Calabi-Yau Varieties: Arithmetic, Geometry and Physics

拟议的计划涉及数论(算术代数几何)和理论物理(弦理论)的十字路口的问题。这些结果将增强我们在选定的数学和物理领域，并最终对我们的宇宙的理解。弦理论是一种物理预言，认为宇宙在最基本的层面上是由弦(一维物体)组成的，而不是点粒子(零维物体)。弦理论需要十维时空(而不是我们现实世界的四维时空)。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流形(三重)，它们是我研究的主要问题。这项研究的目的是从数学的角度理解Calabi-Yau流形镜像对称性的物理预测。Calabi-Yau流形是一种紧致复Kaehler流形，它的第一类为零，第一类为零。一维、二维和三维的Calabi-Yau流形分别是椭圆曲线、K3曲面和Calabi-Yau三重流形。镜像对称性是弦理论中的一种预言，即某些Calabi-Yau三重“镜像对”产生相同的物理理论。模(和准模)形式、Hilbert、Siegel和Jacobi模形式以及自同构形式作为几何不变量的生成函数、或配分函数、或镜像对称景观中的镜像映射而出现。我的目标之一是用算术不变量来解释镜像对称性，如Zeta函数和所讨论的Calabi-Yau流形的L级数。在这方面，L系列的自同构问题将被大力追问。这里，“自同构”指的是假定由Calabi-Yau流形产生的(动机)L级数在朗兰兹程序的背景下是自同构的L-级数。在这个项目中，最引人注目的工作是考虑具有形变参数的Calabi-Yau流形族的上述问题。当我们考虑镜像Calabi-Yau流形时，这样的族就出现了。我们应该引入带参数的自同构(模)形式。我这个项目的另一个中心目标是概念性地理解配分函数的模属性，以及Gromov-Witten不变量、Donaldson-Thomas不变量和其他几何不变量的母函数，例如Calabi-Yau流形的瞬子数。这将涉及研究三叶图上的Feynman路积分，这类图的归纳性质，以及它们在Kodaira-Spencer理论和Bershadsky-Cecotti-Ooguri-Vafa(BCOV)理论方面的解释。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。

The proposed program is concerned with problems at the crossroads of number theory (arithmetic algebraic geometry) and theoretical physics (string theory). The outcomes will enhance our understanding in the chosen fields of mathematics and physics, and ultimately of our universe. String theory is a physics prediction that what the universe is made of, is, at the most elementary level, 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 manifolds (threefolds), and they are the main concern in my investigation. The objective of the proposed research is to understand the physical prediction of mirror symmetry for Calabi-Yau manifolds 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 (and quasimodular) forms, Hilbert, Siegel and Jacobi modular forms, and automorphic forms appear, as generating functions of geometric invariants, or as partition functions, or as mirror maps in the mirror symmetry landscape. One of my goals is to interpret mirror symmetry in terms of arithmetic invariants such as zeta-functions and L-series of the Calabi-Yau manifolds in question. In this regard, the automorphy question for the L-series will be vigorously pursued. Here, “automorphy” refers to the presumed fact that the (motivic) L-series arising from Calabi-Yau manifolds are automorphic L-series in the context of the Langlands program. The most compelling endeavour in this project is to consider the above problem for families of Calabi-Yau manifolds with deformation parameters. Such families arise when we consider mirror Calabi-Yau manifolds. We ought to introduce automorphic (modular) forms with parameters. Another central goal of my project is the conceptual understanding of the modular properties of the partition functions, and of the generating functions of the Gromov-Witten invariants, the Donaldson-Thomas invariants and other geometric invariants, such as instanton number, for Calabi-Yau manifolds. This will involve studying, among other things, Feynman path integrals on trivalent graphs, the inductive nature of such graphs, and their interpretation in terms of Kodaira-Spencer theory and Bershadsky-Cecotti-Ooguri-Vafa (BCOV) theory. It is imperative to lay solid mathematical foundations for string theory, for the benefit of both mathematicians and string theorists.