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

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。这个建议包括几个项目，使用代数拓扑技术来研究几何拓扑和几何所产生的问题。这种类型的研究中最令人兴奋和活跃的领域是“字符串拓扑”， 研究模空间的同伦类型，研究拓扑场论。本提案中的项目，有些是长期的，涉及对这些新的和非常有趣的拓扑学领域的调查。 在这个提议中，有一些项目将把弦拓扑应用到几何中，包括 一个辛流形的“量子弦拓扑”的发展与M。施瓦茨，用I. Madsen，以及流形的弦拓扑与余切丛的具有标准辛结构的“Chalaya-Seidel”范畴之间的关系，C. Teleman和A.布伦伯格 还有一些项目将 继续研究和发展这个理论的同伦理论结构，包括与J.D.S. Jones，以及流形的“弦拓扑”A无穷范畴及其Hochschild同调的构造和研究，其中A. Blumberg和C. Teleman. 其他项目包括与马德森合作的共形场论特征类的研究，以及与N。Kitchloo，以解决几何学家F。Lalonde和D. McDuff关于具有辛纤维的丛的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”的新研究领域，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。 在这个提议中，这个理论在各种背景下进行了研究，包括量子场论的数学形式。此外，我们还研究了弦拓扑与更多几何理论之间的关系。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). This proposal consists of several projects using algebraic topological techniques to study questions arising from geometric topology and geometry. Among the most exciting and active areas of this type of study are ``String Topology", the study of the homotopy type of moduli spaces, and the study of topological field theories. The projects in this proposal, some long term, involve investigations into these new and very interesting areas of topology. In this proposal, there are projects that will apply string topology to geometry, including the development of ``Quantum String topology" of a symplectic manifold with M. Schwarz, the study of the cobordism type of moduli spaces of holomorphic curves with I. Madsen, and the relation of the string topology of a manifold to the ``Fukaya-Seidel" category of the cotangent bundle with its canonical symplectic structure,with C. Teleman and A. Blumberg. There are also projects that will to continue to study and develop the homotopy theoretic structure of this theory, including the construction and study of ``higher order" string topology operations with J.D.S. Jones, and the construction and study of the ``string topology $A infinity category" of a manifold, and its Hochschild homology, with A. Blumberg and C. Teleman. Other projects include the study of characteristic classes of conformal field theories, with Madsen, and a project with N. Kitchloo, to address questions asked 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 new 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. In this proposal this theory is studied in a variety of contexts, including the mathematical formalisms of Quantum Field Theory. Moreover the several projects understanding the relationship between string topology and more geometric theories are pursued.