Communications in Mathematical Physics

, Volume 55, Issue 2, pp 117–124 | Cite as

Instantons and algebraic geometry

  • M. F. Atiyah
  • R. S. Ward


Minimum action solutions for SU(2) Yang-Mills fields in Euclidean 4-space correspond, via the Penrose twistor transform, to algebraic bundles on the complex projective 3-space. These bundles in turn correspond to algebraic curves. The implication of these results for the Yang-Mills fields is described. In particular all solutions are rational and can be constructed from a series of AnsätzeA l forl≧1.


Neural Network Statistical Physic Complex System Nonlinear Dynamics Algebraic Geometry 
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  1. 1.
    Atiyah, M.F., Hitchin, N.J., Singer, I.M.: Proc. Nat. Acad. Sci. U.S.74 (1977)Google Scholar
  2. 2.
    Barth, W.: Math. Ann.226, 125–150 (1977)Google Scholar
  3. 3.
    Grauert, H., Mülich, G.: manuscripta math.16, 75–100 (1975)Google Scholar
  4. 4.
    Jackiw, R., Nohl, C., Rebbi, C.: Phys. Rev. D15, 1642–1646 (1977)Google Scholar
  5. 5.
    Jackiw, R., Rebbi, C.: Phys. Letters67B, 189–192 (1977)Google Scholar
  6. 6.
    Maruyama, M.: Nagoya Math. J.58, 25–68 (1975)Google Scholar
  7. 7.
    Newlander, A., Nirenberg, L.: Ann. Math.65, 391–404 (1957)Google Scholar
  8. 8.
    Penrose, R.: The twistor programme. Rept. Math., Phys., to appearGoogle Scholar
  9. 9.
    Schwarz, A.S.: Phys. Letters67B, 172–174 (1977)Google Scholar
  10. 10.
    Serre, J.P.: Ann. Inst. FourierVI, 1–42 (1956)Google Scholar
  11. 11.
    Ward, R.S.: Phys. Letters61A, 81–82 (1977)Google Scholar
  12. 12.
    Witten, E.: Some exact multi-instanton solutions of classical Yang-Mills theory (Preprint, 1976)Google Scholar

Copyright information

© Springer-Verlag 1977

Authors and Affiliations

  • M. F. Atiyah
    • 1
  • R. S. Ward
    • 1
  1. 1.Mathematical InstituteUniversity of OxfordOxfordEngland

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