Moduli Spaces: Their Topology and Representations

这个项目的目标是在适当的意义上研究曲线模空间及其近亲的（同调群）的“表示环”。这些对象是非平凡拓扑量子场论的严格构造。在曲线模空间的情况下，相应的表示是上同场理论（在Kontsevich和Manin的意义上），其中最引人注目的例子来自光滑射影变的Gromov-Witten不变量。研究者将使用从同伦理论（操作数）、代数几何（模空间）和量子场论（费曼图）中借来的技术，研究由这些模空间的几何产生的这些结构的正则族的性质。最近，数学和理论物理之间的许多相互作用都围绕着拓扑量子场论，量子场论的可观测值对应于拓扑不变量。通过研究Gromov-Witten不变量，以及从黎曼曲面到Kahler流形的映射空间的拓扑不变量，我们可以严格地构造出这类非平凡的理论。这个项目的目的是研究这些理论（及其相关理论）的自然族，这些理论（及其相关理论）产生于这些空间的几何形状，并了解它们在几何形状暗示的自然运算下的性质。成功应该同时说明所涉及的数学和物理。＊＊＊

9803427 Kimura This project's goal is to study the "representation ring," in the appropriate sense, of the (homology groups of the) moduli space of curves and its close cousins. Such objects are rigorous constructions of nontrivial topological quantum field theories. In the case of the moduli space of curves, the corresponding representations are cohomological field theories (in the sense of Kontsevich and Manin), the most dramatic examples of which arise from Gromov-Witten invariants of a smooth projective variety. The investigator will study properties of canonical families of such structures arising from the geometry of these moduli spaces, using techniques borrowed from homotopy theory (operads), algebraic geometry (moduli spaces), and quantum field theory (Feynman diagrams). Recently, much of the interaction between mathematics and theoretical physics has revolved about topological quantum field theories, quantum field theories whose observables correspond to topological invariants. A rigorous construction of a nontrivial class of such theories is now possible through the study of Gromov-Witten invariants, topological invariants associated to a certain space of maps from Riemann surfaces into a Kahler manifold. The purpose of this project is to study natural families of such theories (and their relatives) that arise from the geometry of these spaces and to understand their properties under natural operations suggested by the geometry. Success should shed light on both the mathematics and the physics involved. ***