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)提供资金的。这项建议由几个项目组成，使用代数拓扑技术来研究几何拓扑和几何产生的问题。在这类研究中最令人兴奋和活跃的领域是“弦拓扑学”，研究模空间的同伦型，以及拓扑场论的研究。这项建议中的项目，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。在这个建议中，有一些项目将把弦拓扑应用到几何中，包括与M.Schwarz一起发展辛流形的“量子弦拓扑”，与I.Madsen研究全纯曲线的模空间的余边型，以及与C.Teleman和A.Blumberg一起研究流形的弦拓扑与余切丛的“Fukaya-Seidel”范畴及其标准辛结构的关系。还有一些项目将继续研究和发展这一理论的同伦理论结构，包括与J.D.S.Jones一起构造和研究“高阶”弦拓扑运算，与A.Blumberg和C.Teleman一起构造和研究流形的“弦拓扑$A无穷范畴”及其Hochschild同调。其他项目包括与Madsen一起研究共形场论的特征类，以及与N.Kitchloo合作的一个项目，以解决几何学家F.Lalonde和D.McDuff提出的关于带有辛纤维的丛的Serre谱序列的问题。这项建议包括几个项目，这些项目调查被称为“弦拓扑学”的新研究领域，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。在这个提议中，这个理论是在不同的背景下研究的，包括量子场论的数学形式。此外，还继续进行了几个理解弦拓扑与更多几何理论之间关系的项目。

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.