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

本课题的主要研究对象是由Salamon提出的四维流形上反自对偶瞬子的拉格朗日边值问题。这是由带边界的三维流形上的Chern-Simons泛函自然产生的，从而为带边界的三维流形上的Floer同调提供了一种新的途径。此外，这种新的Floer同调的构建是一个程序的第一步，该程序可能导致Atiyah-Floer猜想的证明。后者将同调三球面的Floer同调与平坦连通模空间中相应的拉格朗日Floer同调联系联系起来，后者是由三维流形的Heegaard分裂产生的。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的不同不变量之间关系的重要一步。Atiyah-Floer猜想有一个将Ozsvath和Szabo的新Heegaard Floer同调与Seiberg-Witten不变量联系起来的类似方法。本程序还旨在了解这些新的不变量与Atiyah-Floer猜想中的Floer同调之间的关系。更广泛地说，本程序旨在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。在过去的二十年里，从Donaldson关于光滑四维流形的工作开始，取得了重要的进展，该工作基于反自对偶瞬子(这大致是电磁场方程的一个特例)，以及Gromov关于辛流形中的伪全纯曲线的工作(全纯复函数的推广)。在这两门学科中，弗洛尔在80年代末介绍了他对无限维摩尔斯理论的新方法。在Morse理论中，有限维空间的性质是通过这个空间上的梯度向量场的零点和流线来理解的。在Floer的背景下，这个空间是无限维的，但它是由具有某种额外结构的底层有限维流形中的某些对象(如路径)产生的。相应的Floer理论从与之相关的偏微分方程解的空间中提取关于这个潜在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，其中基础流形有一个边界，这就产生了一个偏微分方程的边界条件。这是理解来自相同基础流形的不同Floer理论之间的关系的程序中的一步。

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.