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对偶现象之一。它起源于理论物理，并对数学的各个领域产生了很大的影响，特别是微分几何和代数几何。研究对象是被称为Calabi-Yau流形的几何形状，它们在不同的数学和物理环境中自然出现。镜像对称预言，给定任何Calabi-Yau流形，就会存在一个镜像流形，这样两者之间就会有一个不断增长的几何关系列表，其中包含了深奥而精细的不变量，而这些不变量在其他方面是不相关的。尽管这一学科不断取得进展，但镜像流形存在的全部范围和镜像对称的潜在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，为研究镜像对称带来了新的途径。目标是构思和追求一个完整的非阿基米德镜像对称理论，这将导致现有方法无法获得的新结果，以及超出当前范围的应用。该奖项也将支持研究生和本科生。本文通过列举Calabi-Yau流形非阿基米德SYZ环面振动中有边界的解析曲线来研究镜像存在问题。除了最小模型程序外，还将使用非阿基米德蒙日-安普雷方程来探索这种纤维的新结构。基于广义尾条件，建立了具有边界的非阿基米德曲线的格罗莫夫紧性定理的类比，它依赖于目标空间的新的解析运算。曲线计数的虚拟基本类的构造将依赖于我们以前关于派生的非阿基米德几何的工作。PI将证明从曲线计数中得到的镜像代数的各种性质，包括结合性，收敛半径和奇点估计。局部镜像代数被期望粘合在一起，形成使用壁交叉公式的全局镜像品种。一个长期的目标是表明镜子的结构是一个对折，最好地展示镜子的对偶性。PI也致力于镜像对称以外的应用，特别是在表示理论中的簇代数和双民族几何中KSBA稳定对的模。PI期望与基于对数几何的Gross-Siebert程序进行富有成效的未来互动。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。