這篇論文包含了兩個部分。在第一個部分，我們將會提及經典的結果，即一個 principal U (n)-bundle 上的 flat connections 的模空間與向量叢上的 flat connections 的模空間。更深入地說，當向量叢上具備一個 Hermitian 結構 h ，且該向量叢上的 flat connections 是與 h 相容的，則在上述的兩個模空間存在一個一對一對應。

在第二部分，我們將結構群改為 Sp(n)，並試著找到一個條件，建構出與上述的情況類似的結果。為了達成這一點，我們必須先處理一個問題：怎麼樣的向量叢會與 principal Sp(n)-bundle 對應？大致上地說，該向量叢除了要有 Hermitian 結構外，還需要是所謂的 “symplectic”。至於在連絡（connections）的層次上，一個 flat Sp(n)-connection，在我們的結果下，可以對應到一個向量叢上與 h 相容的，並且滿足一些與 symplectic 結構有特定關係的連絡。

This thesis consists of two parts. In the first part, we recall the classical result on the relation between the moduli space of flat connections over a principal U (n)-bundle and the moduli space of flat connections over a vector bundle. In particular, there is an one-to-one correspondence when the vector bundle has a Hermitian structure h and the flat connections on it are compatible with h.

In the second part, we try to find the precise condition when the structure group of the principal bundle is Sp(n), the (compact) symplectic group. To accomplish that, we need to deal with the question: What kind of complex vector bundle E can correspond to a principal S p (n)-bundle? Roughly speaking, the vector bundle need to be Hermitian and “symplectic”. As for the connection level, a flat Sp(n)-connection can correspond to the connection which is h-compatible and satisfies some relation with the symplectic structure.

1 Introduction 1

2 The Case for U(n) 2
2.1 Some basic facts and lemmas 2
2.2 The one-to-one correspondence between Pr_U(n)(M) and Vec_h(M;n) 3
2.2.1 From Pr_U(n)(M) to Vec_h(M;n) 3
2.2.2 From Vec_h(M;n) to Pr_U(n)(M) 6
2.2.3 Proof of the theorem 2.2 7
2.3 On the connection level 9

3 The Case for Sp(n) 13
3.1 Symplectic structures and property J 13
3.1.1 Symplectic structure on a complex vector bundle 13
3.1.2 The endomorphism J on a Hermitian vector bundle 14
3.2 The one-to-one correspondence between Pr_Sp(n)(M) and Vec_h(M;J;2n) 17
3.2.1 From Vec_h(M;J;2n) to Pr_Sp(n)(M) 17
3.2.2 From Pr_Sp(n)(M) to Vec_h(M;J;2n) 18
3.3 On connection level 18
3.4 Conclusions 21

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