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）的十字路口问题。这些成果将增强我们对数学和物理学领域的理解，并最终增强我们对宇宙的理解。弦论是一种物理学预测，宇宙是由什么组成的，在最基本的水平上，是弦（一维物体），而不是点粒子（零维物体）。弦理论要求10维时空（与我们真实的世界的4维相对）。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-丘流形（三重），它们是我研究的主要对象。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。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（和拟模）形式，希尔伯特，西格尔和雅可比模形式，自守形式出现，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系列的卡-丘流形的问题。在这方面，将大力探讨L系列的自同构问题。这里，“自同构”是指假设的事实，即由卡-丘流形产生的（动机）L-级数在朗兰兹纲领的上下文中是自同构的L-级数。在这个项目中最引人注目的努力是考虑上述问题的家庭的卡-丘流形变形参数。这样的家庭出现时，我们考虑镜像卡-丘流形。我们应该引入带参数的自守（模）形式。我的项目的另一个中心目标是概念上的理解的模块化属性的分区功能，并生成功能的Gromov-Witten不变量，唐纳森-托马斯不变量和其他几何不变量，如瞬子数，卡拉比-丘流形。这将涉及研究，除其他事项外，费曼路径积分的三价图，归纳性质的这种图形，以及他们的解释在Kodaira-Spencer理论和Bershadsky-Cecotti-Ooguri-Vafa（BCOV）理论。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.