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Monopoles and maps from \(S^ 2\) to \(S^ 2\); the topology of the configuration space. (English) Zbl 0594.58053
The article is devoted to the study of those topological properties of the Yang-Mills-Higgs equations configuration space which are relevant for the theorem proved earlier by the author which states that the SU(2) Yang-Mills equations on \(R^ 3\) in the Prasad-Sommerfield limit have an infinite number of nonminimal solutions in each path component of the configuration space. It is shown that the configuration space for the SU(2) Yang-Mills-Higgs equations on \(R^ 3\) is homotopic to the space of smooth maps from \(S^ 2\) to \(S^ 2\) and this configuration space indexes a family of twisted Dirac operators. This Dirac family is used to prove that the configuration space does not retract onto any subspace on which the SU(2) Yang-Mills-Higgs functional is bounded.
Reviewer: B.Konopel’chenko

MSC:
58J90 Applications of PDEs on manifolds
81T08 Constructive quantum field theory
55Q99 Homotopy groups
54C05 Continuous maps
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