Communications in Mathematical Physics

, Volume 89, Issue 2, pp 145–190 | Cite as

On the construction of monopoles

  • N. J. Hitchin


We show that any self-dual SU (2) monopole may be constructed either by Ward's twistor method, or Nahm's use of the ADHM construction. The common factor in both approaches is an algebraic curve whose Jacobian is used to linearize the non-linear ordinary differential equations which arise in Nahm's method. We derive the non-singularity condition for the monopole in terms of this curve and apply the result to prove the regularity of axially symmetric solutions.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Adler, M., van Moerbeke, P.: Linearization of Hamiltonian systems, Jacobi varieties and representation theory. Adv. Math.38, 318–379 (1980)Google Scholar
  2. 2.
    Atiyah, M.F.: Geometry of Yang-Mills fields (Fermi Lectures). Scuola Normale Superiore, Pisa (1979)Google Scholar
  3. 3.
    Atiyah, M.F., Drinfeld, V.G., Hitchin, N.J., Manin, Yu.I.: Construction of instantons. Phys. Lett.65A, 185–187 (1978)Google Scholar
  4. 4.
    Atiyah, M.F., Hitchin, N.J., Singer, I.M.: Self-duality in four dimensional Riemannian geometry. Proc. R. Soc. London A362, 425–461 (1978)Google Scholar
  5. 5.
    Atiyah, M.F., Ward, R.S.: Instantons and algebraic geometry. Commun. Math. Phys.55, 117–124 (1977)Google Scholar
  6. 6.
    Corrigan, E., Goddard, P.: A 4n-monopole solution with 4n-1 degrees of freedom. Commun. Math. Phys.80, 575–587 (1981)Google Scholar
  7. 7.
    Hitchin, N.J.: Linear field equations on self-dual spaces. Proc. R. Soc. London A370, 173–191 (1980)Google Scholar
  8. 8.
    Hitchin, N.J.: Monopoles and geodesics. Commun. Math. Phys.83, 579–602 (1982)Google Scholar
  9. 9.
    Jaffe, A., Taubes, C.: Vortices and monopoles. Boston: Birkhäuser (1980)Google Scholar
  10. 10.
    Nahm, W.: All self-dual multimonopoles for arbitrary gauge groups (preprint), TH. 3172-CERN (1981)Google Scholar
  11. 11.
    Prasad, M.K.: Yang-Mills-Higgs monopole solutions of arbitrary topological charge. Commun. Math. Phys.80, 137–149 (1981)Google Scholar
  12. 12.
    Rawnsley, J.H.: On the Atiyah-Hitchin vanishing theorem for certain cohomology groups of instanton bundles. Math. Ann.241, 43–56 (1979)Google Scholar
  13. 13.
    Schmid, W.: Variation of Hodge structure: the singularities of the period mapping. Invent. Math.22, 211–319 (1973)Google Scholar
  14. 14.
    Wavrik, J.J.: Deforming cohomology classes. Trans. Am. Math. Soc.181, 341–350 (1973)Google Scholar

Copyright information

© Springer-Verlag 1983

Authors and Affiliations

  • N. J. Hitchin
    • 1
  1. 1.St. Catherine's CollegeOxfordEngland

Personalised recommendations