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.

这个项目的中心是一个名为“辛拓扑”的领域，这是几何学的一部分，起源于天体力学的发展，其独特的数学特征出现在20世纪80年代。这一领域是当今研究活动蓬勃发展的主题。 PI，沿着他的研究生和合作者，将研究辛拓扑和几何的其他两个部分之间的联系，每个表面上都很不同。第一个连接是代数几何，并通过已知的现象的隐喻名称“镜像对称”。在镜像对称中，来自辛拓扑的对象以变换的形式重新出现在镜子世界中的代数形式中。辛对象之间的相互关系和代数对象之间的相互关系以惊人的方式精确匹配。镜像对称是在1990年左右由从事弦论研究的物理学家提出的，多年来，数学家们只能通过例子来验证它，但无法解释它。PI认为，在镜像对称提出四分之一个世纪后，是时候证明镜像对称如何工作的基本定理了。 第二个连接是三维和四维的几何。 虽然项目的两个部分涉及数学的不同部分，但它们共享共同的技术工具，即“伪全纯曲线”和“Floer上同调”理论。该项目的一个重要部分是支持研究生在该项目的两个方面工作的培训。该项目中提出的两个研究方向都基于辛Floer上同调，辛拓扑中的一个工具被证明与代数拓扑中的奇异上同调一样精辟和适应性强。 一股探讨辛拓扑和代数几何之间的连接结构方面称为镜像对称，在卡-丘（CY）流形的设置。同调镜像对称和CY流形的福谷范畴占据了中心位置。该建议的关键是调查镜像对称的不同表述之间的逻辑关系：镜像对的构造;同调镜像对称; Hodge理论镜像对称;以及全纯曲线的枚举。另一条链产生于弗洛尔上同调的辛和规范理论版本之间的关系。 该建议是发展一个新的Floer上同调理论的3流形使用相同的机制，作为巨大的生产Heegaard Floer同调的Ozsvath-Szabo，但工作不是在对称产品的Heegaard表面（做了一个复杂的曲线），就像在Heegaard Floer同调中所做的那样，而是在Heegaard曲面上的“稳定对”空间中，由秩为2的全纯向量丛及其全纯截面组成。这个理论可能与Heegaard Floer同调理论本身和瞬子Floer理论都有密切的关系，并可能阐明这些理论之间的关系。