Symplectic Floer cohomology, mirror symmetry and gauge theory

这个项目的中心是一个叫做“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。今天，这一领域是活跃的研究活动的主题。这位PI将与他的研究生和合作者一起，研究辛拓扑和几何的另外两个部分之间的联系，每个部分表面上都有很大的不同。第一种联系是与代数几何有关的，它是通过被比喻为“镜像对称”的现象来实现的。在镜像对称中，辛拓扑中的对象在镜面世界中以变换形式重现为代数形式。辛对象之间的相互关系和代数对象之间的相互关系以一种令人震惊的方式精确匹配。镜像对称是在1990年左右由从事弦理论的物理学家提出的，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，在它成立四分之一个世纪后，是时候证明基本定理了，这些基本定理概念化地描述了镜像对称是如何工作的。第二个是与3维和4维几何的联系。尽管该项目的两个部分涉及不同的数学部分，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。这项建议的一个重要部分是支持对从事这两个项目的研究生的培训。本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用。其中一篇文章探讨了在Calabi-Yau(CY)流形的背景下，辛拓扑和称为镜像对称的代数几何之间的联系的结构方面。同调镜像对称性和CY流形的Fukaya范畴占据了中心舞台。该建议的关键是研究不同镜面对称性公式之间的逻辑关系：镜面对的构造；同调镜面对称性；霍奇理论镜面对称性；以及全纯曲线的计数。另一条线索来自于Floer上同调的辛理论和规范理论之间的关系。我们的建议是发展一个新的三维流形的Floer上同调理论，使用与Ozsvath-Szabo的大量产生的Heegaard Floer同调相同的机制，但不是像在Heegaard Floer同调中所做的那样，在Heegaard曲面的对称乘积(做一条复杂的曲线)中工作，而是在Heegaard曲面上的“稳定对”的空间中工作，该空间由秩为2的全纯向量丛及其全纯截面组成。这一理论可能既与Heegaard Floer同调本身，又与瞬子Floer理论有密切的关系，并可能阐明这些理论之间的关系。

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.