THE SELF-DUALITY EQUATIONS ON A RIEMANN SURFACE
THE SELF-DUALITY EQUATIONS ON A RIEMANN SURFACE
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DOI：
10.1112/plms/s3-55.1.59
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发表时间：
1987-07
影响因子：
1.8
通讯作者：
N. Hitchin
中科院分区：
文献类型：
--
作者：
N. Hitchin
本文研究自对偶杨-米尔斯方程的一类特殊解。数学物理中的自对偶方程最初是在欧氏四维空间上定义的。物理上相关的解是有限作用的解--所谓的"瞬子"。通过在一个方向上施加平移不变性，相同的方程可以在维度上减少到欧几里得3-空间。这些方程也有物理上的关联--在三维空间中有有限作用的解是"磁链"。如果我们把简化过程再进一步，考虑在两个平移下不变的解，我们得到一组平面上的方程。然而，这里没有明确的物理意义，事实上，寻找有限作用量解的尝试失败了。然而，这些是我们要考虑的方程。尽管在R2中缺乏有趣的解，但方程具有重要的性质-共形不变性-这使得它们可以通过共形映射（即黎曼曲面）在R2上建模的流形上定义。这里我们将考虑定义在紧致黎曼曲面上的自对偶方程的解。实际上存在解，正如我们将展示的，所有解的模空间是一个具有极其丰富几何结构的流形，这将是我们研究的重点。它以一种和谐的方式汇集了黎曼几何、拓扑、代数几何和辛几何的主题。阐明同一对象的所有这些方面占了本文的篇幅。自对偶方程是来自规范理论的方程;在几何上，它们是根据主丛上的联系定义的。虽然主丛的群可以任意选择以使方程有意义，但我们在这里将注意力限制在SU（2）或SO（3）的最简单情况。这有两个原因。第一，也是最明显的，是它简化了计算，避免了使用归纳过程，而归纳过程是考虑高阶一般李群所固有的。第二个原因是SU（2）的解与黎曼曲面的内部结构有着密切的关系。作为结果的后果，我们将证明有关解决方案的自我对偶方程，我们了解一些关于模空间的复杂结构的表面本身，即Teichmuller空间。
In this paper we shall study a special class of solutions of the self-dual Yang-Mills equations. The original self-duality equations which arose in mathematical physics were defined on Euclidean 4-space. The physically relevant solutions were the ones with finite action-the so-called 'instantons'. The same equations may be dimensionally reduced to Euclidean 3-space by imposing invariance under translation in one direction. These equations also have physical relevance-the solutions which have finite action in three dimensions are the 'magnetic monopoles'. If we take the reduction process one step further and consider solutions which are invariant under two translations, we obtain a set of equations in the plane. Here, however, there is no clear physical meaning and, indeed, attempts to find finite action solutions have failed. Nevertheless, these are the equations we shall consider. Despite the lack of interesting solutions in R2, the equations have the important property-conformal invariance-which allows them to be defined on manifolds modelled on R2 by conformal maps, namely Riemann surfaces. We shall consider here solutions of the self-duality equations defined on a compact Riemann surface. There are in fact solutions, as we shall show, and the moduli space of all solutions turns out to be a manifold with an extremely rich geometric structure which will be the focus of our study. It brings together in a harmonious way the subjects of Riemannian geometry, topology, algebraic geometry, and symplectic geometry. Illuminating all these facets of the same object accounts for the length of this paper. The self-duality equations are equations from gauge theory; geometrically they are defined in terms of connections on principal bundles. While the group of the principal bundle may be chosen arbitrarily for the equations to make sense, we restrict attention here to the simplest case of SU(2) or SO(3). There are two reasons for this. The first, and most obvious, is that it simplifies calculations and avoids the use of inductive processes which are inherent in the consideration of a general Lie group of higher rank. The second reason is that solutions for SU(2) have an intimate relationship with the internal structure of the Riemann surface. As a consequence of results we shall prove about solutions to the self-duality equations, we learn something about the moduli space of complex structures on the surface itself, namely Teichmuller space.