Floer homology for immersed Lagrangian submanifolds

大多数现代几何学研究的是某种空间。微分几何中考虑的空间称为流形，这种空间局部看起来像n维欧几里德空间，但整体上却有一个有趣的形状。如果流形是封闭的，没有边，则它是紧致的。甜甜圈的表面是一个紧凑的二维流形。流形M的子流形N是流形M的子集，它本身就是流形，通常比M的维度小。有两种类型：嵌入子流形和浸入子流形，它们可能不相交(交叉)。人们通常认为流形具有一些额外的几何结构，例如黎曼度量，它告诉你流形中路径的长度，或者辛结构，它告诉你流形的二维子流形的面积。辛流形是力学的数学公式的基础，也是许多经典物理的基础。他们本身也很有趣。数学家喜欢它们是因为它们是极少数具有无限维局部对称性的结构之一，这赋予了辛几何一种不同寻常的、完全全局的味道。拉格朗日子流形是辛流形的一种特殊的子流形。给出辛流形M的两个紧致的嵌入拉格朗日子流形N，N*，在一定条件下可以定义Floer同调群Hf(N，N*)，它们粗略地说是有限维向量空间。这个定义非常困难。为此，在M上选择一个辅助复结构J，并计算M中具有N和N*并的边界(边)的J-全纯二维圆盘D。关于HF(N，N*)值得注意的是，它与J的选择无关，并且在拉格朗日子流形之间移动N和N*也是不变的。它编码了一些关于拉格朗日子流形的神秘的、不平凡的信息，人们无法通过任何其他已知的方式获得。它是辛几何中的一个强有力的工具。研究的主要目的是将Floer同调群Hf(N，N*)的定义推广到浸入拉格朗日子流形N，N*，并理解定义它们的条件(定义障碍)。这涉及到的新技术问题涉及J-全纯圆盘D，其边界通过N或N*中的自交点，以及什么是正确的代数设置来包括和计数这些来得到行为良好的群HF(N，N*)。我们还想了解N和N*在不改变HF(N，N*)的浸没拉格朗日子流形之间的允许运动。除了辛几何学家感兴趣外，我们相信这些结果将对弦理论物理学家感兴趣的关于特殊拉格朗日几何和Calabi-Yau流形的几个主要猜想有重要的应用。重点是，只有当一个人在正确的拉格朗日类中工作，而嵌入的非奇异拉格朗日不是一个足够大的类时，这些猜想才可能是正确的。有很好的证据表明，要考虑的正确类别可能是沉浸式拉格朗日，其Floer同调是畅通无阻的，但为了理解这意味着什么，我们首先需要一个沉浸式拉格朗日的Floer同调理论，我们希望发展这一理论。

Most of modern geometry studies some kind of space. The spaces considered in differential geometry are called manifolds , spaces which locally look like n-dimensional Euclidean space but globally have an interesting shape. A manifold is compact if it is closed up, with no edges. The surface of a doughnut is a compact 2-dimensional manifold. A submanifold N of a manifold M is a subset of M which is itself a manifold, usually of smaller dimension than M. There are two kinds: embedded submanifolds, which may not intersect (cross) themselves, and immersed submanifolds, which may. One usually considers manifolds with some extra geometric structure, such as a Riemannian metric , which tells you the lengths of paths in the manifold, or a symplectic structure , which tells you the areas of 2-dimensional submanifolds of the manifold. Symplectic manifolds are the foundation of the mathematical formulation of mechanics, and so of much of classical physics. They are also very interesting in their own right. Mathematicians like them because they are one of very few structures with an infinite-dimensional amount of local symmetry, which gives symplectic geometry an unusual, entirely global flavour. Lagrangian submanifolds are a special kind of submanifold of a symplectic manifold. Given two compact, embedded Lagrangian submanifolds N, N* of a symplectic manifold M, one can under certain conditions define the Floer homology groups HF(N,N*), which are roughly speaking finite-dimensional vector spaces. The definition is very difficult. To do it, one chooses an auxiliary complex structure J on M and counts J-holomorphic 2-dimensional discs D in M with boundary (edge) in the union of N and N*. The remarkable thing about HF(N,N*) is that it is independent of the choice of J, and is also unchanged by moving N and N* around amongst Lagrangian submanifolds. It encodes some mysterious, nontrivial information about Lagrangian submanifolds one cannot get at in any other known way. It is a powerful tool in symplectic geometry. The main aim of the research is to extend the definition of Floer homology groups HF(N,N*) to immersed Lagrangian submanifolds N, N*, and to understand the conditions under which they can be defined ( obstructions to their definition). The new technical problems this involves have to do with J-holomorphic discs D whose boundary passes through self-intersection points in N or N*, and what is the right algebraic set-up for including and counting these to get well-behaved groups HF(N,N*). We also want to understand the allowed motions of N and N* amongst immersed Lagrangian submanifolds which do not change HF(N,N*). As well as being interesting to symplectic geometers, we believe these results will have important applications to several major conjectures about special Lagrangian geometry and Calabi-Yau manifolds, which are of interest to physicists working in String Theory. The point is that these conjectures can only be true if one works in the right class of Lagrangians, and embedded nonsingular Lagrangians are not a large enough class. There is good evidence that the right class to consider may be immersed Lagrangians whose Floer homology is unobstructed, but to understand what this means we first need a theory of Floer homology for immersed Lagrangians, which we hope to develop.