FRG: Collaborative Research: In and Around Theory X

在过去的一代中，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，一组六维量子场论正成为引起数学极大兴趣的对象。这个重点研究小组汇集了在数学和物理的不同部分工作的研究人员来研究这些理论。它们最初是作为弦理论的极限出现的，通常被称为“超共形（2,0）理论”，以引起人们对其对称性的注意。更简单的称谓“X理论”强调了我们对它知之甚少。这里进行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。其次，我们将利用X理论的期望性质及其对低维的紧化来推导几何表示理论中的新猜想和新的组织原则。六维量子X理论与21世纪几何学、拓扑学和几何表示理论的一系列中心主题之间快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始，在这场革命中，X理论起着主导的物理作用。在揭示其结构及其后果方面取得的进展将产生广泛的影响。物理学长期以来一直推动着数学的发展，过去30年是一个特别富有成果的时期。数学进入基础物理理论的深度不断加深，同时这些理论的结构和预测对数学的影响也日益深刻。这个项目是挖掘这一智力丰富的思想组合的众多努力之一。我们对X理论的追求将不可避免地照亮更广泛的思想圈子，并有助于对当代物理学的数学理解。过去几代人在数学和物理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、复杂的医疗工具，以及更多的东西，它们的存在都归功于这一领域的基础研究，这些研究可以追溯到一个多世纪以前。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.