Calabi-Yau Manifolds and Mirror Symmetry

该提案涉及数论和弦论的十字路口问题。弦理论是一种物理预言，即宇宙是由弦(一维 对象)，而不是点粒子(零维对象)。弦理论需要十维空间 时空(而不是我们现实世界的四维空间)。这是，粗略地说，因为 更多的维度可以容纳更多可能的弦振动。额外的六维物体 在弦理论中被称为Calabi-Yau三重，他们是本研究的主要对象。 这项拟议研究的目标是了解镜像对称性的物理预测 Calabi-Yau流形及其结果，从数学的观点来看。阿卡拉比-尤 流形是一种紧致复Kaehler流形，具有零优先为零的第一类为零 贝蒂号码。一维、二维和三维的Calabi-Yau流形分别是椭圆的 曲线、K3曲面和Calabi-Yau三重。 镜像对称性是弦理论中的一种预言，即某些Calabi-Yau镜像对是三重的 产生相同的物理理论。模形式、Siegel和Jacobi模形式和自同构 形式要么作为几何不变量的生成函数出现，要么作为配分函数出现。 一 我的研究目标之一是用算术不变量来解释镜像对称，例如 Calabi-Yau流形的Zeta函数和L级数在这方面，自同构问题 对于L系列的追捧也将如火如荼。在这里，“自同构”指的是(动机) 从Calabi-Yau流形出发的L级数是自同构的L级数 程序。目前，最紧迫的问题是解决家庭的自我同构问题 具有三次上同调的所有Hodge数等于1的Calabi-Yau流形。这样的家庭 从而产生了四维伽罗瓦表示。一个非常粗糙的猜测是，这些不可约的 四维伽罗瓦表示应对应于权3和权的Siegel模形式 Sp(4，Z)的某些准模子群上的亏格2。 另一个中心目标是从概念上理解各种模块化形式的外观 在配分函数和Gromov-Witten不变量的母函数中， Donaldson-Thomas不变量、Gopakumar-Vafa不变量和其他几何或物理不变量 Calabi-Yau流形的不变量(例如，BPS状态计数)。这是势在必行的 弦理论的坚实的数学基础，对数学家和 弦理论家。 我计划通过这个项目培训HQP(博士后研究员和研究生)。我的方法是给他们每个人指定具体的例子，以确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.