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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.

微分几何是研究多维弯曲空间称为“流形”。球面或环面（圆环面）的表面是二维流形的例子。人们常常在流形上加上额外的几何结构。“辛流形”是具有“辛结构”的流形。它们是力学的数学基础的一部分--物体如何运动的基本物理学--和量子力学。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，其维数为M的一半，以某种方式与辛结构相容。它们是辛几何中的中心对象。拓扑学研究包括流形在内的空间的“形状”，专注于拉伸或弯曲空间时不变的定性性质。二维流形的“洞数”（亏格）是一个拓扑不变量。球面上没有洞，而环面上只有一个洞，所以它们作为拓扑空间是不同的，空间M的一个重要拓扑不变量是它的“同调”H（M）。由Henri Poincare在世纪提出，H（M）是一个代数对象，它度量了M中的“洞数”。如果M是一个空间，“循环空间”LM是M中所有循环（圆）的空间。例如，如果M是我们生活的三维空间，那么（嵌入的）循环就是结。回路空间在拓扑学中很重要，物理学中的弦论也很重要，弦论将基本粒子建模为微小的“弦回路”，而不是点。1999年，Chas和Sullivan发现，如果M是流形，那么回路空间LM的同调H（LM）带有一些额外的代数结构，这些结构来自M中回路族的交叉--基本上，H（M）上有一个乘法。这是意想不到的和令人兴奋的，并已发展成为一个主题称为“弦拓扑”。同调H（M）是使用一个更基本的对象C（M）定义的，称为“链复合体”。有许多不同的方法来定义链复形C（M），但它们都计算相同的同调H（M）。弦拓扑中的一个困难是，到目前为止，还没有很好的方法来定义链复形C（LM）计算H（LM）上的弦拓扑乘法-它只能直接定义在同调H（LM）上。弦拓扑学家认为，如果可以在链C（LM）的水平上定义运算，那么强拓扑中更深层次的代数结构就会被揭示，从而导致拓扑共形场论的定义（TCFT，量子物理学中的一个数学对象）。我们的第一个项目是定义一个计算H（LM）的链复合体C（LM），在此基础上，我们可以定义链上的弦拓扑乘法的一个版本，从而构建TCFT（至少是亏格0部分）。我们将使用PI最近定义的称为“Kuranishi同源”的新同源理论来实现这一点。仓西同调具有这样的性质，即与“横截性”有关的问题通常会消失。由于定义链上的弦拓扑运算的困难主要在于横截性，Kuranishi同调是一个很好的选择。辛几何中的一个主要工具是研究“J-全纯曲线”，即M中的二维流形C，其边界dC是N中的一维流形（回路）。因此，存在从J-全纯曲线族到循环空间LN的自然映射，将C映射到dC。我们的第二个项目是证明福谷的一些猜想。这就是说，使用（M，N）中的J-全纯曲线族，我们可以定义C（LN）中的链，该链满足涉及第一个项目中定义的链级弦拓扑运算的方程。通过结合这一事实的拓扑LN，我们希望证明重要的新的限制拓扑的拉格朗日子流形在简单的辛流形，如平坦空间和射影空间。