威腾猜想及其推广和物理中的弦对偶
摘要
近几十年来，理论物理，特别是弦理论，已经取得了巨大的进展，并且深刻地影响着数学的发展。本论文的主要目的就是阐述近些年来，数学和物理相互交织所产生的深刻结果。选择了两个当前数学物理最前沿的课题进行论述。
第一个课题是关于Witten猜想，对应于本论文的第一章。这一章主要阐述了Witten猜想以及它的两种推广形式的历史发展和它们的解决情况。这一章各个章节的主要内容是：第一节，回忆了Witten在物理上为什么提出该猜想以及它在数学上的几种等价表述形式。第二节，主要讨论了现有的关于Witten猜想的五种不同的证明，本人将它们分为三种类型，也就是，通过矩阵积分，Hurwitz数和Weil-Peterson体积的这三种方法。这一节给出了这些证明的主要思想。第三节，主要讨论Witten猜想的两种推广形式，Virasoro猜想和Witten's r-Spin猜想的表述，并且描述了后者的主要证明思路。最后第四节是本章的总结，提出了作者对这个问题的一些看法。
第二个课题是讨论关于不同的物理理论之间的对偶性，以及由此所产生的数学上的深刻结果，这部分内容对应于本论文的第二章。这章的主要内容是描述了物理学家怎样从物理上提出这些猜想，接着给出它们在数学上的表述形式，以及数学家在数学上提供的精确证明。第二章各节的主要内容是：第一节，首先给出本章所要用到的关于拓扑弦的基本知识以及它跟超弦的联系。第二节给出了本章需要涉及到的数学基础。第三节和第四节是本章的主体部分，将阐述拓扑弦之间的对偶性理论，其中包括，拓扑A—模型和拓扑B—模型之间的对偶性(镜像对称)；大N对偶(Chern—Simons／弦理论)。然后简要给出了这些对偶所产生的数学猜想和它们在数学上的证明，其中包括，镜像猜想，Gopakumar-Vafa猜想，Labastida-Marino-Ooguri-Vafa猜想，Marino-Vafa猜想。在本章的最后一节，提到了本论文并未涉及到的关于拓扑弦的其他方面的内容，比如和四维规范场的对偶性等。
The last two decades, theoretic physics, particularly the string theory, has made greatprogresses and impacted the development of mathematics deeply. The goal of this thesisis to pursue the interactions between mathematics and physics. Here, I choose two topicswhich is very important in the development of mathematic physics.
The first one is about the Witten conjecture which is chapter 1 of this thesis. Thischapter is devoted to illustrate the historical development of the Witten conjecture and itsgeneralizations. Chapter 1 is arranged as follows: In section 1, we recall the descriptionsof the Witten conjecture. In section 2, we discuss the different proofs for the Wittenconjecture. I divide all the proofs so far we have known to three types, and show their mainideas to prove this conjecture. In section 3, we discuss two generalizations of the Wittenconjecture, i.e. Virasoro conjecture and the Witten's r-spin conjecture. Particularly, I willshow the proof of the Witten's r-spin conjecture precisely. Finally, In section 4, I give theconclusion for this chapter.
The second one is the dualities between the different physics. The topological stringtheory is the simplest string theory which is very important both in mathematics andphysics. Their interactions have produced many good mathematics. In chapter 2, I willshow the developments in this field. I mainly focus on the conjectures proposed by physicistthrough dualities and their mathematical proofs. This chapter is organized as follows:First, In section 1, I show the basis for the topological string and it's relation to thesuperstring. Then, some related mathematics needed in this chapter are given in section2. Section 3 and section 4 are the main body of this chapter. I will describe the dualitiesbetween the different topological strings (A-model and B-model) and the large N-dualities.Here we mainly discuss the conjectures and its solutions including the mirror symmetry, Gopakumar-Vafa conjecture, Labastida-Marino-Ooguri-Vafa conjecture and Marino-Vafaconjecture. In the last section, I will refer to some other relations which have not mentionedhere. The applications of topological string are showed in [NV].
第一个课题是关于Witten猜想，对应于本论文的第一章。这一章主要阐述了Witten猜想以及它的两种推广形式的历史发展和它们的解决情况。这一章各个章节的主要内容是：第一节，回忆了Witten在物理上为什么提出该猜想以及它在数学上的几种等价表述形式。第二节，主要讨论了现有的关于Witten猜想的五种不同的证明，本人将它们分为三种类型，也就是，通过矩阵积分，Hurwitz数和Weil-Peterson体积的这三种方法。这一节给出了这些证明的主要思想。第三节，主要讨论Witten猜想的两种推广形式，Virasoro猜想和Witten's r-Spin猜想的表述，并且描述了后者的主要证明思路。最后第四节是本章的总结，提出了作者对这个问题的一些看法。
第二个课题是讨论关于不同的物理理论之间的对偶性，以及由此所产生的数学上的深刻结果，这部分内容对应于本论文的第二章。这章的主要内容是描述了物理学家怎样从物理上提出这些猜想，接着给出它们在数学上的表述形式，以及数学家在数学上提供的精确证明。第二章各节的主要内容是：第一节，首先给出本章所要用到的关于拓扑弦的基本知识以及它跟超弦的联系。第二节给出了本章需要涉及到的数学基础。第三节和第四节是本章的主体部分，将阐述拓扑弦之间的对偶性理论，其中包括，拓扑A—模型和拓扑B—模型之间的对偶性(镜像对称)；大N对偶(Chern—Simons／弦理论)。然后简要给出了这些对偶所产生的数学猜想和它们在数学上的证明，其中包括，镜像猜想，Gopakumar-Vafa猜想，Labastida-Marino-Ooguri-Vafa猜想，Marino-Vafa猜想。在本章的最后一节，提到了本论文并未涉及到的关于拓扑弦的其他方面的内容，比如和四维规范场的对偶性等。
The last two decades, theoretic physics, particularly the string theory, has made greatprogresses and impacted the development of mathematics deeply. The goal of this thesisis to pursue the interactions between mathematics and physics. Here, I choose two topicswhich is very important in the development of mathematic physics.
The first one is about the Witten conjecture which is chapter 1 of this thesis. Thischapter is devoted to illustrate the historical development of the Witten conjecture and itsgeneralizations. Chapter 1 is arranged as follows: In section 1, we recall the descriptionsof the Witten conjecture. In section 2, we discuss the different proofs for the Wittenconjecture. I divide all the proofs so far we have known to three types, and show their mainideas to prove this conjecture. In section 3, we discuss two generalizations of the Wittenconjecture, i.e. Virasoro conjecture and the Witten's r-spin conjecture. Particularly, I willshow the proof of the Witten's r-spin conjecture precisely. Finally, In section 4, I give theconclusion for this chapter.
The second one is the dualities between the different physics. The topological stringtheory is the simplest string theory which is very important both in mathematics andphysics. Their interactions have produced many good mathematics. In chapter 2, I willshow the developments in this field. I mainly focus on the conjectures proposed by physicistthrough dualities and their mathematical proofs. This chapter is organized as follows:First, In section 1, I show the basis for the topological string and it's relation to thesuperstring. Then, some related mathematics needed in this chapter are given in section2. Section 3 and section 4 are the main body of this chapter. I will describe the dualitiesbetween the different topological strings (A-model and B-model) and the large N-dualities.Here we mainly discuss the conjectures and its solutions including the mirror symmetry, Gopakumar-Vafa conjecture, Labastida-Marino-Ooguri-Vafa conjecture and Marino-Vafaconjecture. In the last section, I will refer to some other relations which have not mentionedhere. The applications of topological string are showed in [NV].
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