Algebraic Topology, Representation Theory, and Theoretical Physics
代数拓扑、表示论和理论物理
基本信息
- 批准号:0603964
- 负责人:
- 金额:$ 50.26万
- 依托单位:
- 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:
- 关键词:
项目摘要
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.
国际和平研究所将继续两条主线的研究。第一个是与Michael Hopkins和Constantin Teleman共同建立的,它建立在我们的定理的基础上,用一定扭曲的等变K-理论环来识别循环群表示论中的Verlinde环。Verlinde环是三维拓扑量子场论--Chern-Simons理论的一部分,我们将尝试用K理论在这个方向上作进一步的构造。在另一个方向上,Verlinde环可以用对应图和K理论中的积分来构造。要做到这一点,模空间的一致取向是必要的,在这方面,Madsen-Tillman谱的出现将得到进一步的探索。对曲面族和“开弦”的扩展可能会与其他数学相联系,例如冯·诺依曼代数,并可能开辟新的联系。第二条研究主线,与不同的合作者联合,涉及弦理论和M理论中的拓扑思想。例如,我们希望与格雷格·摩尔和格雷厄姆·西格尔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、Heisenberg群、二次形式和指数定理。还有一些有趣的问题涉及D-膜和B-场的各个方面,其中一些问题影响了目前关于弦理论解的景观的想法。此外,还有许多学生项目与这些主题以及狄拉克算符的几何理论有关。这项研究是始于20世纪70年代中期的代数拓扑学和量子场论相互作用的一部分。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。从物理学中涌现的新思想启发了数学的发展和新的方向。特别是,在过去的15年里,量子场论的一个新的拓扑面已经发展起来,不仅在拓扑学上产生了影响,而且在几何学的其他部分也产生了影响。我们的一些努力致力于利用代数拓扑中的积分过程来阐明拓扑量子场论的积分过程--前者是明确定义的,而后者尚未在必要的一般性数学上得到理解。这笔赠款还支持教育活动,包括周六早上数学小组,这是一个非常成功的初中生和高中生外展计划。
项目成果
期刊论文数量(0)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
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Daniel Freed的其他文献
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{{ truncateString('Daniel Freed', 18)}}的其他基金
Conference: Arithmetic quantum field theory
会议:算术量子场论
- 批准号:
2400553 - 财政年份:2024
- 资助金额:
$ 50.26万 - 项目类别:
Standard Grant
RTG: Unified Training in Geometry and Topology
RTG:几何和拓扑的统一训练
- 批准号:
1148490 - 财政年份:2012
- 资助金额:
$ 50.26万 - 项目类别:
Continuing Grant
FRG: Collaborative Research: In and Around Theory X
FRG:协作研究:X 理论及其周边
- 批准号:
1160461 - 财政年份:2012
- 资助金额:
$ 50.26万 - 项目类别:
Standard Grant
EMSW21-RTG: Unified Approach to Training in Geometry
EMSW21-RTG:几何训练的统一方法
- 批准号:
0636557 - 财政年份:2007
- 资助金额:
$ 50.26万 - 项目类别:
Continuing Grant
Mathematical Sciences: Topology, Geometry and Physics
数学科学:拓扑、几何和物理
- 批准号:
9626698 - 财政年份:1996
- 资助金额:
$ 50.26万 - 项目类别:
Continuing Grant
Mathematical Sciences: Topology, Geometry, and Physics
数学科学:拓扑、几何和物理
- 批准号:
9307446 - 财政年份:1993
- 资助金额:
$ 50.26万 - 项目类别:
Continuing Grant
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