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# Complex monopoles I: The Haydys monopole equation

 dc.contributor.author Nagy, Akos dc.contributor.author Oliveira, Gonçalo dc.date.accessioned 2019-07-02T16:34:13Z dc.date.available 2019-07-02T16:34:13Z dc.identifier.uri https://hdl.handle.net/10161/19072 dc.description.abstract 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. dc.subject math.DG dc.subject math.DG dc.subject math-ph dc.subject math.MP dc.subject 53C07, 58D27, 58E15, 70S15 dc.title Complex monopoles I: The Haydys monopole equation dc.type Journal article duke.contributor.id Nagy, Akos|0819340 dc.date.updated 2019-07-02T16:34:12Z pubs.organisational-group Trinity College of Arts & Sciences pubs.organisational-group Duke pubs.organisational-group Mathematics duke.contributor.orcid Nagy, Akos|0000-0002-1799-7631
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