, pp 443–451

Geometrical structure and ultraviolet finiteness in the supersymmetric σ-model

  • Luis Alvarez-Gaume
  • Daniel Z. Freedman


A complete geometrical classification of supersymmetric σ-models is given. Extended supersymmetry requires covariantly constant complex structures, and Kahler and hyperkahler manifolds play a special role. As an application of the classification, it is shown that a particular class of these models is on-shell ultraviolet finite to all orders in perturbation theory.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Zumino, B.: Phys. Lett.87 B, 203 (1979)Google Scholar
  2. 2.
    Alvarez-Gaumé, L., Freedman, D.Z.: Phys. Lett.94 B, 171 (1980)Google Scholar
  3. 3.
    Alvarez-Gaumé, L., Freedman, D.Z.: Phys. Rev. D22, 846 (1980)Google Scholar
  4. 4.
    Townsend, P.K., Roĉek, M.: CERN Preprint Th. 2914 (1980)Google Scholar
  5. 5.
    Alvarez-Gaumé, L.: MIT Preprint Nucl. Phys. B (to appear)Google Scholar
  6. 6.
    Friedan, D.: Phys. Rev. Lett.45, 1057 (1980)Google Scholar
  7. 7.
    This result supercedes that of Ref. 1 in that hermiticity is proved rather than assumedGoogle Scholar
  8. 8.
    Calabi, E.: Ann. Soc. l'E.N.S.12, 266 (1979)Google Scholar
  9. 9.
    Gibbons, G.W., Hawking, S.W.: Phys. Lett.78 B, 430 (1978); Hitchin, N.: Proc. Cambridge Philos. Soc.85, 465 (1979)Google Scholar
  10. 10.
    Freedman, D.Z., Townsend, P.K.: Phys. B (to appear)Google Scholar
  11. 11.
    Lichnerowicz, A.: Théorie globale des connections et des groupes d'holonomie, published by Consiglio Nazionale delle Ricerche 1955Google Scholar
  12. 12.
    See Ref. 11, Chap. III for a proof and further detailsGoogle Scholar
  13. 13.
    Chevalley, C.: Theory of Lie Groups, p. 185. Princeton, New Jersey: Princeton Univ. Press 1946Google Scholar
  14. 14.
    Ref. 11, pp. 258–261Google Scholar
  15. 15.
    Alvarez-Gaumé, L., Freedman, D.Z., Mukhi, S.: M.I.T. Preprint (1980)Google Scholar
  16. 16.
    Weinberg, S.: Gravitation and Cosmology, p. 290. New York: John Wiley & Sons, 1972, Eq. (10.9.3) differs from (23) below by an irrelevant infinitesimal diffeomorphismGoogle Scholar
  17. 17.
    DeWit, B., Grisaru, M.T.: Phys. Rev. D20, 2082 (1979)Google Scholar
  18. 18.
    Poggio, E., Pendleton, H.: Phys. Lett.72 B, 200 (1977);Google Scholar
  19. 18a.
    Jones, D.R.T.: Phys. Lett.72 B, 199 (1977)Google Scholar
  20. 19.
    Grisaru, M.T., Roĉek, M., Siegel, W.: Phys. Rev. Lett.45, 1063 (1980)Google Scholar

Copyright information

© Springer-Verlag 1981

Authors and Affiliations

  • Luis Alvarez-Gaume
    • 1
    • 2
  • Daniel Z. Freedman
    • 1
    • 2
  1. 1.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA
  2. 2.Institute for Theoretical PhysicsState University of New YorkStony BrookUSA

Personalised recommendations