Gauge theory and Mirror Symmetry

本研究项目是关于低维拓扑规范理论的实例研究。在特殊情况下，规范理论被认为强调对宇宙的数学描述；例如，粒子物理的标准模型是4维的规范理论。在过去的半个世纪里，对量子物理的许多数学研究都集中在做出准确的实验预测所需的分析上，但直到最近一二十年，才被忽略的是，该理论的拓扑(与距离无关)特征由一个潜在的代数决定，该代数具有一些新的和未被探索的特征。PI将在低维空间研究与这个代数有关的基本问题。主要项目旨在通过镜像对称测试技术对拓扑规范理论(所谓的A模型)进行严格描述，将定义为相干同伦系统的“软”拓扑问题转换为解析或代数几何的刚性问题。PI提出了3维纯规范理论的镜像，它控制规范和2维规范理论，如由具有哈密顿群作用的辛流形产生的规范理论。这项研究的另一个组成部分涉及三维规范理论，特别是Chern-Simons理论，它似乎介于软A模型和刚性B模型之间。最后，将进行一个涉及表示理论(代数曲线的几何朗兰兹对偶)的小项目：简单花边群的一般朗兰兹核的构造，它依赖于上同调和代数几何技巧，但本课题是从4维规范理论的电磁对偶中获得信息和启发的。

This research project concerns the study of examples of topological gauge theories in low dimensions. Gauge theory, in special cases, is believed to underline the mathematical description of the universe; for example, the standard model for particle physics is a gauge theory in 4 dimensions. Much mathematical research into quantum physics over the last half-century has focussed on the analysis required to make accurate experimental predictions, but what was missed until the last decade or two was that topological (distance-independent) features of the theory are governed by an underlying algebra, which has some novel and unexplored features. The PI will study fundamental questions pertaining to this algebra in low dimension.The main project aims at a rigid description of topological gauge theory (the so-called A-model) by the tested technique of mirror symmetry, which converts "soft" topological questions, defined up to coherent systems of homotopies, into rigid questions of analytic or algebraic geometry. The PI has a proposal for the mirror of pure gauge theory in 3 dimensions, which controls the gauging and 2-dimensional gauge theories such as arise from symplectic manifolds with Hamiltonian group actions. Another component of the research involves gauge theory in 3 dimensions, specifically Chern-Simons theory, which seems to lie half-way between the soft A- and the rigid B-model. Finally, a small project involving representation theory (geometric Langlands duality for algebraic curves) will be undertaken: the construction of a generic Langlands kernel for simply laced groups, which relies on cohomological and algebro-geometric techniques, but the subject is informed and inspired by electric-magnetic duality for gauge theory in 4 dimensions.