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

这项建议包括几个项目，使用代数拓扑技术来研究弦拓扑学中出现的问题，模空间的拓扑学，以及拓扑量子场论的研究。弦拓扑理论最早由Chas和Sullivan于1999年提出，现在涉及到流形的路径和环空间上的大量丰富结构。在这份提案中，有一些项目将把弦拓扑学与几何和拓扑场论联系起来。这包括与A.Blumberg和C.Teleman的一个联合项目，该项目将把弦拓扑学与由于Costello和Hopkins-Lurie而产生的拓扑场论的最新分类理论联系起来。特别地，他们将构造和研究流形的弦拓扑范畴，并计算它的Hochschild同调。他们将调查这一类别的``Calabi-Yau‘性质。在这项研究中，涉及到Koszul对偶在拓扑场论中的作用的有趣的问题出现了，作者将解决这些问题。科恩和他的合作者还有一个目标是将流形的余切丛的辛场理论与该流形的弦拓扑进行比较。该建议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合作解决几何学家F.Lalonde和D.McDuff关于带有辛纤维的丛的Serre谱序列的猜想。该建议包括几个项目，调查被称为“弦拓扑学”的研究领域，以及相关问题。弦拓扑是由查斯和沙利文于1999年首次提出的一种理论，它研究路径、环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。这个想法是为了了解背景空间中的环路(或路径)如何随时间演变。环路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曲面来测量的，这些曲面横跨这些循环。在这个项目中，科恩倾向于研究与拓扑量子场理论相关的理论的各个方面，以及与辛几何相关的那些方面。

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.