Partially Wrapped Fukaya Categories and Functoriality in Mirror Symmetry

镜像对称中的部分包裹深谷范畴和函子性

• 批准号: 2202984
• 负责人: Denis Auroux
• 金额: $ 53.91万
• 依托单位: Harvard University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-07-01 至 2025-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2202984&HistoricalAwards=false
• 关键词: Partially; Wrapped; Fukaya; Categories; Functoriality

In modern geometry it is often useful to think of spaces in non-local terms rather than relying on classical concepts such as position. This is especially true of symplectic geometry -- the geometry of phase spaces of classical mechanics, where points are "too small" to be relevant, and it is more natural to consider objects called Lagrangian submanifolds. These objects and their interactions are encoded by an algebraic structure (or "non-commutative space") called the Fukaya category. A remarkable mathematical conjecture inspired by ideas from theoretical physics, "homological mirror symmetry", asserts that Fukaya categories are in fact often equivalent to honest (commutative) spaces of the sort studied in algebraic geometry. This research project studies versions of Fukaya categories for symplectic manifolds equipped with one or more (commuting) functions and/or relative to certain directions at infinity. The rich structure of these categories, coming from the additional data, yields new ways of understanding the effect of various geometric constructions on the Fukaya category of a symplectic manifold; this in turn should greatly extend the range of settings in which homological mirror symmetry can be verified. The project will also provide research opportunities for several graduate students and generally aim to make this research area more accessible to the broader mathematical community. From a technical standpoint, the first goal of this project is to develop a better geometric setup for partially wrapped Fukaya categories as they arise in mirror symmetry, to arrive at a formulation where computations are possible and the expected structural features are manifest. The other main goal is to use these foundations to give a general proof of homological mirror symmetry for (not necessarily Calabi-Yau) complete intersections in toric varieties, and to study canonical bases of their coordinate rings. Finally, this project will also bring conceptual clarity to the field by unifying different proposed constructions of partially wrapped Fukaya categories, finding new instances of functoriality in mirror symmetry, and studying the interplay between geometric constructions and categorical ones.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.

在现代几何形状中，通常以非本地术语来思考空间，而不是依靠诸如位置之类的经典概念很有用。符号几何形状尤其如此 - 古典力学的相位空间的几何形状，其中点“太小”而无法相关，并且考虑使用称为Lagrangian Submanifolds的对象是更自然的。这些对象及其相互作用由称为福卡亚类别的代数结构（或“非共同空间”）编码。一个出色的数学猜想，灵感来自理论物理学的思想“同源镜对称性”，断言福卡亚类别实际上通常等于代数几何学研究的那种诚实（交换性）空间。该研究项目研究了具有一个或多个（通勤）功能和/或相对于Infinity在Infinity的某些方向的富加亚类别的版本。这些类别的丰富结构来自其他数据，产生了理解各种几何结构对符合歧管的福卡亚类别的影响的新方法。反过来，这应该大大扩展可以验证同源对称性的设置范围。该项目还将为几位研究生提供研究机会，通常旨在使该研究领域更广泛地访问更广泛的数学社区。从技术的角度来看，该项目的第一个目标是为它们在镜像对称性中出现的部分包裹的福卡亚类别开发更好的几何设置，以达到一个可能的计算，并显示出预期的结构特征。另一个主要目标是使用这些基础为（不一定是calabi-yau）在圆磨品种中提供同源镜子对称性的一般证明，并研究其坐标环的规范底座。最后，该项目还将通过统一部分包裹的福卡亚类别的不同拟议的结构来为现场带来概念上的清晰度，在镜像对称性中找到新的功能性实例，并研究几何结构和分类之间的相互作用。该奖项反映了NSF的法定任务，并通过评估范围来反映了对范围的支持，并通过评估了范围的范围。