Manton涡旋方程与反自对偶联络
Manton's Vortex Equations and Anti-Self-Dual Connections
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中文总结 AI 辅助
本文研究黎曼曲面上Manton涡旋方程与反自对偶联络的对应关系，通过构造等变全纯丛和不变度量建立双向联系，并给出三种等变结构。
中文摘要 AI 辅助
我们研究了黎曼曲面$X$上Manton五个涡旋方程中的四个方程与反自对偶联络的关系。从每个涡旋方程的解出发，我们在$X\times\mathbb{P}^1$、$X\times\Delta$或$X\times\mathbb{C}$上构造了一个等变全纯丛，并赋予一个不变的Hermite或伪Hermite度量，其相伴联络是反自对偶的。等变结构分别由$SU(2)$、$SU(1,1)$和$SE(2)$的作用给出。反之，我们证明了当相伴联络为反自对偶时，一个带有不变（伪）Hermite度量的等变全纯丛会产生相应涡旋方程的解。
英文摘要
We study four of Manton's five vortex equations on a Riemann surface $X$ in relation to anti-self-dual connections. From a solution of each vortex equation, we construct an equivariant holomorphic bundle over $X$$\times$$\mathbb{P}^1$, $X$$\times$$Δ$, or $X$$\times$$\mathbb{C}$, together with an invariant Hermitian or pseudo-Hermitian metric whose associated connection is anti-self-dual. The equivariant structures are given by the actions of $SU(2)$, $SU(1,1)$, and $SE(2)$, respectively. Conversely, we show that an equivariant holomorphic bundle equipped with an invariant (pseudo-)Hermitian metric gives rise to a solution of the corresponding vortex equation when the associated connection is anti-self-dual.
发表机构
- Research Institute for Mathematical Sciences, Kyoto University（京都大学数学科学研究所）
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