Moduli Spaces: Their Topology and Representations

9803427木村这个项目的目标是在适当的意义上研究曲线的模空间及其近亲的“表示环”(同调群)。这样的物体是非平凡的拓扑量子场论的严格构造。在曲线的模空间中，相应的表示是上同调场理论(在Kontsevich和Manin的意义下)，最戏剧性的例子来自光滑射影簇的Gromov-Witten不变量。研究人员将借用同伦理论(算术)、代数几何(模空间)和量子场论(费曼图)的方法，研究由这些模空间的几何产生的这种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，即其观测值与拓扑不变量相对应的量子场论。现在，通过研究Gromov-Witten不变量，可以严格地构造这类理论的非平凡类。Gromov-Witten不变量是与从Riemann曲面到Kahler流形的映射的特定空间相关联的拓扑不变量。这个项目的目的是研究这些理论的自然族(及其亲属)，这些理论产生于这些空间的几何，并理解它们在几何所建议的自然运算下的性质。成功应该会让人们明白所涉及的数学和物理知识。***

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. ***