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

本课题的主要研究对象是Salamon提出的四维流形上反自对偶瞬子的一个新的拉格朗日边值问题。这自然地出现在三边界流形上的Chern-Simons泛函上，并导致了三边界流形上Floer同调的一种新方法。此外，这个新的弗洛尔同调的构建是一个程序的第一步，该程序可能会导致Atiyah-Floer猜想的证明。后者将同调三球面的弗洛尔同调与平坦联络模空间中相应的拉格朗日弗洛尔同调联系起来，平坦联络模空间是由三流形的Heegaard分裂产生的。这个猜想是一个长期存在的问题，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猜想有一个类似物，将Ozsvath和Szabo的新Heegaard Floer同调与Seiberg-Witten不变量联系起来。本课程的目的是了解这些新的不变量和Atiyah-Floer猜想中的Floer同调之间的关系，更一般地说，本课程的目的是更好地理解和阐述规范理论的分析基础和Floer同调的构造。本项目属于辛几何和低维拓扑之间相互作用的一般领域。在这些领域取得了重要进展，在过去的二十年开始工作的唐纳森光滑四维流形，这是基于反自对偶瞬子（这大致是一个特殊情况下的电磁场方程），并与工作格罗莫夫pseudoholomorphic曲线辛流形（一个推广的全纯复函数）。在这两个主题弗洛尔介绍了在八十年代后期他的新方法无限维莫尔斯理论。 在莫尔斯理论中，有限维空间的性质被理解为这个空间上的梯度向量场的零点和流线。在弗洛尔的上下文中，这个空间是无限维的，但源于一个具有某种额外结构的有限维流形中的某些对象（如路径）。相应的Floer理论从与之相关的偏微分方程的解空间中提取关于这个基础流形及其额外结构的信息。本项目的目的是定义一个Floer理论，其中基础流形有一个边界，这引起了偏微分方程的边界条件。这是一个程序中的一个步骤，以了解不同的弗洛尔理论之间的关系，产生于相同的基础流形。

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.