On the construction of monopoles for the classical groups. (English) Zbl 0682.32026

In recent years, magnetic monopoles have been extensively studied in at least three different ways, namely by a direct analytic approach, by twistor methods and by an extension, due to W. Nahm, of the ADHM construction for instantons. In the case when the gauge groups is SU(2), a detailed account of these methods and the relationship between them has been given by N. J. Hitchin [Commun. Math. Phys. 83, 579-602 (1982; Zbl 0502.58017); ibid. 89, 145-190 (1983; Zbl 0517.58014)]. The object of this paper is to extend these ideas to the other classical groups, establishing in each case an equivalence between three types of data, namely (A) generic monopoles with maximum symmetry breaking at infinity, (B) generic solutions of Nahm’s equations satisfying appropriate boundary conditions and (C) spectral data.
Reviewer: P.E.Newstead


32L25 Twistor theory, double fibrations (complex-analytic aspects)
53C80 Applications of global differential geometry to the sciences
81T08 Constructive quantum field theory
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
53C05 Connections (general theory)
20G45 Applications of linear algebraic groups to the sciences
Full Text: DOI


[1] Adams, M., Harnad, J., Hurtubise, J.: Isospectral Hamiltonian flows in finite and infinite dimensions: II. Integration of flows. Preprint · Zbl 0717.58051
[2] Atiyah, M. F., Hitchin, N. J.: The geometry and dynamics of magnetic monopoles. Princeton: P.U.P. 1988 · Zbl 0671.53001
[3] Donaldson, S. K.: Nahm’s equations and the classification of monopoles. Commun. Math. Phys.96, 387-408 (1984) · Zbl 0603.58042
[4] Jaffe, A., Taubes, C. H.: Vortices and monopoles. PPh2, Boston: Birkhaüser 1980 · Zbl 0457.53034
[5] Hartshorne, R.: Algebraic geometry. GTM vol.52. Berlin Heidelberg, New York: Springer 1977 · Zbl 0367.14001
[6] Hitchin, N. J.: Monopoles and geodesics. Commun. Math. Phys.83, 579-602 (1982) · Zbl 0502.58017
[7] Hitchin, N. J.: On the construction of monopoles. Commun. Math. Phys.89, 145-190 (1983) · Zbl 0517.58014
[8] Hitchin, N. J.: Linear field equations on self dual spaces. Proc. R. Soc. Lond.A370, 173-191 (1980) · Zbl 0436.53058
[9] Hitchin, N. J., Murray, M. K., Spectral curves and the ADHM construction. Commun. Math. Phys.118, 463-474 (1988) · Zbl 0646.14021
[10] Hurtubise, J. C.: The asymptotic Higgs field of a monopole. Commun. Math. Phys.97, 381-389 (1985) · Zbl 0582.58040
[11] Hurtubise, J. C.: Monopoles and rational maps: A note on a theorem of Donaldson. Commun. Math. Phys.100, 191-196 (1985) · Zbl 0591.58037
[12] Hurtubise, J. C.: The classification of monopoles for the classical groups. Commun. Math. Phys.120, 613-641 (1989) · Zbl 0824.58015
[13] Murray, M. K.: Non-Abelian magnetic monopoles. Commun. Math. Phys.96, 539-565 (1984) · Zbl 0582.58038
[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
[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) · Zbl 0486.35072
[16] Taubes, C. H.: Min-max theory for the Yang-Mills-Higgs equations. Commun. Math. Phys.97, 473-540 (1985) · Zbl 0585.58016
[17] Ward, R. S.: Ansätze for self-dual Yang-Mills fields. Commun. Math. Phys.80, 563-574 (1981)
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