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研究生：	林碩 Shuo Lin
論文名稱：	Mirror Symmetry and The Quintic Model
指導教授：	姚美琳 Mei-Lin Yau
口試委員:
學位類別：	碩士 Master
系所名稱：	理學院 - 數學系 Department of Mathematics
論文出版年：	2016
畢業學年度：	104
語文別：	英文
論文頁數：	87
中文關鍵詞：	鏡像對稱、五次式、卡拉比-丘流形
外文關鍵詞：	mirror symmetry, quintic, Calabi-Yau manifold
相關次數：	點閱：16 下載：0
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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。

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.

摘要                                                    i
Abstract                                               ii

Contents

Background on Complex Geometry                        1
1 Sheaf Language                                      1
2 Complex Manifolds and Holomorphic Bundles           5
3 Kahler Manifolds and Hodge Theory                  12
4 Chern Classes                                      19

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

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

Variation and Degeneration of Hodge Structures       45
1 Variation of Hodge Structures and (1,2)-Yukawa Coupling                                               45
2 Period Maps                                        49
3 Degeneration of Hodge Structures                   51

Mirror Conjecture                                    62

Quintic, Revisit                                     63
1 Batyrev's Construction and The Mirror Family of Quintic                                                63
2 Complex Moduli Space of $\check{X}$                66
3 The Periods of $\check{X}$                         67
4 Picard-Fuchs Equations                             70
5 Equations with Regular Singular Points             72
6 Canonical Coordinates                              75
7 The (1,2)-Yukawa Coupling on $\check{X}$           77

References                                              80
                                

A. Bertram, Another way to enumerate rational curves with torus actions, Inventiones mathematicae 142 (2000), 487-512.

G. Bredon, Topology and Geometry, Graduate Texts in Mathematics 139, Springer-Verlag, 1993.

P. Candelas, X. de la Ossa, P. Green, and L. Parkes, A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory, Nuclear Physics B359 (1991), 21-74.

A. Cannas da Silva, Lectures on Symplectic Geometry, Lecture Notes in Mathematics 1764, Springer-Verlag, 2001.

E. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, 1955.

D. Cox, J. Little, and H. Schenck, Toric Varieties, Graduate Studies in Mathematics, Volume 124, American Mathematical Society, 2011.

A. Givental, Equivariant Gromov-Witten invariants, International Mathematics Research Notice 13 (1996), 613-663.

P. Griffiths and J. Harris, Principles of Algebraic Geometry, Wiley, 1978.

M. Gross, D. Huybrechts, and D.D. Joyce, Calabi-Yau Manifolds and Related Geometries, Universitext, Springer-Verlag, 2003.

R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics 52, Springer-Verlag, 1977.

K. Hori, S. Katz, A. Klemm et al., Mirror Symmetry, Clay Mathematics Monographs, Vol. 1, AMS, Clay Math. Institute, 2003.

D. Huybrechts, Complex Geometry, Universitext, Springer-Verlag, 2005.

D.D. Joyce, Riemannian Holonomy Groups and Calibrated Geometry, Oxford Graduate Texts in Mathematics 12, Oxford Press, 2006.

S. Katz, Enumerative Geometry and String Theory, Student Mathematical Library, vol. 32, AMS, IAS, 2006.

B. Lian, K. Liu, and S.-T. Yau, Mirror principle. I, Asian Journal of Mathematics 1 (1997), 729-763.

D. McDuff and D. Salamon, J-holomorphic Curves and Symplectic Topology, American Mathematical Society colloquium publications, 2004.

D. Morrison, Picard-Fuchs equations and mirror maps for hypersurfaces, Essays on mirror manifolds (S.-T. Yau, ed.), International Press, Hong Kong, 1992, pp. 241-264.

C. Voisin, Hodge Theory and Complex Algebraic Geometry I, Cambridge Studies in Advanced Mathematics 76, Cambridge University Press, 2002.

C. Voisin, Hodge Theory and Complex Algebraic Geometry II, Cambridge Studies in Advanced Mathematics 77, Cambridge University Press, 2003.

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