镜像对称是现代数学物理中的重要课题，数学家和物理学家从多个方面对其 进行了研究，例如通过它来计算一般的五次三维流形 (quintic threefold) 上的有理 曲线的数目，或者研究 Calabi-Yau 流形的凝聚层的导出范畴与它的一个镜像流形 的导出深谷范畴之间的联系等。本文是关于这些领域的一篇读书报告，将简要介 绍与镜对称有关的计数几何学，特别是 Givental 在其文章 [1] 中给出的射影空间的 镜对称猜想的证明。我们首先介绍与 Gromov-Witten 不变量相关的基本定义和性 质，并利用它们来定义量子上同调。然后，我们会说明 Givental 如何利用量子上 同调来引入量子微分方程，之后我们会利用它来给出镜对称猜想的数学形式。接 下来，我们会介绍证明镜对称猜想所需要的一些数学工具，特别是等变上同调和 局部化方法，这一方法可以大大化简模空间上 Gromov-Witten 不变量的计算，从而 使镜对称猜想的证明成为可能。最后，我们先利用量子微分方程来给出射影空间 上镜对称猜想的数学形式，再分情况介绍其证明：首先是较简单的 Fano 情形，将 一定的初始条件代入通过局部化方法给出的递推公式即可得出；然后是较为复杂 的 Calabi-Yau 情形，此时对于 Fano 情形中得出的递推公式而言初始条件并不够用， 因此我们将会推导所需要的额外限制，以及给出较为复杂的变换以联系镜像对称 关系两边的不变量，最终完成猜想的证明。
Mirror symmetry is an important topic in modern mathematical physics which is researched by mathematicians and physicists from different aspects. For example, some people used it as a tool to predict the number of rational curves on a generic quintic threefold, while some other people researched the relationship between the derived category of coherent sheaves of a Calabi-Yau 3-fold and the derived Fukaya category of its mirror pair.In this article, we will introduce enumerative geometries about mirror symmetry, especially Givental’s proof of the mirror theorem of projective spaces. We will first introduce some definitions and properties about Gromov-Witten invariants, and use them to define quantum cohomology. Then following Givental’s approach, we will introduce quantum differential equations, which will be used to give the mathematical statement of mirror conjecture, using quantum cohomology. Next, we are to introduce some useful tools which are necessary for Givental’s proof, especially equivariant cohomology and the technique of localization, which can significantly simplify the computation of GromovWitten invariants of moduli spaces and hence make the proof possible. Lastly, we will use quantum differential equations to provide the mathematical statement of the mirror theorm and then prove it case by case: the Fano case, which is simplier and can be proved by a recursion formula derived from localization; and then the Calabi-Yau case, which requires much more work since the recursion formula is not enough and the mirror map is much more complex.
