Geometric Representation Theory, Topology, and Mathematical Physics

我对数学物理学、拓扑学和表象论之间的联系感兴趣，特别是后两者在量子场论和弦论中的体现。更具体地说，我目前正在研究一个特定类别的等变矩阵分解，它来自弦理论中的LandauGinzburg或B模型，和一个特定类别的半单李群及其循环群的表示之间的联系。这种联系是通过一个狄拉克型算子建立的，该算子产生了扭曲的等变Fredholm丛，当群是紧的时，它与扭曲的K-理论有关系。这种等价性也可以被看作是Verlinde公式的分类，因此对3D拓扑Chern Simons理论有影响，考虑到各种TFT的连接和对偶的令人眼花缭乱的网络，它可能与所有类型的数学物理有关。我目前的项目的目的有三个：首先，我正在发展的分析方法扩展狄拉克算子技术的情况下，非紧的真实的半单李群，并证明了一个类似的等价范畴。其次，我是一个新的证明Kirillov字符公式紧凑的群体，循环紧凑的群体，并缓和表示的真实的半单群不利用哈里什-钱德拉的工作轨道积分-而是直接遵循的性质，这狄拉克运营商。最后，我使用代数方法，使用半无限上同调和顶点算子代数，开发非紧群的循环群的表示的好类，这些表示可以作为推广这个分类的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。在这个项目中使用的新方法围绕数学物理。几年前，我的导师和他的合作者在弦理论的启发下，引入了矩阵分解来研究表示。作为应用，利用Dirac族的Chern特征标给出了紧群的Kirillov特征标公式的一个新的证明，该公式的优点是不作太大的改动，即可推广到Loop群和真实的半单群.其他的方法包括半无限上同调和顶点算子代数，这两种方法都有物理学的起源，我使用的是结合非紧半单李群的环群的标志簇的几何来开发离散级数的类似物，这个项目属于EPSRC拓扑研究领域的福尔斯

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