Gauge Theory on Manifolds with Special Holonomy

规范理论是一门起源于数学和高能物理的交界处的学科。在特殊情况下，规范理论被认为是宇宙的数学描述的基础；例如，粒子物理的标准模型是四维规范理论。这门学科还与许多数学领域有很深的联系，包括偏微分方程组、表示论、代数几何、微分几何和拓扑学。然而，到目前为止，规范理论的大部分研究和几乎所有的应用都集中在二维、三维和四维。在这个项目中，PI打算在更高的维度上推动我们对规范理论的理解。PI的结果预计将对几何分析产生广泛的影响，因为定义规范理论的基本方程系统(杨-米尔斯方程)在控制解方面提出了额外的挑战，而这些解在较低的维度中不存在。PI的工作也将通过综合研究和培训产生影响，因为PI计划让本科生参与与其项目相关的研究机会。PI的研究项目包括三个部分。该项目的第一部分将把这本规范理论和复代数几何之间的词典扩展到厄米-杨-米尔斯奇点联系的背景下，并根据相应的自反层的复代数几何具体地理解奇点处的精细结构。在这个项目的第二部分，PI将发展一类名为一维奇异集的Yang-Mills联络的形变理论，并用它来构造由代数几何输入的奇异G2-瞬子的具体例子。PI研究项目的第三部分将进一步探讨推广的Seiberg-Witten方程(特别是Andriy Haydys和PI共同工作中发现的方程)在高维规范理论中所起的作用。如上所述，高维规范理论取得进展的主要障碍是我们缺乏对非紧致性问题的理解，这是因为杨-米尔斯方程从五维开始变得超临界，并导致不可移除的奇点和气泡现象的形成。

Gauge theory is a subject that originated at the interface of mathematics and high energy physics. In special cases, gauge theory is believed to underlie the mathematical description of the universe; for example, the standard model for particle physics is a gauge theory in four dimensions. The subject also has deep links with many areas of mathematics, including partial differential equations, representation theory, algebraic geometry, differential geometry and topology. However, most research in gauge theory and almost all applications so far have focused on dimensions two, three and four. In this project, the PI intends to push the boundary of our understanding of gauge theory in higher dimensions. The PI's results are expected to have wide implications for geometric analysis, because the fundamental system of equations that define gauge theory (the Yang-Mills equations) poses additional challenges, in terms of controlling the solutions, that are not present in lower dimensions. The PI's work will also have impact through integrated research and training, as the PI plans to involve undergraduates in research opportunities related to his project.The PI's research project consists of three parts. The first part of the project will extend this dictionary between gauge theory and complex algebraic geometry to the context of Hermitian-Yang Mills connections with singularities, and specifically understand the fine structure at the singularities in terms of the complex algebraic geometry of the corresponding reflexive sheaves. In the second part of this project, the PI will develop a deformation theory for a certain class of Yang-Mills connections called G2-instantons with one-dimensional singular set, and use this to construct concrete examples of singular G2-instantons from algebro-geometric input. The third part of the PI's research project will further explore the role that generalized Seiberg-Witten equations (specifically the ones discovered in joint work of Andriy Haydys and the PI) play in gauge theory in higher dimensions. As alluded to above, the main obstacle to progress in gauge theory in higher dimension is our lack of understanding of the non-compactness issues arising from the fact that the Yang-Mills equations become super-critical starting in dimension five and leading to the formation of non-removable singularities and bubbling phenomena.