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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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