五维接触单极子：刚性及齐次分类
Contact Monopoles in Dimension Five: Rigidity and Homogeneous Classification
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中文总结 AI 辅助
本文在五维接触度量流形上引入接触单极子，证明紧致情形下其具有平行Higgs场，并利用Wang定理对单位切球丛$T_1S^3$上的齐次接触单极子进行分类，得到非平坦$U(1)$及非交换$SO(3)$解。
中文摘要 AI 辅助
我们在接触度量$5$-流形上引入接触单极子并研究其基本几何性质。我们通过Kähler曲面上的Boothby--Wang纤维化构造自然例子，并将该方程与Sasakian $5$-流形上的接触瞬子联系起来。特别地，我们证明具有平行Higgs场的反自对偶接触瞬子提供接触单极子。我们还证明了一个紧致刚性结果：在紧致接触度量$5$-流形上，每个光滑接触单极子都具有平行Higgs场并满足约化曲率方程$F_A\wedge\omega=0$。在Sasakian情形下，这将在通常的横截类型条件下将紧致接触单极子与反自对偶接触瞬子联系起来。我们还解释了为什么相关的形变复形在通常意义下不是椭圆的。然后我们专注于定向Riemannian $3$-流形的单位切球丛，赋予其典范接触度量结构。在此设定下，我们相对于典范适应标架推导出接触单极子方程的局部形式，并证明来自底空间的纯拉回构型是刚性的。最后，对于齐次Sasakian流形$T_1S^3\cong SO(4)/SO(2)$，我们使用Wang定理将不变接触单极子方程约化为有限维代数系统，并对齐次$U(1)$-和$SO(3)$-丛的不变解进行分类。在交换情形中，分类产生非零拓扑荷的非平坦齐次$U(1)$接触单极子。在非交换情形中，除$m=1$外的所有权重都约化为最大环面，而权重$m=1$给出一个真正的非交换齐次$SO(3)$接触单极子分支。
英文摘要
We introduce contact monopoles on contact metric $5$-manifolds and study their basic geometric properties. We construct natural examples from Boothby--Wang fibrations over Kähler surfaces and relate the equation to contact instantons on Sasakian $5$-manifolds. In particular, we show that anti-self-dual contact instantons with a parallel Higgs field provide contact monopoles. We also prove a compact rigidity result: on compact contact metric $5$-manifolds, every smooth contact monopole has parallel Higgs field and satisfies the reduced curvature equation $F_A\wedgeω=0$. In the Sasakian case, this relates compact contact monopoles to anti-self-dual contact instantons under the usual transverse type condition. We also explain why the associated deformation complex is not elliptic in the usual sense. We then focus on unit tangent sphere bundles of oriented Riemannian $3$-manifolds, endowed with their canonical contact metric structures. In this setting, we derive a local form of the contact monopole equation with respect to the canonical adapted frame and prove that pure pullback configurations from the base are rigid. Finally, for the homogeneous Sasakian manifold $T_1S^3\cong SO(4)/SO(2)$, we use Wang's theorem to reduce the invariant contact monopole equation to finite-dimensional algebraic systems and classify the invariant solutions for homogeneous $U(1)$- and $SO(3)$-bundles. In the abelian case, the classification yields non-flat homogeneous $U(1)$ contact monopoles for nonzero topological charge. In the non-abelian case, all weights except $m=1$ reduce to the maximal torus, while the weight $m=1$ gives a genuinely non-abelian branch of homogeneous $SO(3)$ contact monopoles.
发表机构
- University of Cauca（考卡大学）
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