我的論文是關於在具有邊界的黎曼曲面Hitchin方程解上取得的quasi-Hamiltonian結 構。我們讓K是一個緊緻且連通的李群，且讓Σ是一個緊緻且連通的黎曼曲面， 更甚者，我們再假設Σ的屬(genus)是比一大的且Σ是具有邊界的。我們要研究的 是具有邊界的黎曼曲面Hitchin方程解的模空間，且讓它的結構群是K。在我的論 文裡面，我們遇到下面兩種情況:第一種情況，我們讓K是單連通且Σ有任意 數量的邊界連通區。第二種情況，我們讓K不是單連通的。然而在這個情況 裡，我們的討論實際上又分成下面兩種例子:第一種例子是Σ只有一個邊界連 通區，第二種例子則是Σ有一個以上的邊界連通區。我們將會在兩種情況中解 釋我們是如何在以上兩種情況的模空間得到quasi-Hamiltonian結構。
我們也會在第四章中稍微討論一下當我們的Σ是一個圓環面，也就是Σ的屬 (genus)是一的情況。我們假定唐納森和柯萊特的定理是對的，我們可以比較他 的兩個超凱勒結構，並給兩個實際上的例子。

My thesis is about attaining the quasi-Hamiltonian structures on the moduli spaces of solutions to the Hitchin’s equations over Riemann surfaces with boundaries. Let K be a compact connected Lie group, and Σ be a compact Riemann surface of genus bigger than one with non-empty boundary components. We study the moduli spaces of solutions to the Hitchin’s equations with structure group K over Σ. In my thesis, we look at two situations. First, we consider the case when K is simply connected and Σ has arbitrary number of boundary components. Second, we consider the case when K is not simply connected. In this case, the discussion is divided into two subcases: one is with one boundary component; the other is with more than one boundary components. We explain how to obtain quasi-Hamiltonian structures on the moduli spaces in both cases.
We also study the moduli spaces when the based manifolds are closed tori in chapter four. Assuming the theorem of Donaldson and Corlette is valid in this case, we compare two hyperka ̈hler structures on the moduli spaces, and give two concrete examples.

1. Introduction 5
2. Preliminary 7
2.1. Symplectic Reduction and Hyperka ̈hler Reduction 7
2.2. Hamiltonian Loop Group Spaces and Quasi-Hamiltonian Spaces 8
2.3. Hitchin’s Equations 10
3. Hitchin’s Equations for Abelian Groups on Orientable Surfaces without Boundary 13
4. Hitchin’s Equations on Tori 15
4.1. Hyperka ̈hler Structures 15
4.2. Examples 20
5. Hitchin’s Equations for Simply Connected Groups on Oriented Surfaces with Boundaries 30
5.1. Ka ̈hler structures 30
5.2. Hamiltonian loop group construction 35
5.3. Quasi-Hamiltonian construction 41
6. Hitchin’s Equations for Non Simply Connected Groups on Oriented Surfaces with Boundaries 51
6.1. Oriented Surfaces with One boundary Component 51
6.2. Oriented Surfaces with More than One Components 58
7. Induced Quasi-Hamiltonian Structures From Larger Groups 70
7.1. The Case of Homomorphism Images 70
7.2. The Case of Subgroups 77
Appendix A. Some Special Solutions to Hitchin’s Equations 81
Appendix B. The Weyl Groups of A Compact Lie Group and Its Complexification 86
Appendix C. Some Lemmas About Connections 88
References 92

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