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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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