Calabi-Yau Manifolds and Mirror Symmetry

该提案涉及数论和弦理论交叉点的问题。弦论是一种物理学预测，宇宙是由弦（一维物体）组成的，而不是点粒子（零维物体）。弦理论要求10维时空（而不是我们真实的世界的4维）。粗略地说，这是因为更多的维度可以容纳更多可能的弦振动。弦论中额外的六维物体被称为卡-丘三重体，它们是本研究的主要对象。这项研究的目的是从数学的角度来理解卡-丘流形镜像对称的物理预测及其后果。一个Calabi-Yau流形是一个紧致复Kaehler流形，它的第一个Chern类为零，第一个Betti数为零。一维、二维和三维的卡-丘流形分别是椭圆曲线、K3曲面和卡-丘三重。镜像对称是弦论中的一个预言，即卡-丘三重的某些“镜像对”产生相同的物理理论。模形式，西格尔和雅可比模形式，自守形式出现，或者作为几何不变量的生成函数，或者作为分区函数。 我的研究目标之一是解释镜像对称的算术不变量，如ζ-函数和L-系列的卡-丘流形。在这方面，将大力追求L系列的自同构问题。这里，“自同构”指的是这样一个事实，即从卡-丘流形产生的（动机）L-级数是自同构的L-级数，就像在朗兰兹纲领的上下文中一样。目前，最紧迫的问题是解决所有三阶上同调的Hodge数等于1的Calabi-Yau流形族的自同构问题。这样的家庭产生四维伽罗瓦表示。一个非常粗略的猜想是，这些不可约的四维伽罗瓦表示应该对应于Sp（4，Z）的某些仿模子群上的权为3且亏格为2的西格尔模形式。 另一个中心目标是概念上理解卡-丘流形的配分函数中各种模形式的出现，以及Gromov-Witten不变量、Donaldson-Thomas不变量、Gopakumar-Vafa不变量和其他几何或物理不变量（例如BPS状态计数）的生成函数。为了数学家和弦理论家的利益，必须为弦理论奠定坚实的数学基础。我计划通过这个项目培养博士后研究员和研究生。我的方法是给他们每个人分配具体的例子，以制定和理解这个项目的主要目标。并最终导致新的数学发现。

The proposal is concerned with problems at the crossroads of number theory and string theory. String theory is a physics prediction that what the universe is made of, is, strings (one-dimensional objects), rather than point particles (zero-dimensional objects). String theory demands ten-dimensional space-time (as opposed to the dimension four of our real world). This is, roughly speaking, because more dimensions can accommodate more possible string vibrations. The extra six-dimensional objects in string theory are known as Calabi-Yau threefolds, and they are the main objects of this investigation. The goal of the proposed research is to understand the physical prediction of mirror symmetry for Calabi-Yau manifolds, and its consequences, from a mathematical point of view. A Calabi-Yau manifold is a compact complex Kaehler manifold with vanishing first Chern class and zero first Betti number. Calabi-Yau manifolds of dimension one, two, and three are, respectively, elliptic curves, K3 surfaces, and Calabi-Yau threefolds. Mirror symmetry is a prediction in string theory that certain "mirror pairs" of Calabi-Yau threefolds yield identical physical theories. Modular forms, Siegel and Jacobi modular forms, and automorphic forms appear, either as generating functions of geometric invariants, or as partition functions. One of my research goals is to interpret mirror symmetry in terms of arithmetic invariants such as zeta-functions and L-series of the Calabi-Yau manifolds. In this connection, the automorphy question for the L-series will be vigorously pursued. Here, "automorphy" refers to the fact that the (motivic) L-series arising from Calabi-Yau manifolds are automorphic L-series as in the context of the Langlands program. At the moment, the most pressing issue is to address the automorphy question for families of Calabi-Yau manifolds with all Hodge numbers of the third cohomology equal to one. Such families give rise to four-dimensional Galois representations. A very crude conjecture is that these irreducible four-dimensional Galois representations should correspond to Siegel modular forms of weight 3 and genus 2 on some paramodular subgroups of Sp(4, Z).  Another central goal is the conceptual understanding of the appearance of various modular forms in the partition functions, and of the generating functions of the Gromov-Witten invariants, the Donaldson-Thomas invariants, the Gopakumar-Vafa invariants and other geometric or physical invariants (e.g, BPS state counting numbers), for Calabi-Yau manifolds. It is imperative to lay solid mathematical foundations for string theory, for the benefit of both mathematicians and string theorists. I plan to train HQP (postdoctoral fellows and graduate students) through this project. My approach will be to assign concrete examples to each of them to work out and understand the main goal of this project. and eventually lead to new mathematical discoveries.