Symplectic Floer cohomology, mirror symmetry and gauge theory

辛弗洛尔上同调、镜像对称和规范理论

• 批准号: 1406418
• 负责人: Timothy Perutz
• 金额: $ 19.28万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-08-01 至 2017-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1406418&HistoricalAwards=false
• 关键词: Symplectic; Floer; cohomology; mirror; symmetry

This project centers on a field called "symplectic topology", a part of geometry with roots in the development of celestial mechanics, whose distinctive mathematical features emerged in the 1980s. This field is today the subject of vigorous research activity. The PI, along with his graduate students and collaborators, will study connections between symplectic topology and two other parts of geometry, each superficially quite different. The first connection is to algebraic geometry, and is through the phenomenon known by the metaphorical name of "mirror symmetry". In mirror symmetry, objects from symplectic topology reappear in transformed form into algebraic form in a looking-glass world. The inter-relations between the symplectic objects and those between the algebraic objects match precisely, in an astounding way. Mirror symmetry was proposed around 1990 by physicists working in string theory, and for years, mathematicians could verify it in examples but not explain it. The PI contends that a quarter-century after its inception, the time has come to prove basic theorems conceptualizing how mirror symmetry works. The second connection is to geometry in dimensions 3 and 4. Though the two parts of the project touch on different parts of mathematics, they share common technical tools, the theories of "pseudo-holomorphic curves" and "Floer cohomology". A significant part of the proposal is to support the training of graduate students working on the two aspects of the project.The two strands of the research proposed in this project are both based on symplectic Floer cohomology, a tool in symplectic topology that is proving as incisive and adaptable as singular cohomology is in algebraic topology. One strand explores structural aspects of the connection between symplectic topology and algebraic geometry known as mirror symmetry, in the setting of Calabi-Yau (CY) manifolds. Homological mirror symmetry, and the Fukaya category of a CY manifold, take center-stage. Key to the proposal is the investigation of logical relationships between different formulations of mirror symmetry: constructions of mirror pairs; homological mirror symmetry; Hodge-theoretic mirror symmetry; and enumeration of holomorphic curves. The other strand arises from the relationship between the symplectic and gauge-theoretic versions of Floer cohomology. The proposal is to develop a new Floer cohomology theory for 3-manifolds using the same mechanism as the hugely productive Heegaard Floer homology of Ozsvath-Szabo, but working not in a symmetric product of the Heegaard surface (made a complex curve), as one does in Heegaard Floer homology, but rather in a space of "stable pairs" on the Heegaard surface, consisting of a rank 2 holomorphic vector bundle together with a holomorphic section thereof. This theory is likely to have close relations both to Heegaard Floer homology itself, and to instanton Floer theory, and may illuminate the relations between those theories.

该项目集中在一个名为“象征性拓扑”的领域，这是几何学的一部分，其根源在天体力学的发展中，其独特的数学特征在1980年代出现。如今，该领域是剧烈研究活动的主题。 PI以及他的研究生和合作者将研究象征性拓扑与几何学的其他两个部分之间的联系，每个几何学上都完全不同。第一个联系是代数几何形状，并且是通过“镜面对称性”的隐喻名称所知的现象。在镜像对称性中，来自象征性拓扑的对象以转化形式转化为镜头世界中的代数形式。符号对象与代数对象之间的相互关联以惊人的方式符合。镜像对称性是由弦理论的物理学家左右提出的，多年来，数学家可以在示例中对其进行验证，但不能解释它。 PI认为，在其成立之后，一个世纪以来，现在是时候证明基本定理概念化了镜子对称性的工作原理。 第二个连接是与几何形状3和4。该提案的重要部分是支持研究该项目两个方面的研究生的培训。该项目中提出的研究的两条链都基于Symblectic Floer Coomomology，这是一种符号拓扑的工具，这种工具被证明是尖锐的，并且像奇异的共同体一样敏锐而适应性，在代数拓扑中是奇异的。 一条链探讨了在Calabi-yau（CY）歧管的环境中，符号拓扑与代数几何形状与代数几何形状之间的联系的结构方面。同源镜对称性和CY歧管的福卡亚类别为中心阶段。该建议的关键是对镜子对称的不同公式之间的逻辑关系的研究：镜对的结构； 同源镜对称；霍奇理论镜对称性；和尸体曲线的枚举。另一个链来自浮子共同体的符号和仪表理论版本之间的关系。 该提议是要使用与ozsvath-szabo的巨大生产力的Heegaard浮子同源性相同的机制来开发一种新的浮动提出理论，而不是在Heegaard表面的对称产品中（制造一个复杂的曲线），就像在Heegaard Floer同源物中一样，而在台面上是一个稳定的strace of Heegaard曲线，而是在台面上构成的。全态矢量束以及其圆形截面。该理论可能与Heegaard Floer同源性本身以及Instanton Floer理论都有密切的关系，并可能阐明这些理论之间的关系。