AI 中文总结
该研究引入粗糙加性函数概念，将粗糙路径理论扩展到分布 1 - 形式线积分。通过随机微分方程定义受控规范变换等，利用奇异椭圆偏微分方程及相关方法求解，应用于单位正方形上的杨 - 米尔斯测度，得到有最优正则性的规范固定表示。
AI 中文摘要
我们引入了粗糙加性函数的概念，它将粗糙路径理论扩展到分布 1 - 形式的线积分。在规范理论背景下，我们用此概念通过随机微分方程定义受控规范变换和全纯性。我们的一个主要结果是基于粗糙加性函数的分布联络的粗糙版本的乌伦贝克紧致性。主要成分是一个奇异椭圆偏微分方程以获得库仑规范。我们使用正则结构和受隐函数定理启发的方法求解此偏微分方程。令人惊讶的是，求解该方程所需的模型完全由粗糙加性函数确定，这是一个更简单且几何上更自然的对象。我们将结果应用于单位正方形上的杨 - 米尔斯测度，表明它具有具有最优正则性的规范固定表示。尽管本文专注于单位正方形，但我们期望结果能应用于更一般的曲面。
英文摘要
We introduce the concept of rough additive functions, which extends rough paths theory to line integrals of distributional 1-forms. In the context of gauge theory, we use this notion to define controlled gauge transformations and holonomies via RDEs. One of our main results is a rough version of Uhlenbeck compactness for distributional connections based on rough additive functions. The main ingredient is a singular elliptic PDE to obtain a Coulomb gauge. We solve this PDE using regularity structures and a method inspired by the implicit function theorem. Surprisingly, the model needed to solve the equation is determined entirely from rough additive functions, which is a simpler and geometrically more natural object. We apply our results to the Yang-Mills measure on the unit square, showing that it has a gauge-fixed representation with optimal regularity. Although we focus on the unit square in this article, we expect our results to apply to more general surfaces.
Comments107 pages, 7 figures
