双周期瞬子的Nahm变换
Nahm transform of doubly-periodic instantons
摘要：
本文研究了四维欧氏空间上反自对偶Yang-Mills方程在两个方向上具有周期性的有限能量解，即所谓的双周期瞬子。我们建立了一个循环的想法，涉及这些对象的等效分析和代数几何描述。在第一章中，我们提供了一个概述的问题，并陈述了主要的结果，证明在论文中。在第二章中，我们研究了我们所关心的联络的渐近性态，证明了耦合Dirac算子是Fredholm算子。打好这些基础后，我们准备解决的论文的主要议题，双周期瞬子的Nahm变换的建设。通过结合微分几何和全纯方法，我们在第三章到第五章中证明了双周期瞬子与某些奇异Higgs对，即定义在椭圆曲线上的希钦方程的亚纯解，具有双射性。这个思想的循环在第8章中终于结束了。我们首先介绍了Friedman、Morgan和Witten提出的一个结构，它将每个双周期瞬子与一个由黎曼面加上其上的线丛组成的谱对联系起来。另一方面，Hitchin证明了希格斯对等价于一组类似的数据。我们表明，弗里德曼，摩根和威滕光谱对与双周期瞬子相一致的希钦光谱对与其Nahm变换。(This论文包含数学DG/9909069、数学DG/9910120和数学AG/9909146的论文。
英文摘要：
This work concerns the study of certain finite-energy solutions of the anti-self-dual Yang-Mills equations on Euclidean 4-dimensional space which are periodic in two directions, so-called doubly-periodic instantons. We establish a circle of ideas involving equivalent analytical and algebraic-geometric descriptions of these objects. In the first introductory chapter we provide an overview of the problem and state the main results to be proven in the thesis. In chapter 2, we study the asymptotic behaviour of the connections we are concerned with, and show that the coupled Dirac operator is Fredholm. After laying these foundations, we are ready to address the main topic of the thesis, the construction of a Nahm transform of doubly-periodic instantons. By combining differential-geometric and holomorphic methods, we show in chapters 3 through 5 that doubly-periodic instantons correspond bijectively to certain singular Higgs pairs, i.e. meromorphic solutions of Hitchin's equations defined over an elliptic curve. The circle of ideas is finally closed in chapter 8. We start by presenting a construction due to Friedman, Morgan and Witten that associates to each doubly-periodic instanton a spectral pair consisting of a Riemann surface plus a line bundle over it. On the other hand, it was shown by Hitchin that Higgs pairs are equivalent to a similar set of data. We show that the Friedman, Morgan and Witten spectral pair associated with a doubly-periodic instanton coincides with the Hitchin spectral pair associated with its Nahm transform. (This thesis contains the papers math.DG/9909069, math.DG/9910120 and math.AG/9909146.)
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