Flag Superspaces and Supersymmetric Yang-Mills Equations

  • Yurii Ivanovic Manin
Part of the Progress in Mathematics book series (PM, volume 36)

Abstract

The self-dual Yang-Mills and Einstein equations have a simple geometric meaning, since they imply the vanishing of a part of the curvature tensor of a connection. This connection, the physicists’ gauge potential, is given either on an external vector bundle (the Yang-Mills case) or on the spinorial bundle (the Einstein case) over space-time. After a suitable base change, the relevant part of the curvature becomes the total curvature of the lifted connection along the leaves of a foliation. At least locally (with respect to the initial base manifold), this foliation is a libration and the self-dual field in question can be represented by the vector bundle of horizontal sections along the leaves on the base space of the foliation (Yang-Mills) or by the base space itself (Einstein). This representation is called the Penrose transform. The idea is closely related to the classical Radon transform. One of Penrose’s discoveries was the possibility of using the rigidity of the holomorphic geometry to effectively construct the solutions of the differential equations by geometric means. A mathematician may profitably consult M. F. Atiyah [1] and the references cited therein.

Keywords

Free Sheaf Closed Embedding Tautological Bundle Covariant Differential Trivial Monodromy 
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.

References

  1. [1]
    Atiyah, IMF., Geometry of Yang-Mills fields, Lezioni Fermiani, Pisa, 1979.Google Scholar
  2. [2]
    van Nieuwenhuizen, P., Supergravity, Phys. Reports, 68, (1981), 189–398.MathSciNetCrossRefGoogle Scholar
  3. [3]
    Ferber, A., Supertwistors and conformal supersymmetry, Nuclear Physics, B 132 (1978), 55–64.MathSciNetGoogle Scholar
  4. [4]
    Witten, E., An interpretation of classical Yang-Mills theory, Phys. Letters, 77B (1978), 394–400.CrossRefGoogle Scholar
  5. [5]
    Leites, D.A., Introduction to the theory of supermanifolds, Uspekhi 53: 1 (1980), (in Russian).Google Scholar
  6. [6]
    Kostant, B., Graded manifolds, graded Lie theory, and prequantization, Springer Lecture Notes in Math. 570 (1977), 177–306.MathSciNetCrossRefGoogle Scholar
  7. [7]
    Isenberg, J., Yasskin Ph. B., Green, P.S., Non-selfdual gauge fields, Phys. Letters, 78B, (1978), 464–468.Google Scholar
  8. [8]
    Manin, Yu. I., Gauge fields and holomorphic geometry, in Sovremennye Problemy Matematiki, 17 (1981), Moscow, VINITI (in Russian).Google Scholar
  9. [9]
    Henkin, G.M., Manin, Yu. I., Twistor description of classical YangMills-Dirac fields, Phys. Letters 95B (1980), 405–408.MathSciNetCrossRefGoogle Scholar
  10. [10]
    Henkin, G.M., Manin Yu. I., On the cohomology of twistor flag spaces, Compositio Math. 44, (1981), 103–111.MathSciNetMATHGoogle Scholar
  11. [11]
    Deligne, P., Milne, J.S., Ogus, A., Shin, K., Hodge cycles, motives and Shimura varieties, Springer Lecture Notes in Math. 900 (1982).Google Scholar

Copyright information

© Springer Science+Business Media New York 1983

Authors and Affiliations

  • Yurii Ivanovic Manin
    • 1
  1. 1.Steklov Institute of MathematicsMoscowUSSR

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