FRG: Collaborative Research: In and Around Theory X

在上一代的四维经典自对偶Yang-Mills方程及其降维对代数几何、微分几何、低维拓扑、可积系统和非线性偏微分方程产生了深远的影响。 现在，一系列六维量子场论正成为数学界的热门话题。 这个重点研究小组汇集了研究人员在数学和物理的不同部分工作，研究这些理论。 它们最初是作为弦理论的极限而出现的，通常被称为“超共形（2，0）-理论”，以引起人们对它们对称性的注意。 “X理论”这个简单的名称强调了我们所知甚少。 在此开展的项目有两个总体目标。 首先，我们将通过应用过去25年来对拓扑和共形量子场论的详细而深刻的数学理解，对X理论的结构进行研究。 第二，我们将利用理论X的预期性质及其对低维的紧化来推导几何表示理论中的新结构和新组织原则。 六维量子X理论与21世纪世纪几何学、拓扑学和几何表象理论中的一系列中心话题之间迅速发展的相互作用网络表明，我们正在看到一场新的革命的开端，在这场革命中，X理论扮演着主导性的物理角色。物理学长期以来一直在推动数学的发展，过去30年是一个特别富有成果的时期。 数学的深度进入基本物理理论已经稳步加强，同时这些理论的结构和预测对数学产生了越来越深刻的影响。 这个项目是挖掘这种智力丰富的想法组合的许多努力之一。 我们对X理论的追求将不可避免地照亮一个更广泛的思想圈，并有助于对当代物理学的数学理解。 过去几代人在物理学-物理学界面上的工作为现代世界提供了动力：我们的计算机、GPS系统、交通工具、先进的医疗工具等等，都要归功于这个领域的基础研究，而这个领域的基础研究可以追溯到一个世纪。 虽然我们无法预测当前的基础研究将如何影响未来，但我们可以肯定地说，这种影响将是深远的。

Over the last generation the four-dimensional classical self-dual Yang-Mills equations and their dimensional reductions have had profound consequences for algebraic geometry, differential geometry, low-dimensional topology, integrable systems, and nonlinear PDE. Now a family of six-dimensional quantum field theories is emerging as an object of acute mathematical interest. This Focused Research Group brings together researchers working in diverse parts of mathematics and physics to study these theories. They originally arose as limits of string theories and are usually called 'superconformal (2,0)-theories' to call attention to their symmetries. The simpler appellation 'Theory X' emphasizes how little is known. The projects undertaken here have two overall goals. First, we will make inroads on the structure of Theory X by applying the detailed and profound mathematical understanding of topological and conformal quantum field theories obtained over the past 25 years. Second, we will use expected properties of Theory X and its compactifications to lower dimensions to deduce new conjectures and new organizing principles in geometric representation theory. The rapidly developing web of interactions between the six-dimensional quantum Theory X and a host of central topics in twenty-first century geometry, topology, and geometric representation theory indicates that we are seeing the beginnings of a new revolution, one in which Theory X plays the dominant physical role. Progress towards unraveling its structure and its consequences will have broad ramifications.Physics has long fueled developments in mathematics, and the past 30 years have been a particularly fruitful period. The depth of mathematics which enters fundamental physical theories has steadily intensified, and at the same time the structure and predictions of these theories have had increasingly profound impacts on mathematics. This project is one of many efforts to mine this intellectually fertile mix of ideas. Our pursuit of Theory X will inevitably illuminate a much broader circle of ideas and contribute to the mathematical understanding of contemporary physics. The work of past generations at the mathematics-physics interface fuels the modern world: our computers, GPS systems, transportation, sophisticated medical tools, and much more owe their existence to basic research in this area which stretches back well over a century. While we cannot predict how current basic research will impact the future, we can say with certainty that the effect will be far-reaching.