A Canonical Construction of Mirrors for Polarized Calabi-Yau Manifolds

偏振卡拉比-丘流形镜的规范结构

• 批准号: 1561632
• 负责人: Sean Keel
• 金额: $ 60.61万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-06-01 至 2022-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1561632&HistoricalAwards=false
• 关键词: Canonical; Construction; Mirrors; Polarized; Calabi

This award supports research in algebraic geometry. Focused primarily on varieties defined by polynomial equations, algebraic geometry is an ancient subject that plays a key role in numerous fields of mathematics, both pure and applied, as well as in physics. The main theme of this project is that a broad class of geometric objects, so called Calabi-Yau varieties, come with a natural system of coordinates. Informally speaking, creatures living on a Calabi-Yau variety should be able to perceive natural, intrinsic quantities whose values determine their precise position. As these geometric objects play a fundamental role in diverse areas of mathematics, these intrinsic quantities should play a similar fundamental role in theoretical physics, particularly in superstring theory. Since string theory models suggest that extra dimensions of spacetime comprise a Calabi-Yau variety, this study suggests there are corresponding fundamental intrinsic quantities, not yet understood, in our world. This research project aims to deepen and advance knowledge in this field.The main objective of the proposed research is to continue the investigator's collaborative work to generalize the classical theory of theta functions for abelian varieties to polarized Calabi-Yau varieties, both open (i.e. log) and compact. More precisely, the goal is to give a canonical basis for the vector space of global sections, and a formula for the structure constants for the multiplication rule in the coordinate ring, expressed in the canonical basis, as counts of holomorphic disks on the mirror. The existence of such generalized theta functions points to the existence of a geometrically meaningful compactification of the moduli space, vastly generalizing the compactificaton of polarized abelian varieties and the theory of the secondary polytope, and at the same time suggests a synthetic construction of the mirror in terms of the canonically described ring. The project includes a detailed scheme for constructing the compactification in dimension two, and the synthetic construction of the mirror in full generality, by counting rigid analytic disks.

该奖项支持代数几何的研究。代数几何主要关注由多项式方程定义的变量，是一门古老的学科，在数学的许多领域，无论是纯粹的还是应用的，以及在物理学中都起着关键作用。这个项目的主题是一大类几何物体，即所谓的Calabi-Yau品种，具有自然的坐标系统。非正式地说，生活在卡拉比-丘品种上的生物应该能够感知自然的、内在的数量，这些数量的值决定了它们的精确位置。由于这些几何对象在数学的各个领域发挥着基础作用，这些内在量应该在理论物理中发挥类似的基础作用，特别是在超弦理论中。由于弦理论模型表明，额外的时空维度包含了卡拉比-丘变化，这项研究表明，在我们的世界中存在着相应的、尚未被理解的基本内在量。这个研究项目旨在加深和推进这一领域的知识。本研究的主要目标是继续研究者的合作工作，将经典的阿贝尔变量theta函数理论推广到极化的Calabi-Yau变量，包括开放（即对数）和紧化。更准确地说，目标是给出全局截面的向量空间的正则基，以及坐标环中乘法规则的结构常数的公式，用正则基表示，作为镜像上全纯磁盘的计数。这类广义函数的存在表明模空间存在一个几何上有意义的紧化，极大地推广了极化阿贝尔变体的紧化和次多面体理论，同时提出了一种基于正则描述环的镜像的综合构造。该项目包括构造二维紧化的详细方案，以及通过计算刚性分析盘在完全一般情况下的镜像的综合构造。