Algebraic Topology, Representation Theory, and Theoretical Physics

代数拓扑、表示论和理论物理

• 批准号: 0603964
• 负责人: Daniel Freed
• 金额: $ 50.26万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-06-01 至 2012-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0603964&HistoricalAwards=false
• 关键词: Algebraic; Topology; Representation; Theory; Theoretical

The PI will continue two main lines of research. The first, joint withMichael Hopkins and Constantin Teleman, builds on our theorem identifying theVerlinde ring in the representation theory of loop groups with a certaintwisted equivariant K-theory ring. The Verlinde ring appears as part of a3-dimensional topological quantum field theory, Chern-Simons theory, and wewill attempt to make further constructions in this direction using K-theory.In another direction, the Verlinde ring may be made using correspondencediagrams and integration in K-theory. Consistent orientations of modulispaces are necessary to carry this out, and the appearance of Madsen-Tillmannspectra in this regard will be further explored. Extensions to families ofsurfaces and "open strings" potentially connect with other mathematics, forexample von Neumann algebras, and may open up new links. The second mainline of research, joint with a variety of collaborators, concerns topologicalideas in string theory and M-theory. For example, with Greg Moore and GraemeSegal we hope to construct completely the quantum theory of a generalizedself-dual field. This will mix theta functions, Heisenberg groups, quadraticforms, and the index theorem in a novel manner. There are also interestingquestions involving various aspects of D-branes and the B-field, some ofwhich impact current ideas about the landscape of solutions to string theory.In addition, there are many student projects which relate to these topics aswell as to the geometric theory of Dirac operators.This research is part of an interaction between algebraic topology andquantum field theory which began in the mid 1970s. The flow of ideas isbidirectional. Existing mathematics is applied to solve particular problemsin physical theories. New ideas emerging from the physics inspiredevelopments and new directions in the mathematics. In particular, over thepast 15 years a new topological side to quantum field theory has beendeveloped and has had ramifications not only in topology but other parts ofgeometry as well. Some of our efforts are devoted to using the integrationprocesses in algebraic topology to shed light on the integration processes intopological quantum field theory--the former are well-defined whereas thelatter have as yet to be understood mathematically in the necessarygenerality. The grant also supports educational activities, including theSaturday Morning Math Group, a highly successful outreach program for middleand high school students.

PI 将继续开展两条主要研究方向。 第一个是与迈克尔·霍普金斯和康斯坦丁·泰勒曼联合提出的，建立在我们的定理基础上，该定理确定了环群表示论中的 Verlinde 环与某个扭曲等变 K 理论环。 Verlinde 环作为三维拓扑量子场论 Chern-Simons 理论的一部分出现，我们将尝试使用 K 理论在这个方向上进行进一步的构造。在另一个方向上，Verlinde 环可以使用对应图和积分来制作K理论。 要实现这一点，需要模空间的一致方向，并且将进一步探讨 Madsen-Tillmannspectra 在这方面的出现。 曲面族和“开弦”的扩展可能与其他数学（例如冯·诺依曼代数）联系起来，并可能开辟新的联系。 第二条研究主线与众多合作者共同关注弦理论和 M 理论中的拓扑思想。 例如，我们希望与 Greg Moore 和 GraemeSegal 一起完整地构建广义自对偶场的量子理论。 这将以一种新颖的方式混合 theta 函数、海森堡群、二次型和指数定理。 还有涉及 D 膜和 B 场各个方面的有趣问题，其中一些影响了当前关于弦理论解决方案的想法。此外，还有许多与这些主题以及几何理论相关的学生项目这项研究是 20 世纪 70 年代中期开始的代数拓扑和量子场论之间相互作用的一部分。 思想的流动是双向的。 现有的数学被用来解决物理理论中的特定问题。 物理学中出现的新思想激发了数学的发展和新方向。 特别是，在过去 15 年里，量子场论的一个新的拓扑方面已经发展起来，并且不仅对拓扑产生了影响，而且对几何的其他部分也产生了影响。 我们的一些努力致力于使用代数拓扑中的积分过程来阐明拓扑量子场论中的积分过程——前者是明确定义的，而后者尚未在必要的普遍性上得到数学上的理解。 这笔赠款还支持教育活动，包括周六上午数学小组，这是一个针对中学生和高中生的非常成功的外展项目。