A Canonical Construction of Mirrors for Polarized Calabi-Yau Manifolds

该奖项支持代数几何的研究。代数几何主要集中在由多项式方程定义的变量上，是一门古老的学科，在数学的许多领域中起着关键作用，无论是纯数学还是应用数学，以及物理学。这个项目的主题是，一个广泛的几何对象，所谓的卡拉比-丘品种，来与一个自然的坐标系。 非正式地说，生活在卡-丘类型上的生物应该能够感知自然的、内在的量，这些量的值决定了它们的精确位置。由于这些几何对象在不同的数学领域中扮演着基础性的角色，这些内禀量在理论物理学中也应该扮演着类似的基础性角色，特别是在超弦理论中。由于弦理论模型表明时空的额外维度包含卡-丘多样性，这项研究表明，在我们的世界中存在着尚未理解的相应基本内禀量。本研究项目旨在深化和推进这一领域的知识。拟议研究的主要目标是继续研究者的合作工作，将阿贝尔簇的经典theta函数理论推广到极化Calabi-Yau簇，包括开放（即log）和紧致。更确切地说，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，以正则基表示，作为镜像上全纯圆盘的计数。这种广义theta函数的存在表明模空间存在一个几何意义的紧化，极大地推广了极化阿贝尔簇的紧化和次多面体理论，同时也表明了一个关于正则描述环的镜像的综合构造。该项目包括一个详细的计划，用于构建在二维紧化，并在充分的一般性的综合建设的镜子，通过计数刚性解析磁盘。

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.