Gauge theory and Mirror Symmetry

本研究计画是关于低维拓扑规范理论的研究。在特殊情况下，规范理论被认为强调了宇宙的数学描述;例如，粒子物理学的标准模型是四维规范理论。在过去的半个世纪里，许多对量子物理的数学研究都集中在进行准确的实验预测所需的分析上，但直到最近一二十年，人们才发现，理论的拓扑（与距离无关）特征是由一个潜在的代数决定的，这个代数有一些新颖的和未探索的特征。PI将研究与低维代数相关的基本问题。主要项目旨在通过镜像对称的测试技术对拓扑规范理论（所谓的A模型）进行严格的描述，该技术将定义为同伦相干系统的“软”拓扑问题转化为解析几何或代数几何的严格问题。PI有一个关于3维纯规范理论的镜像的提议，它控制着规范和2维规范理论，例如从辛流形与哈密顿群作用产生的规范理论。研究的另一个组成部分涉及三维规范理论，特别是陈-西蒙斯理论，它似乎位于软A模型和刚性B模型之间。最后，一个涉及表示理论（代数曲线的几何朗兰兹对偶）的小项目将被承担：简单花边群的通用朗兰兹核的构建，它依赖于上同调和代数几何技术，但该主题受到四维规范理论的电磁对偶的启发。

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.