Instanton Floer Homology with Lagrangian Boundary Conditions and the Atiyah-Floer Conjecture

本课题的主要研究对象是Salamon提出的四流形上反自对偶实例的一个新的拉格朗日边值问题。这是由带边界的三流形上的chen - simons泛函自然产生的，并为带边界的三流形上的Floer同调提供了一种新的方法。此外，这个新的Floer同调的构造是可能导致证明Atiyah-Floer猜想的程序的第一步。后者将同调三球的花同调与由三流形的heegard分裂产生的平面连接模空间中相应的拉格朗日花同调联系起来。这个猜想是一个长期悬而未决的问题，它的解决将是理解同调三球不同不变量之间关系的重要一步。Atiyah-Floer猜想有一个关于Ozsvath和Szabo的Heegaard Floer同调与Seiberg-Witten不变量的类比。这个程序还旨在了解这些新的不变量和阿蒂亚-弗洛尔猜想中的弗洛尔同调之间的关系。更一般地说，本课程旨在更好地理解和阐述规范理论的分析基础和花同调的构造。这个项目属于辛几何和低维拓扑之间相互作用的一般领域。从donaldson关于光滑四维流形的研究(基于反自对偶实例（大致是电磁场方程的一种特例）开始，到Gromov关于辛流形中的伪全纯曲线（全纯复函数的一种推广）的研究，在过去的二十年里，这些领域取得了重要进展。在这两门学科中，弗洛尔在八十年代末介绍了他对无限维莫尔斯理论的新方法。在莫尔斯理论中，有限维空间的性质是根据该空间上的梯度向量场的零点和流线来理解的。在Floer的语境中，这个空间是无限维的，但它是由潜在的有限维流形中的某些对象（如路径）产生的，并带有一些额外的结构。相应的Floer理论从与其相关的偏微分方程的解空间中提取有关该底层流形及其额外结构的信息。这个项目的目的是定义一个Floer理论，其中底层流形有一个边界，这就产生了偏微分方程的边界条件。这是一个程序的一个步骤，以了解不同的弗洛尔理论之间的关系，产生于相同的底层流形。

The main object of this project is a new Lagrangian boundary value problem for anti-self-dual instantons on four-manifolds proposed by Salamon. This appears naturally from the Chern-Simons functional on three-manifolds with boundary and leads to a new approach for Floer homology on three-manifolds with boundary. The construction of this new Floer homology moreover is a first step in a program that might lead to a proof of the Atiyah-Floer conjecture. The latter relates the Floer homology of a homology three-sphere to a corresponding Lagrangian Floer homology in a moduli space of flat connections, which arise from a Heegaard splitting of the three-manifold. This conjecture is a longstanding open question and its solution would be an important step towards understanding the relations between different invariants of homology three-spheres.The Atiyah-Floer conjecture has an analogue relating the new Heegaard Floer homology by Ozsvath and Szabo to Seiberg-Witten invariants. This program also aims to understand the relation between these new invariants and the Floer homologies in the Atiyah-Floer conjecture.More generally, this program aims to achieve a better understanding and exposition of the analytic foundations of gauge theory and the construction of Floer homologies.This project belongs into the general realm of interaction between symplectic geometry and low dimensional topology. Important progress in these areas has been made in the last twenty years starting with the work ofDonaldson on smooth four-dimensional manifolds, which was based on anti-self-dual instantons (which roughly are a special case of the electromagnetic field equations), and with the work of Gromov on pseudoholomorphic curves in symplectic manifolds (a generalization of holomorphic complex functions). In both subjects Floer introduced in the late eighties his new approach to infinite dimensional Morse theory. In Morse theory, properties of a finite dimensional space are understood in terms of the zeros and flow lines of gradient vector fields on this space.In Floer's context, this space is infinite dimensional but arises from certain objects (like paths) in an underlying finite dimensional manifold with some extra structure. The corresponding Floer theory extracts informations about this underlying manifold and its extra structure from the space of solutions of a partial differential equation associated to it.The aim of this project is to define a Floer theory, where the underlying manifold has a boundary, and this gives rise to a boundary condition for the partial differential equation. This is one step in a program to understand the relation between different Floer theories that arise from the same underlying manifolds.