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Nahm’s equations and the classification of monopoles. (English) Zbl 0603.58042
In the study of Higgs fields over fibre bundles based on $$R^ 3$$ with bundle group SU(2), monopoles occur. These are parametrised either by continuous variables or by moduli spaces $$M^ k$$, $$k\geq 0$$. k is called the topological charge. Monopoles also occur in twistor geometry. The present paper uses the connection established by Nahm between the above two, to settle certain questions on the moduli spaces for $$k=2$$. The Nahm equations which are ordinary differential equations for matrix valued functions, are split into two, one real and one complex. The complex equation alone is invariant under a complex group. The real equation is shown to admit a unique solution on each orbit of the complex group. The equivalence classes of solutions of the complex equation are classified using algebraic methods. The author verifies the conjecture of M. F. Atiyah and M. K. Murray [Ph. D. Thesis, Oxford (1983)] relating moduli spaces for $$k=2$$ to rational functions on $${\mathbb{C}}P^ 1={\mathbb{C}}\cup \{\infty \}$$.