String Topology, Field Theories, and the Topology of Moduli Spaces

弦拓扑、场论和模空间拓扑

• 批准号: 1104555
• 负责人: Ralph Cohen
• 金额: $ 35.67万
• 依托单位: Stanford University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-07-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1104555&HistoricalAwards=false
• 关键词: String; Topology; Field; Theories; Moduli

This proposal consists of several projects using algebraic topological techniques to study questions arising in String Topology, the topology of moduli spaces, and the study of Topological Quantum Field Theories. The theory of String Topology, first introduced by Chas and Sullivan in 1999, now involves a vast array of rich structure on spaces of paths and loops of manifolds. In this proposal, there are projects that will relate string topology to geometry and topological field theory. This includes a joint project with A. Blumberg and C. Teleman that will relate string topology to the recent classification theories of topological field theories due to Costello and Hopkins-Lurie. In particular they will construct and study a string topology category of a manifold, and compute its Hochschild homology. They will investigate the ``Calabi-Yau" properties of this category. In this study interesting questions involving the role of Koszul duality in topological field theory arise, which the authors will address. Cohen and his collaborators also have a goal of comparing the symplectic field theory of the cotangent bundle of a manifold, with the string topology of that manifold. Other projects in this proposal include a collaboration with M.Schwarz on the development of the ``Quantum String topology" of a symplectic manifold, and a project with N. Kitchloo, to address conjectures by the geometers F. Lalonde and D. McDuff on the Serre spectral sequence for bundles with symplectic fibers.This proposal consists of several projects investigating the area of research known as "String Topology", as well as related questions. String topology, a theory that was first introduced by Chas and Sullivan in 1999, studies structures on spaces of paths, loops, and surfaces. This structure was motivated by formalisms in string theory in physics. The idea is to understand how loops (or paths) in a background space can evolve in time. Loops can evolve by changing in size and even breaking apart. These phenomena are measured by studying surfaces mapping to the background space, that span these loops. In this project, Cohen tends to study various aspects of the theory related to topological quantum field theories, as well as those related to symplectic geometry.

本提案包括几个项目，使用代数拓扑技术来研究弦拓扑，模空间拓扑和拓扑量子场论的研究中出现的问题。弦拓扑理论是由Chas和Sullivan于1999年首次提出的，现在涉及到路径空间和流形环路上的大量丰富结构。在这个建议中，有一些项目将弦拓扑与几何和拓扑场理论联系起来。这包括与a . Blumberg和C. Teleman的一个联合项目，该项目将弦拓扑与最近由Costello和Hopkins-Lurie提出的拓扑场论的分类理论联系起来。特别地，他们将构造和研究流形的弦拓扑范畴，并计算其Hochschild同调。他们将研究这一类别的“Calabi-Yau”属性。在这项研究中，有趣的问题涉及到科祖尔对偶在拓扑场理论中的作用，作者将解决。Cohen和他的合作者还有一个目标，就是比较流形的余切束的辛场论，和该流形的弦拓扑。该提案中的其他项目包括与M.Schwarz合作开发辛流形的“量子弦拓扑”，以及与N. Kitchloo合作解决几何学者F. Lalonde和D. McDuff关于辛纤维束的Serre谱序列的猜想。该提案由几个项目组成，研究被称为“弦拓扑”的研究领域，以及相关问题。弦拓扑是由Chas和Sullivan于1999年首次提出的理论，研究路径、环路和曲面空间上的结构。这种结构是由物理学中弦理论的形式主义所激发的。这个想法是为了理解背景空间中的循环（或路径）是如何随时间演变的。循环可以通过改变大小甚至分裂来进化。这些现象是通过研究映射到背景空间的表面来测量的，这些表面跨越了这些循环。在这个项目中，Cohen倾向于研究与拓扑量子场论相关的理论的各个方面，以及与辛几何相关的理论。