Complex monopoles I: The Haydys monopole equation

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We study complexified Bogomolny monopoles using the complex linear extension of the Hodge star operator, these monopoles can be interpreted as solutions to the Bogomolny equation with a complex gauge group. Alternatively, these equations can be obtained from dimensional reduction of the Haydys instanton equations to 3 dimensions, thus we call them Haydys monopoles. We find that (under mild hypotheses) the smooth locus of the moduli space of finite energy Haydys monopoles on $\mathbb{R}^3$ is a hyperk"ahler manifold in 3-different ways, which contains the ordinary Bogomolny moduli space as a complex Lagrangian submanifold---an (ABA)-brane---with respect to any of these structures. Moreover, using a gluing construction we find an open neighborhood of the normal bundle of this submanifold which is modeled on a neighborhood of the tangent bundle to the Bogomolny moduli space. This is analogous to the case of Higgs bundles over a Riemann surface, where the (co)tangent bundle of holomorphic bundles canonically embeds into the Hitchin moduli space. These results contrast immensely with the case of finite energy Kapustin--Witten monopoles for which we show a vanishing theorem in the second paper of this series [11]. Both papers in this series are self contained and can be read independently.





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