Mathematical Sciences: Floer Homotopy, Kontsevich-Gromov- Witten Theory, and Quantum Cohomology

在过去的几年里，Kaehler流形的自由环空间的Floer同调一直受到相当多的关注，部分原因是它定义了一个二维拓扑场论。最近，Cohen、Jones和Segal定义了一个弗洛尔同伦型的基本概念，可以将其解释为拓扑场论的“普遍变形”，物理学家称之为二维拓扑引力理论。这个flower同伦型是一个相当神秘的mu -代数谱。在这个项目中，研究者为它构建了一个推测模型，他探索了一些方法，从同伦理论的角度来理解弦物理学家的结构。整个课题对同伦理论和全局分析的未来具有深远的意义。19世纪发展起来的力学理论是建立在“最小作用原理”的基础上的：例如，一束光沿着使其飞行时间最短的路径运动。物理学家理查德·费曼（Richard Feynman）用所有可能路径的空间积分理论重新解释了这些想法；他的思想现在是我们理解量子力学的基础。不幸的是，这种费曼路径积分的理论从来没有变得严格；的确，现在已经知道，经典积分理论的朴素推广不能构成现代物理学中出现的积分的充分基础。然而，在几何领域，最近已经很清楚费曼积分理论的思想可以用来解决纯数学的经典问题，并且有证据表明，许多几何问题在某种意义上是“驯服”的，因此可以严格地建立费曼积分理论的类似物。尽管在许多方面受到限制，但这些几何测试问题为理解物理学直接感兴趣的“动态”问题提供了非常清晰和极其重要的数据，它们是我们对费曼积分一致理论的最佳指导。在这个项目中，研究者从代数拓扑的角度，对一类相当宽泛的拓扑场论进行了推测性描述，这些理论源于费曼积分技术在复杂流形几何中的应用。＊＊＊

9504234 Morava The Floer homology of the free loopspace of a Kaehler manifold has been the subject of considerable attention in the last few years, in part because it defines a two-dimensional topological field theory. Recently Cohen, Jones, and Segal have defined an underlying notion of Floer homotopy type, which can be interpreted as the `universal deformation' of this topological field theory to what physicists call a theory of two-dimensional topological gravity. This Floer homotopy type is a rather mysterious MU-algebra spectrum. In this project the investigator constructs a conjectural model for it, and he probes some of the ways in which it provides an understanding of the constructions of string physicists from a homotopy-theoretical point of view. The whole subject has profound implications for the future of homotopy theory and global analysis. The theory of mechanics developed in the ninteenth century was based on 'principles of least action': a ray of light, for example, follows the path which minimizes its time of flight. The physicist Richard Feynman reinterpreted these ideas in terms of a theory of integration over the space of all possible paths; his ideas are now fundamental to our understanding of quantum mechanics. Unfortunately, the theory of such Feynman path integrals has never been made rigorous; indeed, it is now known that no naive generalization of the classical theory of integration can form an adequate basis for the integrals which arise in modern physics. In geometry, however, it has become clear recently that ideas from the theory of Feynman integrals can be used to solve classical problems of pure mathematics, and there is evidence that many geometric problems are in some sense 'tame' enough so that an analogue of the theory of Feynman integrals can be established rigorously. Although restricted in many ways, these geometrical test questions provide very clear and extremely important data for understanding the 'dyna mical' problems of direct interest to physics, and they are our best guide to a consistent theory of Feynman integrals. In this project the investigator sketches a conjectural description, in terms of algebraic topology, for a rather lapge class of topological field theories, which arise from the application of Feynman integral techniques to the geometry of complex manifolds. ***