Mathematical Sciences: Two-dimensional Topological Field Theories and Complex Cobordism

数学科学：二维拓扑场论和复配边

• 批准号: 9119954
• 负责人: Jack Morava
• 金额: $ 16.1万
• 依托单位: Johns Hopkins University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1992
• 资助国家: 美国
• 起止时间: 1992-02-15 至 1995-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9119954&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Two; dimensional; Topological

Under support from DMS 8713718, a problem of paramount interest was to generalize elliptic cohomology so as to encompass curves of higher genus. Y. Shimizu and the investigator found a solution to this problem (over the rational numbers), based on the idea that affine connections on elliptic curves (which do not generalize to higher genus) should be interpreted as special cases of the projective connections (which exist for all Riemann surfaces). This allowed them to associate in a natural way to any chiral conformal field theory (in the sense of Segal), the topological genus defined by the projective connection (or stress-energy tensor) of the field theory. They were able to identify this genus in the case of the charge-one fermion and to show that it reduced (in the case of the elliptic curve) to the elliptic genus. A corollary of this construction is that the complex cobordism ring supports a canonical quadratic differential, closely related in some way to a quadratic differential supported by the fundamental bosonic representation of the Virasoro algebra. The relation between these two quadratic differentials seems to be of fundamental importance and will be a major emphasis of the current project. In recent years mathematical constructions inspired by theories used to make sense of subatomic particles have found repeated application to purely geometric problems. To their mutual profit physicists and mathematicians are interacting more than ever in exploring these connections, and the investigator is a major player in this game.

在DMS 8713718的支持下， 有趣是推广椭圆上同调， 高等属的曲线 Y.清水和调查员发现 解决这个问题（在有理数上），基于 椭圆曲线上的仿射连接（不 一般化到更高的属）应该被解释为特殊的 投影联络的情况（存在于所有黎曼 表面）。 这使他们能够以自然的方式联系起来， 任何手征共形场论（在西格尔意义上）， 由射影联络定义的拓扑亏格（或 能量张量（Stress-Energy Tensor）。 他们能够 在荷一费米子的情况下， 为了表明它减少（在椭圆曲线的情况下）到 椭圆亏格 这种结构的一个推论是， 复配边环支持标准二次型 微分，在某种程度上与二次函数密切相关 基本玻色子支持的微分 Virasoro代数的表示。 的关系 这两个二次微分似乎是基本的 这一点很重要，也将是当前项目的重点。 近年来，受理论启发的数学结构 用来解释亚原子粒子的理论 应用于纯几何问题。 为了他们的共同利益 物理学家和数学家的互动比以往任何时候都多， 探索这些联系，调查人员是一个主要的 玩家在这个游戏中。