Symmetric monopoles. (English) Zbl 0846.53016

Monopoles considered in this paper are pairs \((A, \varphi)\), consisting of an \( su(2)\)-valued connection 1-form \(A\) on \(\mathbb{R}^3\) and an \(su (2)\)-valued Higgs field \(\varphi\), which satisfy the Bogomolny equations \(^*F_A = \nabla_A \varphi\) with suitable additional boundary conditions at infinity. The moduli space of framed monopoles of topological charge \(k\) modulo gauge equivalence is a hyper-Kähler manifold \({\mathcal M}_k\) of dimension \(4k\) [M. Atiyah and N. J. Hitchin, The geometry and dynamics of magnetic monopoles, Princeton Univ. Press (1988; Zbl 0671.53001)].
This interesting paper is concerned with classes of \(k\)-monopole solutions which are invariant under various symmetry groups. As it is hard to study monopoles directly in terms of the fields \((A, \varphi)\), the authors carry out their analysis mostly in terms of other types of mathematical objects associated in a unique way to monopoles: the spectral curve arising from the twistorial approach [N. J. Hitchin, Commun. Math. Phys. 89, 145-190 (1983; Zbl 0517.58014)], the rational curve defined by S. K. Donaldson [ibid. 96, 387-407 (1984; Zbl 0603.58042)], and Nahm’s equations, resulting by a clever adaptation of ADHM construction of instantons. Monopoles invariant under inversion in a fixed plane, monopoles having cyclic symmetry, and monopoles invariant under symmetry groups of regular solids are carefully analyzed.
Although a \(k\)-monopole can be physically described as the superposition of \(k\) unit charge magnetic monopoles, it has a well-defined centre and total phase. In connection with these notions, strongly centered \(k\)-monopoles are defined in Section 5, and it is proved that their moduli space is a geodesic submanifold of the whole moduli space \({\mathcal M}_k\).
Major results are the proof of the existence of a tetrahedrally symmetric 3-monopole and of an octahedrally symmetric 4-monopole, together with the explicit description of the corresponding spectral curves. On the other hand, in spite of some evidence to the contrary, it is proved the nonexistence of an icosahedrally symmetric monopole of charge 6.
In the last section of this rich paper, the authors describe the rational maps associated to strongly centered cyclic symmetric \(k\)-monopoles and discover a novel type of nonplanar \(k\)-monopole scattering process.


53C07 Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills)
32L05 Holomorphic bundles and generalizations
81T13 Yang-Mills and other gauge theories in quantum field theory
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