A Non-Archimedean Approach to Mirror Symmetry

镜像对称的非阿基米德方法

• 批准号: 2302095
• 负责人: Tony Yue Yu
• 金额: $ 24.92万
• 依托单位: California Institute of Technology
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-07-01 至 2026-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2302095&HistoricalAwards=false
• 关键词: Non; Archimedean; Approach; Mirror; Symmetry

Mirror symmetry is one of the most mysterious dualities in mathematics. It originated from theoretical physics, and generated great impact on various areas of mathematics, in particular differential geometry and algebraic geometry. The objects of study are geometric shapes called Calabi-Yau manifolds that appear naturally in different contexts of mathematics and physics. Mirror symmetry predicts that given any Calabi-Yau manifold, there exists a mirror manifold, such that an ever-growing list of geometric relations hold between the two, involving deep and elaborate invariants that are otherwise unrelated. Despite the continual progress in the subject, both the full extent of existence of mirror manifolds and the underlying mathematical mechanism of mirror symmetry remain unsolved today. This project brings a new approach to the study of mirror symmetry, based on latest developments from non-archimedean geometry. The goal is to conceive and pursue a full-fledged theory of non-archimedean mirror symmetry, which will lead to new results unattainable from existing methods, as well as applications beyond the current scope. This award will also support graduate and undergraduate students. The mirror existence problem will be studied via the enumeration of analytic curves with boundaries in non-archimedean SYZ torus fibrations of Calabi-Yau manifolds. New constructions of such fibrations will be explored using non-archimedean Monge-Ampère equations in addition to the minimal model program. An analog of the Gromov compactness theorem for non-archimedean curves with boundaries needs to be established, based on generalized tail conditions, which depend on novel analytic surgeries of the target spaces. The construction of virtual fundamental classes for the curve counts will rely on our previous works on derived non-archimedean geometry. PI will prove various properties of the mirror algebra obtained from the curve counts, including associativity, radius of convergence and singularity estimates. The local mirror algebras are expected to glue together to form the global mirror variety using wall-crossing formulas. A long-term goal is to show that the mirror construction is an involution, the best exhibition of mirror duality. PI also aims for applications outside mirror symmetry, in particular towards cluster algebras in representation theory and the moduli of KSBA stable pairs in birational geometry. PI expects fruitful future interactions with the Gross-Siebert program based on logarithmic geometry.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

镜像对称是数学中最神秘的二元性之一。它起源于理论物理学，对数学的各个领域，特别是微分几何和代数几何产生了巨大的影响。研究对象是被称为卡拉比-丘流形的几何形状，它们自然出现在不同的数学和物理背景中。镜像对称预测，给定任何卡拉比-丘流形，都存在一个镜像流形，使得两者之间存在不断增长的几何关系列表，涉及深刻而复杂的不变量，否则这些不变量是不相关的。尽管该主题不断取得进展，但镜像流形的全部存在范围和镜像对称的基本数学机制至今仍未得到解决。该项目基于非阿基米德几何的最新发展，为镜像对称研究带来了一种新方法。目标是构思和追求成熟的非阿基米德镜像对称理论，这将带来现有方法无法获得的新结果，以及超出当前范围的应用。该奖项还将支持研究生和本科生。镜像存在问题将通过枚举卡拉比-丘流形的非阿基米德 SYZ 环面纤维中具有边界的解析曲线来研究。除了最小模型程序之外，还将使用非阿基米德 Monge-Ampère 方程探索此类纤维的新结构。需要基于广义尾部条件建立具有边界的非阿基米德曲线的格罗莫夫紧致性定理的模拟，这取决于目标空间的新颖分析手术。曲线计数的虚拟基础类的构造将依赖于我们之前对导出的非阿基米德几何的工作。 PI 将证明从曲线计数获得的镜像代数的各种属性，包括关联性、收敛半径和奇点估计。局部镜像代数预计将使用穿墙公式粘合在一起形成全局镜像簇。长期目标是表明镜子的构造是一种内卷，是镜子二元性的最佳展示。 PI 还致力于镜像对称之外的应用，特别是表示论中的簇代数和双有理几何中 KSBA 稳定对的模。 PI 期望未来与基于对数几何的 Gross-Siebert 项目进行富有成效的互动。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。