Skip to main content
SpringerLink
Log in

On the construction of monopoles for the classical groups

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

ForG a classical group, an equivalence is exhibited between:

  1. A)

    G monopoles over ℝ3, with maximal symmetry breaking at infinity,

  2. B)

    families of (rank (G)) algebraic curves inTℙ1, along with divisors on those curves, satisfying certain constraints,

  3. C)

    solutions of Nahm's equations over (rank(G)) intervals, satisfying the appropriate boundary conditions.

A) and B) are linked by twistor techniques, B) and C) via the Krichever method for solving non-linear differential equations, and A) and C) via the ADHMN construction, providing a unified picture of techniques for solution. Amongst other things, an asymptotic formula for the Higgs field of the monopole is computed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Adams, M., Harnad, J., Hurtubise, J.: Isospectral Hamiltonian flows in finite and infinite dimensions: II. Integration of flows. Preprint

  2. Atiyah, M. F., Hitchin, N. J.: The geometry and dynamics of magnetic monopoles. Princeton: P.U.P. 1988

    Google Scholar 

  3. Donaldson, S. K.: Nahm's equations and the classification of monopoles. Commun. Math. Phys.96, 387–408 (1984)

    Google Scholar 

  4. Jaffe, A., Taubes, C. H.: Vortices and monopoles. PPh2, Boston: Birkhaüser 1980

    Google Scholar 

  5. Hartshorne, R.: Algebraic geometry. GTM vol.52. Berlin Heidelberg, New York: Springer 1977

    Google Scholar 

  6. Hitchin, N. J.: Monopoles and geodesics. Commun. Math. Phys.83, 579–602 (1982)

    Google Scholar 

  7. Hitchin, N. J.: On the construction of monopoles. Commun. Math. Phys.89, 145–190 (1983)

    Google Scholar 

  8. Hitchin, N. J.: Linear field equations on self dual spaces. Proc. R. Soc. Lond.A370, 173–191 (1980)

    Google Scholar 

  9. Hitchin, N. J., Murray, M. K., Spectral curves and the ADHM construction. Commun. Math. Phys.118, 463–474 (1988)

    Google Scholar 

  10. Hurtubise, J. C.: The asymptotic Higgs field of a monopole. Commun. Math. Phys.97, 381–389 (1985)

    Google Scholar 

  11. Hurtubise, J. C.: Monopoles and rational maps: A note on a theorem of Donaldson. Commun. Math. Phys.100, 191–196 (1985)

    Google Scholar 

  12. Hurtubise, J. C.: The classification of monopoles for the classical groups. Commun. Math. Phys.120, 613–641 (1989)

    Google Scholar 

  13. Murray, M. K.: Non-Abelian magnetic monopoles. Commun. Math. Phys.96, 539–565 (1984)

    Google Scholar 

  14. Nahm, W.: The construction of all self-dual multi-monopoles bys the ADHM method, in Monopoles in quantum field theory. Craigie et al. (ed.). Singapore: World Scientific 1982

    Google Scholar 

  15. Taubes, C. H.: The existence of multi-monopole solutions to the non-abelian Yang-Mills Higgs equations for arbitrary simple gauge groups. Commun. Math. Phys.80, 343 (1981)

    Google Scholar 

  16. Taubes, C. H.: Min-max theory for the Yang-Mills-Higgs equations. Commun. Math. Phys.97, 473–540 (1985)

    Google Scholar 

  17. Ward, R. S.: Ansätze for self-dual Yang-Mills fields. Commun. Math. Phys.80, 563–574 (1981)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics, Institute for Advanced Study, 08540, Princeton, N.J., USA

    Jacques Hurtubise

  2. Department of Mathematics, McGill University, Burnside Hall, 805 Sherbrooke St. W., H3A 2K6, Montreal, Quebec, Canada

    Jacques Hurtubise

  3. Department of Mathematics. R.S. Phys. S., The Australian National University, GPO Box 4, 2601, Canberra, ACT, Australia

    Michael K. Murray

Authors
  1. Jacques Hurtubise
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Michael K. Murray
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by A. Jaffe

Research supported in part by NSERC grant A8361 and FCAR grant EQ3518

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hurtubise, J., Murray, M.K. On the construction of monopoles for the classical groups. Commun.Math. Phys. 122, 35–89 (1989). https://doi.org/10.1007/BF01221407

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01221407

Keywords

Search

Navigation