Mirror Symmetry and The Quintic Model

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林碩(Shuo Lin) 查詢紙本館藏

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論文名稱

(Mirror Symmetry and The Quintic Model)

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摘要(中)

鏡像對稱源於弦論，它是指一對卡拉比-丘流形$X$與$check{X}$之間的一種非常特殊的關係。粗略地說，鏡像對稱交換了$X$上關於其複結構的訊息和$check{X}$上關於其辛結構的訊息，反之亦然。在本文中，我們簡要介紹對於了解最基本的鏡像對稱所需要的數學工具，並在最後仔細地研究五次式這個例子。

我們首先回顧複幾何基本的背景知識，接著介紹關於卡拉比-丘流形的經典結果，比如形變，BTT定理，凱勒錐。進而我們研究複模空間以及凱勒模空間上的一些基本結構，並把成對的鏡像流形相應的模空間上的訊息對等起來，給出一般性的鏡像對稱猜想的敘述。最終，我們將這一想法應用到由五次式所給出的卡拉比-丘三維流形上。

摘要(英)

Mirror symmetry is a mysterious relationship between pairs of Calabi-Yau manifolds $X$ and $check{X}$, arising from string theory. Roughly speaking, it exchanges things related to the complex structure of $X$ with things related to the symplectic structure of $check{X}$, and vice versa. In this article, we will give an introduction to the tools needed to understand the elementary mirror symmetry conjecture. It will end with a detailed working out of the example of the quintic.

We begin with the necessary background on complex geometry, and then introduce some of the classical results about Calabi-Yau manifold including deformation, BTT theorem and K"ahler cone. Next we study some essential structures on the complex moduli space and the K"ahler moduli space. We equate the data of structures arising on these two sides for mirror pairs of Calabi-Yau manifolds, and state the general mirror symmetry conjecture, and finally using this idea carry out the example of quintic Calabi-Yau 3-fold.

關鍵字(中)

★ 鏡像對稱
★ 五次式
★ 卡拉比-丘流形

關鍵字(英)

★ mirror symmetry
★ quintic
★ Calabi-Yau manifold

論文目次

摘要 i
Abstract ii

Contents

1 Background on Complex Geometry 1
1.1 Sheaf Language 1
1.2 Complex Manifolds and Holomorphic Bundles 5
1.3 Kahler Manifolds and Hodge Theory 12
1.4 Chern Classes 19

2 The Classical Geometry of Calabi-Yau Manifolds 25
2.1 Complex Structure Deformation and Bogomolov-Tian-Todorov Theorem 25
2.2 Kahler Cone and Complexified Kahler Moduli Space 32
2.3 A Basic Example of Calabi-Yau Manifolds: Quintic 33

3 J-holomorphic Curves and Gromov-Witten Invariants 39
3.1 J-holomorphic Curves 39
3.2 Gromov-Witten Invariants and (1,1)-Yukawa Coupling 43

4 Variation and Degeneration of Hodge Structures 45
4.1 Variation of Hodge Structures and (1,2)-Yukawa Coupling 45
4.2 Period Maps 49
4.3 Degeneration of Hodge Structures 51

5 Mirror Conjecture 62

6 Quintic, Revisit 63
6.1 Batyrev′s Construction and The Mirror Family of Quintic 63
6.2 Complex Moduli Space of $check{X}$ 66
6.3 The Periods of $check{X}$ 67
6.4 Picard-Fuchs Equations 70
6.5 Equations with Regular Singular Points 72
6.6 Canonical Coordinates 75
6.7 The (1,2)-Yukawa Coupling on $check{X}$ 77

References 80

參考文獻

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指導教授

姚美琳(Mei-Lin Yau)

審核日期

2016-1-26

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