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

这个建议包括几个项目，使用代数拓扑技术来研究弦拓扑中出现的问题， 模空间的拓扑，拓扑量子场论的研究。弦拓扑理论由查斯和沙利文在1999年首次提出，现在涉及到流形的路径和回路空间上的大量丰富结构。 在这一建议中， 将弦拓扑学与几何学和拓扑场论联系起来的项目。 这包括与A.布隆伯格和C. Teleman，将弦拓扑与最近的分类理论的拓扑场论，由于科斯特洛和霍普金斯-卢里。 特别是他们将构建和研究弦拓扑范畴的流形， 计算它的Hochschild同调 他们将研究这一类别的“Calabi-Yau”性质。 在这项研究中，有趣的问题涉及的作用Koszul对偶拓扑场论出现，作者将解决。 科恩和他的合作者也有一个 目标是比较流形的余切丛的辛场论与该流形的弦拓扑。 该提案中的其他项目包括与M.施瓦茨合作， 辛流形的“量子弦拓扑”的发展，以及 与N. Kitchloo，以解决几何学家F。Lalonde和D. McDuff关于辛纤维丛的Serre谱序列的研究。这个建议包括几个项目，研究被称为“弦拓扑”的研究领域，以及相关的问题。 弦拓扑学是Chas和Sullivan在1999年首次提出的一个理论，研究路径、环和曲面空间上的结构。 这种结构是由物理学中的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）如何随时间演变。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曲面来测量的，这些曲面跨越这些环。 在这个项目中，科恩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。

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.