杨-米尔斯拉格朗日量的变分
The Variations of Yang-Mills Lagrangian
摘要：
杨-米尔斯理论是在高能物理和数学之间的界面上发展起来的。众所周知，杨-米尔斯理论和规范理论对现代微分几何和代数几何的发展产生了深远的影响。人们可以引用唐纳森不变量在四维微分拓扑，希钦小林猜想有关的存在厄米-爱因斯坦度量的全纯束在凯勒流形和芒福德稳定的复杂几何或格罗莫夫威滕不变量在辛几何.虽然规范理论在几何学中的影响是相当臭名昭著的，但人们有时往往会忘记，杨-米尔斯理论在过去几十年中也是偏微分方程非线性分析的基本进展的核心。这门迷你课程的目的是介绍这个重要的拉格朗日量的变化。我们将提出分析问题，如杨-米尔斯极小的存在性和正则性，或如临界点的模空间的紧化一般杨-米尔斯拉格朗日。这些笔记对应于作者在2014年6月23日至6月27日期间在韩国高等研究院第9届微分几何暑期学校就该主题提供的迷你课程。
英文摘要：
Yang-Mills theory is growing at the interface between high energy physics and mathematics. It is well known that Yang-Mills theory and Gauge theory in general had a profound impact on the development of modern differential and algebraic geometry. One could quote Donaldson invariants in four dimensional differential topology, Hitchin Kobayashi conjecture relating the existence of Hermitian-Einstein metric on holomorphic bundles over Kähler manifolds and Mumford stability in complex geometry or also Gromov Witten invariants in symplectic geometry...etc. While the influence of Gauge theory in geometry is quite notorious, one tends sometimes to forget that Yang-Mills theory has been also at the heart of fundamental progresses in the non-linear analysis of Partial Differential Equations in the last decades. The purpose of this mini-course is to present the variations of this important lagrangian. We shall raise analysis questions such as existence and regularity of Yang-Mills minimizers or such as the compactification of the moduli space of critical points to Yang-mills lagrangian in general. These notes correspond to a mini-course given by the author on the subject at the 9th summer school in Differential Geometry at the Korean Institute for Advanced Studies between june 23rd and june 27th 2014.
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