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.

[AB] M.F. Atiyah and R. Bott, The Yang-Mills equations over Riemann surfaces, Philos. Trans. Roy. Soc. London Ser. A 308 (1983) 523-615.
[AMM] A. Alekseev, A. Malkin, and E. Meinrenken, Lie group valued moment maps, J. Differ- ential Geom. 48 (1998), no. 3, 445-495.
[C] K. Corlette, Flat G-bundles with canonical metrics, J. Diff. Geom. 28 (1988) 361-382.
[D1] S. K. Donaldson, Anti self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles, Proceedings of the London Mathematical Society (3) 50 (1985),
1-26.
[D2] S.K. Donaldson, Twisted harmonic maps and the self-duality equations, Proc. London
Math. Soc. 55 (1987) 127-131.
[G] W.M. Goldman, The symplectic nature of fundamental groups of surfaces, Adv. Math. 54
(1984), 200-225.
[GM] W. M. Goldman and J. J. Millon, The deformation theory of representations of funda-
mental groups of compact Ka ̈hler manifolds, Publ. Math. IHES 67 (1988), 43-96.
[GX] W.M. Goldman and E. Z. Xia, Rank one Higgs bundles and representations of funda- mental groups of Riemann surfaces, Mem. Amer. Math. Soc. 193 (2008), no. 904, viii+69
pp.
[Hi] N.J. Hitchin, The self-duality equations on a Riemann surface, Proc. London Math. Soc.
55 (1987) 59-126.
[HKLR] N.J. Hitchin, A. Karlhede, U. Lindstro ̈m and M. Rocˇek, Hyperka ̈hler metrics and super-
[HL1] [HL2] [HL3]
symmetry, Commun. Math. Phys. 108 (1987) 535-589.
Nan-Kuo Ho and C.-C. Melissa Liu, Connected components of the space of surface group representations, IMRN (2003), no. 44, 2359-2372.
Nan-Kuo Ho and C.-C. Melissa Liu, Connected components of the space of surface group representations II, Inter. Math. Res. Notices 16 (2005), 959-979.
Nan-Kuo Ho and C.-C. Melissa Liu, Connected components of surface group represen- tations for complex reductive Lie groups, an appendix in Covering Spaces of Character Varieties by S. Lawton and D. Ramras, New York Journal of Mathematics 21 (2015), 383- 416.
[HWW] Nan-Kuo Ho, Graeme Wilkin and Siye Wu, Hitchin’s Equations on a nonorientable manifold, arXiv:1211.0746v3. Comm. Anal. Geom. 26 (2018), no.4, 857-886.
[J1] L. C.Jeffrey, Extended moduli spaces of flat connections on Riemann surfaces, Math. Ann. 298 (1994), no. 4, 667-692.
92
[J2] L. C.Jeffrey, Group cohomology construction of the cohomology of moduli spaces of flat connections on 2-manifolds, Duke Math. J. 77 (1995), no. 2, 407-429.
[JK1] L. C. Jeffrey and F. C. Kirwan, Localization for nonabelian group actions, Topology 34 (1995), no. 2, 291-327.
[JK2] L. C. Jeffrey and F. C. Kirwan, Localization and the quantization conjecture, Topology 36 (1997), no. 3, 647-693.
[JK3] L. C. Jeffrey and F. C. Kirwan, Intersection theory on moduli spaces of holomorphic bundles of arbitrary rank on a Riemann surface, Ann. of Math. (2) 148 (1998), no. 1, 109-196.
[Kn] A.W. Knapp, Lie groups beyond an introduction, 198-311.
[Ko] S. Kobayashi, Curvature and stability of vector bundles, Proc. Japan Acad. Ser. A. Math.
Sci., 58 (1982), 158-162.
[Kr] P. B. Kronheimer, A hyperka ̈hler structure on the cotangent bundle of a complex Lie
group, arXiv:math/0409253 (2004), 11 pp.
[MaWe] J. Marsden and A. Weinstein, Reduction of symplectic manifolds with symmetry, Rep.
Math. Phys. 5 (1974),121-130.
[MeWo1] E. Meinrenken and C. Woodward, Fusion of Hamiltonian loop group manifolds and
cobordism, Preprint, July 1997.
[MeWo2] E. Meinrenken and C. Woodward, Cobordism for Hamiltonian loop group actions and
flat connections on the punctured two-sphere. Math. Z. 231 (1999), no. 1, 133-168.
[T1] M. Thaddeus, Conformal field theory and the cohomology of the moduli space of stable
bundles, J. Differential Geom. 35 (1992), no. 1, 131-149.
[T2] M. Thaddeus, A perfect Morse function on the moduli space of flat connections, Topology
39 (2000), no. 4, 773-787.
[S1] C. T. Simpson, Constructing variations of Hodge structure using Yang- Mills theory and
applications to uniformization, J. Amer. Math. Soc. 1 (1988), 867-918.
[S2] C. T. Simpson, Higgs bundles and local systems, Publ. Math. IHES 75 (1992), 5-95.
[S3] C. T. Simpson, Moduli of representations of the fundamental group of a smooth projec-
tive variety II, Publ. Math. IHES 80 (1995), 5-79.