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The classification of monopoles for the classical groups. (English) Zbl 0824.58015
The moduli spaces of monopoles with maximal symmetry breaking at infinity are computed for the groups $$SU(N)$$, $$SO(N)$$ and $$Sp(N)$$ by using the construction of Nahm. The Nahm’s equations are divided into two parts, one invariant under a real group of gauge transformations, the other under a large complex group $$\mathcal G$$ of gauge transformations. It is shown that each $$\mathcal G$$-orbit contains an essentially unique solution to the real equations. Then, the solutions to the complex equations are classified in terms of rational maps. These maps are interpreted in terms of twistor construction of monopoles. It is concluded that the moduli spaces of monopoles are equivalent to spaces of holomorphic maps from $$\mathbb{P}_ 1$$ into flag manifolds.
Reviewer: G.Zet (Iaşi)

##### MSC:
 58D29 Moduli problems for topological structures 58D15 Manifolds of mappings 81T13 Yang-Mills and other gauge theories in quantum field theory
##### Keywords:
moduli space; Nahm construction; twistor; holomorphic map
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##### References:
 [1] Atiyah, M. F., Hitchin, N. J.: The geometry and dynamics of magnetic monopoles. Princeton, NJ: P.U.P. 1988 · Zbl 0671.53001 [2] Buchdahl, N. P.: Hermitian-Einstein connections and stable vector bundles over compact complex surfaces. Math. Ann. (to appear) · Zbl 0617.32044 [3] Donaldson, S. K.: Nahm’s Equations and the classification of monopoles. Commun. Math. Phys.96, 387-407 (1988) · Zbl 0603.58042 · doi:10.1007/BF01214583 [4] Donaldson, S. K.: Anti-self-dual Yang Mills connections over complex algebraic surfaces and stable vector bundles. Proc. Lond. Math. Soc.50, 1-26 (1985) · Zbl 0547.53019 · doi:10.1112/plms/s3-50.1.1 [5] Donaldson, S. K.: Infinite determinants, stable bundles and curvature. Duke Math. J.54, 1, 231-247 (1987) · Zbl 0627.53052 · doi:10.1215/S0012-7094-87-05414-7 [6] Gravesen, J.: On the topology of spaces of holomorphic maps. Roskilde University preprint · Zbl 0696.58014 [7] Hartman, P.: Ordinary differential equations. (Sec. Edn) Boston, MA: Birkhäuser 1982 · Zbl 0476.34002 [8] Hitchin, N. J.: On the construction of monopoles. Commun. Math. Phys.89, 145-190 (1983) · Zbl 0517.58014 · doi:10.1007/BF01211826 [9] Hurtubise, J. C.: Monopoles and rational maps: A note on a theorem of donaldson. Commun. Math. Phys.100, 191-196 (1985) · Zbl 0591.58037 · doi:10.1007/BF01212447 [10] Hurtubise, J. C., Murray, M. K.: On the construction of monopoles for the classical groups. Commun. Math. Phys. (to appear) · Zbl 0682.32026 [11] Jaffe, A., Taubes, C. H.: Vortices and monopoles. Boston, MA: Birkhäuser 1980 · Zbl 0457.53034 [12] Murray, M. K.: Non-Abelian magnetic monopoles. Commun. Math. Phys.96, 539-565 (1984) · Zbl 0582.58038 · doi:10.1007/BF01212534 [13] Nahm, W.: All self-dual multi-monopoles for arbitrary gauge group. CERN Preprint TH-3172 (1981) [14] Taubes, C. H.: Min-Max theory for the Yang-Mills-Higgs equations. Commun. Math. Phys.97, 473-540 (1985) · Zbl 0585.58016 · doi:10.1007/BF01221215 [15] Uhlenbeck, K. K., Yau, S.-T.: On the Existence of Hermitian-Yang-Mills connections in stable vector bundles. Commun. Pure Appl. Math.39, 257-293 (1986) · Zbl 0615.58045 · doi:10.1002/cpa.3160390714
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