Algebraic Topology, Representation Theory, and Theoretical Physics

PI将继续两条主要研究路线。 第一个是与Michael霍普金斯和Constantin Teleman联合建立的，它建立在我们的定理上，该定理将loop群的表示理论中的Verlinde环与某个扭曲的等变K-理论环相鉴别。 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K理论中的对应图和积分来构造。 一致的方向的modulispaces进行这是必要的，并在这方面的麦德森-蒂尔曼谱的外观将进一步探讨。 曲面族和“开弦”的扩展可能与其他数学，例如冯诺依曼代数，有联系，并可能开辟新的联系。 研究的第二条主线，与各种合作者联合，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米西格尔一起，完全构造广义自对偶场的量子理论。 这将以一种新颖的方式混合θ函数、海森堡群、二次型和指数定理。 还有一些有趣的问题涉及D-膜和B-场的各个方面，其中一些影响了当前关于弦理论解的观点。此外，还有许多与这些主题以及狄拉克算子的几何理论有关的学生项目。这项研究是代数拓扑学和量子场论之间相互作用的一部分，始于20世纪70年代中期。 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殊问题。 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，在过去的15年里，量子场论的一个新的拓扑方面得到了发展，不仅在拓扑学上，而且在几何学的其他部分也有了分支。 我们的一些努力致力于使用积分过程在代数拓扑揭示积分过程在拓扑量子场论-前者是明确定义的，而后者还没有被理解数学在necessarygeneralization。 该补助金还支持教育活动，包括星期六上午数学小组，一个非常成功的初中和高中学生外展计划。

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.