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String Topology, J-holomorphic Curves, and Symplectic Geometry

弦拓扑、J 全纯曲线和辛几何

基本信息

批准号：
EP/J016950/1
负责人：
Dominic Joyce
金额：
$ 32.11万
依托单位：
University of Oxford
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2012
资助国家：
英国
起止时间：
2012 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FJ016950%2F1
关键词：
String Topology holomorphic Curves Symplectic

项目摘要

Differential geometry is the study of multi-dimensional curved spaces called "manifolds". The surfaces of a sphere or a torus (doughnut) are examples of 2-dimensional manifolds. Often one puts extra geometric structures on manifolds. "Symplectic manifolds" are manifolds with a "symplectic structure". They are part of the mathematical foundations of mechanics -- the basic physics of how objects move -- and quantum mechanics. If M is a symplectic manifold, a "Lagrangian submanifold" N in M is a manifold N inside M, with half the dimension of M, compatible with the symplectic structure in a certain way. They are central objects in symplectic geometry. Topology studies the "shape" of spaces including manifolds, focussing on qualitative properties that are unchanged by stretching or bending the space. The "number of holes" (genus) of a 2-dimensional manifold is a topological invariant. The sphere has no holes, and the torus has one hole, so they are different as topological spaces.One important topological invariant of a space M is its "homology" H(M). Introduced by Henri Poincare in the 19th century, H(M) is an algebraic object which measures things like the "number of holes" in M.If M is a space, the "loop space" LM is the space of all loops (circles) inside M. For example, if M is the 3-dimensional space we live in, then (embedded) loops are knots. Loop spaces are important in topology, and also String Theory in Physics, which models elementary particles not as points but as tiny "loops of string".In 1999, Chas and Sullivan discovered that if M is a manifold, then the homology H(LM) of the loop space LM carries some extra algebraic structures coming from intersections of families of loops in M with each other -- basically, H(M) has a multiplication on it. This was unexpected and exciting, and has grown into a subject called "String Topology".The homology H(M) is defined using a more basic object C(M) called a "chain complex". There are many different ways to define a chain complex C(M), but they all compute the same homology H(M). One difficulty in String Topology is that, up to now, there is no nice way to define the String Topology multiplication on a chain complex C(LM) computing H(LM) -- it is only defined directly on homology H(LM). String topologists believe that if one could define the operations on the level of chains C(LM), then deeper algebraic structures in Strong Topology would be revealed, leading to the definition of a Topological Conformal Field Theory (TCFT, a mathematical object coming out of quantum physics) from LM.Our first project is to define a chain complex C(LM) which computes H(LM), upon which we can define a version of the String Topology multiplication on chains, and so construct (at least the genus 0 part of) the TCFT.We will do this using a new homology theory called "Kuranishi homology" recently defined by the PI. Kuranishi homology has the property that problems to do with "transversality" usually disappear. Since the difficulty of defining string topology operations on chains is mostly about transversality, Kuranishi homology is a good choice.Now let M be a symplectic manifold, and N a Lagrangian submanifold. A major tool in symplectic geometry is the study of "J-holomorphic curves", 2-dimensional manifolds C in M whose boundary dC is a 1-dimensional manifold (loop) in N. So there is a natural map from families of J-holomorphic curves to the loop space LN, mapping C to dC.Our second project is to prove some conjectures by Fukaya. These say that using the families of J-holomorphic curves in (M,N), we can define chains in C(LN) satisfying equations involving the chain-level String Topology operations defined in the first project. By combining this with facts about the topology of LN, we expect to prove important new restrictions on the topology of Lagrangian submanifolds in simple symplectic manifolds, such as flat spaces and projective spaces.

微分几何是对称为“流形”的多维弯曲空间的研究。球体或环面（环形）的表面是二维流形的示例。通常人们会在流形上放置额外的几何结构。 “辛流形”是具有“辛结构”的流形。它们是力学（物体如何运动的基本物理学）和量子力学的数学基础的一部分。如果M是辛流形，则M中的“拉格朗日子流形”N是M内部的流形N，其维数是M的一半，在某种程度上与辛结构兼容。它们是辛几何中的中心物体。拓扑学研究包括流形在内的空间“形状”，重点关注拉伸或弯曲空间不会改变的定性属性。二维流形的“孔数”（属）是拓扑不变量。球体没有孔，环面有一个孔，因此它们作为拓扑空间是不同的。空间M的一个重要的拓扑不变量是它的“同源性”H(M)。 H(M) 由 Henri Poincare 在 19 世纪提出，是一个代数对象，用于测量 M 中的“孔数”等内容。如果 M 是一个空间，则“循环空间”LM 是 M 内所有循环（圆）的空间。例如，如果 M 是我们生活的 3 维空间，那么（嵌入的）循环就是结。环空间在拓扑学中很重要，在物理学中的弦理论也很重要，它不是将基本粒子建模为点，而是建模为微小的“弦环”。 1999 年，Chas 和 Sullivan 发现，如果 M 是流形，则环空间 LM 的同源性 H(LM) 带有一些额外的代数结构，这些结构来自 M 中的环族彼此的交集 - 基本上，H(M) 具有乘法。这是出乎意料且令人兴奋的，并且已经发展成为一个名为“弦拓扑”的学科。同调 H(M) 是使用称为“链复合体”的更基本对象 C(M) 来定义的。定义链复数 C(M) 的方法有很多种，但它们都计算相同的同源性 H(M)。弦拓扑的一个困难在于，到目前为止，还没有一种很好的方法来定义链式复数 C(LM) 计算 H(LM) 上的弦拓扑乘法——它只是直接在同调 H(LM) 上定义。弦拓扑学家认为，如果能够定义链 C(LM) 层面上的运算，那么强拓扑中更深层次的代数结构就会被揭示，从而导致 LM 的拓扑共形场论（TCFT，一个来自量子物理的数学对象）的定义。我们的第一个项目是定义一个计算 H(LM) 的链复数 C(LM)，在此基础上我们可以定义弦拓扑的一个版本链上的乘法，从而构建（至少是属 0 部分）TCFT。我们将使用 PI 最近定义的称为“Kuranishi 同源性”的新同源理论来完成此操作。仓西同调具有与“横向性”有关的问题通常会消失的特性。由于定义链上的弦拓扑运算的难度主要在于横截性，因此Kuranishi同调是一个不错的选择。现在令M为辛流形，N为拉格朗日子流形。辛几何的一个主要工具是研究“J-全纯曲线”，M 中的二维流形 C，其边界 dC 是 N 中的一维流形（环）。因此，存在从 J-全纯曲线族到环空间 LN 的自然映射，将 C 映射到 dC。我们的第二个项目是证明 Fukaya 的一些猜想。这些说，使用（M，N）中的J全纯曲线族，我们可以定义C（LN）中的链，满足涉及第一个项目中定义的链级弦拓扑运算的方程。通过将其与 LN 拓扑的事实相结合，我们期望证明简单辛流形（例如平坦空间和射影空间）中拉格朗日子流形拓扑的重要新限制。