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The Haydys monopole equation. (English) Zbl 1450.53034

The authors consider the complexified Bogomolny monopoles by using the complex linear extension of the Hodge star operator. These monopoles can be interpreted as the dimension reduction of the Haydys monopoles from four dimensions to three dimensions. On the other hand, using the conjugate-linear extension of the Hodge star operator, these monopoles can be interpreted as the dimension reduction of the Kapustin-Witten monopoles from four dimensions to three dimensions.
This paper focus on the case \(M=\mathbb R^3\). A finite energy Kapustin-Witten monopole over \(\mathbb R^3\) must be a Bogomolny monopole. In contrast to the Kapustin-Witten monopoles, there exist finite-energy Haydys monopoles which are not Bogomolny monopoles. In fact, the main result in this paper is to show that under some hypothesis, the Haydys moduli space is a Kähler manifold containing the Bogomolny moduli space as a minimal Lagrangian submanifold. This is proved by constructing solutions to Haydys monopoles on \(\mathbb R^3\). Moreover, there is an open neighborhood of this submanifold that is modeled on a neighborhood of the zero section in the tangent bundle to the Bogomolny moduli space. This is analogous to the case of Higgs bundles over a Riemann surface, where the cotangent bundle of the moduli space of vector bundles embeds into the moduli space of Higgs bundles.

MSC:

53C07 Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills)
58D27 Moduli problems for differential geometric structures
58E15 Variational problems concerning extremal problems in several variables; Yang-Mills functionals
70S15 Yang-Mills and other gauge theories in mechanics of particles and systems
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