Calabi-Yau Varieties: Arithmetic, Geometry and Physics

该计划涉及数论（算术代数几何）和理论物理（弦理论）交叉的问题。其结果将增强我们对数学和物理选定领域的理解，并最终增强我们对宇宙的理解。弦理论是一种物理学预测，它认为在最基本的层面上，构成宇宙的是弦（一维物体），而不是点粒子（零维物体）。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对象被称为Calabi-Yau流形（三折），它们是我研究的主要关注点。本研究的目的是从数学的角度来理解Calabi-Yau流形的镜像对称性的物理预测。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（和准模）形式，希尔伯特，西格尔和雅可比模形式，以及自同构形式出现，作为几何不变量的生成函数，或作为配分函数，或作为镜像对称景观中的镜像映射。我的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Calabi-Yau流形的问题。在这方面，l系列的自同构问题将被大力追求。在这里，“自同构”是指在Langlands程序的背景下，由Calabi-Yau流形产生的（动机）l -级数是自同构的l -级数这一假定事实。本项目最引人注目的工作是考虑具有变形参数的Calabi-Yau流形族的上述问题。当我们考虑镜像Calabi-Yau流形时，就会出现这样的家族。我们应该引入带参数的自同构（模）形式。我项目的另一个中心目标是对配分函数的模性质的概念理解，以及对Gromov-Witten不变量、Donaldson-Thomas不变量和其他几何不变量（如瞬数）的生成函数的理解。这将包括研究三价图上的费曼路径积分，这种图的归纳性质，以及它们在Kodaira-Spencer理论和Bershadsky-Cecotti-Ooguri-Vafa （BCOV）理论中的解释。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。

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.