Calabi-Yau Manifolds and Mirror Symmetry

该提案涉及数论和弦论交叉的问题。弦理论是一种物理学预测，宇宙是由弦（一维物体）而不是点粒子（零维物体）组成的。弦理论需要十维时空（与我们现实世界的四维相反）。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。弦理论中额外的六维物体被称为Calabi-Yau三折，它们是本研究的主要对象。本研究的目标是从数学的角度来理解Calabi-Yau流形的镜像对称的物理预测及其后果。Calabi-Yau流形是一种第一chen类消失、第一Betti数为零的紧致复Kaehler流形。1维、2维和3维的Calabi-Yau流形分别是椭圆曲线、K3曲面和Calabi-Yau三倍。镜像对称是弦理论中的一种预测，即某些卡拉比-丘三倍的“镜像对”产生相同的物理理论。模形式，Siegel和Jacobi模形式和自同构形式出现，或者作为几何不变量的生成函数，或者作为配分函数。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 -级数的Calabi-Yau流形。在这方面，l系列的自同构问题将被大力探讨。在这里，“自同构”指的是由Calabi-Yau流形产生的（动机）l -级数在Langlands程序中是自同构的l -级数。目前最紧迫的问题是解决第三上同调的所有Hodge数都等于1的Calabi-Yau流形族的自同构问题。这样的族产生了四维伽罗瓦表示。一个非常粗糙的猜想是，这些不可约的四维伽罗瓦表示应该对应于Sp（4， Z）的某些副模子群上权3和属2的Siegel模形式。另一个中心目标是概念上理解配分函数中各种模形式的外观，以及对于Calabi-Yau流形的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状态计数数）的生成函数。为了数学家和弦理论学家的利益，为弦理论奠定坚实的数学基础势在必行。我计划通过这个项目培养HQP（博士后和研究生）。我的方法是给他们每个人分配具体的例子，以找出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.