Geometric Representation Theory, Topology, and Mathematical Physics

几何表示理论、拓扑学和数学物理

• 批准号: 1789682
• 金额: --
• 依托单位: University of Oxford
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2016
• 资助国家: 英国
• 起止时间: 2016 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-1789682
• 关键词: Geometric; Representation; Theory; Topology; Mathematical

I am interested in the connections between mathematical physics, topology, and representation theory, in particular the incarnations of the last two in quantum field theory and string theory. More specifically, I currently study a connection between a certain category of equivariant matrix factorisations, which comes from the LandauGinzburg or B model in string theory, and a certain category of representations ofsemisimple Lie groups and their loop groups. The connection is established via a Dirac-type operator that gives rise to twisted equivariant Fredholm bundles, which, when the group is compact, give a relation to twisted K-theory. This equivalence can also be seen as categorification of the Verlinde formula and thus has implications for 3D topological Chern Simons theory and, given the dizzying web of connections and dualities of varies TFT's, is likely related to all kinds of mathematical physics.The aim of my current project is threefold: Firstly, I am developing the analytic methods to extend the Dirac operator technique to the case of non-compact real semisimple Lie groups and prove a similar equivalence of categories. Secondly, I am a novel proof of the Kirillov character formula for compact groups, loops in compact groups, and tempered representations of real semisimple groups which does not make use of Harish-Chandra's work on orbital integrals - instead it follows directly from the properties of this Dirac operator. Finally, I am using algebraic methods, using semi-infinite cohomology and vertex operator algebras, to develop nice classes of representations of loop groups of noncompact groups which might serve as the correct setting in which to generalize this categorified Verlinde formula, whose existence is hinted at in the topology and physics literature alike.The new methods used in this project revolve around mathematical physics. The use of matrix factorisations to study representations, inspired by string theory, was introduced by my supervisor and his collaborators a few years ago. As an application, taking chern characters of the Dirac families gives a new proof of the Kirillov character formula for compact groups, which has the advantage of being fit for generalisation to loop groups and real semisimple groups without much change. Other methods include semi-infinte cohomology and vertex operator algebras, which both have physics origins as well, which I use is conjunction with the geometry of the flag variety of loop groups of noncompact semisimple Lie groups to develop analogs of the discrete series.This project falls within the EPSRC Topology research area

我对数学物理、拓扑学和表示理论之间的联系很感兴趣，尤其是后两者在量子场论和弦理论中的体现。更具体地说，我目前研究的是某一类等变矩阵分解（来自于弦理论中的LandauGinzburg模型或B模型）与半单李群及其环群的某一类表示之间的联系。该联系是通过dirac型算子建立的，该算子产生了扭曲等变Fredholm束，当群紧化时，给出了与扭曲k理论的关系。这种等价也可以看作是Verlinde公式的分类，因此对三维拓扑Chern Simons理论有影响，并且考虑到令人眼花缭乱的连接网络和各种TFT的对重性，可能与各种数学物理有关。我目前的项目有三个目的：首先，我正在开发分析方法，将Dirac算子技术扩展到非紧实半单李群的情况，并证明类的类似等价。其次，我对紧群、紧群中的环、实半单群的调质表示的Kirillov特征公式进行了新颖的证明，它没有利用harich - chandra关于轨道积分的工作，而是直接从这个狄拉克算子的性质出发。最后，我使用代数方法，使用半无穷上同调和顶点算子代数，来开发非紧群的环群的漂亮的表示类，这可能作为推广这个分类Verlinde公式的正确设置，它的存在在拓扑和物理文献中都有暗示。这个项目中使用的新方法围绕着数学物理。受弦理论的启发，我的导师和他的合作者几年前介绍了使用矩阵分解来研究表示。作为应用，取Dirac族的陈氏特征，给出了紧群的Kirillov特征公式的一个新的证明，该证明具有适合推广到环群和实半单群的优点。其他方法包括半无穷上同调和顶点算子代数，它们都有物理起源，我将其与非紧半单李群的环群的标志变化的几何相结合，以开发离散级数的类似物。该项目属于EPSRC拓扑研究领域