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 three 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 Kähler manifold containing the ordinary Bogomolny moduli space as a minimal Lagrangian submanifold—an A-brane. Moreover, using a gluing construction we construct an open neighborhood of this submanifold 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 (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 have showed a vanishing theorem in Nagy and Oliveira (Kapustin–Witten equations on ALE and ALF Gravitational Instantons, 2019).

中文翻译：

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Haydys单极方程

我们使用Hodge星算子的复数线性扩展来研究复杂的Bogomolny单极子，这些单极子可以解释为Bogomolny方程组的复数解。另外，这些方程可通过将Haydys瞬子方程的维数缩减为三个维而获得，因此我们将它们称为Haydys单极子。我们发现（在轻度假设下）\（\ mathbb {R} ^ 3 \）上的有限能量Haydys单极子模空间的光滑轨迹是一个Kähler流形，它包含普通的Bogomolny模空间，作为最小的Lagrangian子流形（A轴）。此外，使用胶粘构造，我们构造了该子流形的开放邻域，该子邻域以Bogomolny模空间的切线束中零截面的邻域为模型。这类似于在黎曼曲面上的希格斯束的情况，其中全纯束的（共）切线束典范地嵌入到Hitchin模空间中。这些结果与有限能量的Kapustin–Witten单极子的情况形成了鲜明对比，对此我们在Nagy和Oliveira中显示了一个消失的定理（ALE和ALF引力瞬时子上的Kapustin–Witten方程，2019）。