FRG: Collaborative Research: In and Around Theory X

FRG：协作研究：X 理论及其周边

• 批准号: 1342948
• 负责人: David Nadler
• 金额: $ 21.64万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-07-01 至 2016-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1342948&HistoricalAwards=false
• 关键词: FRG; Collaborative; Research; Around; Theory

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.

在上一代中，四维经典的自动划分阳米尔方程及其尺寸降低对代数几何，差异几何，低维拓扑，可集成的系统和非线性PDE产生了深远的影响。 现在，一个六维量子场理论的家族正在成为急性数学兴趣的对象。 这个集中的研究小组汇集了在数学和物理学不同部分研究这些理论的研究人员。 它们最初是作为字符串理论的限制而出现的，通常称为“超符号（2,0）理论”，以引起人们对它们的对称性的关注。 更简单的称呼“理论X”强调了知之甚少。 这里进行的项目有两个总体目标。 首先，我们将通过应用对过去25年中获得的拓扑和保形量子场理论的详细而深刻的数学理解来介绍理论X的结构。 其次，我们将使用理论X的预期特性及其压缩来降低维度，以推断几何表示理论中的新猜想和新的组织原理。 六维量子理论X与二十一世纪几何学，拓扑和几何表示理论中的许多中心主题之间的迅速发展的相互作用网络表明，我们看到了一场新革命的开始，其中理论X扮演着主导的物理作用。揭开其结构及其后果的进展将产生广泛的影响。物理学长期以来一直在数学方面发展，过去30年一直是一个特别富有成果的时期。 进入基本物理理论的数学深度已经稳步增强，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。 该项目是挖掘这种思想肥沃的思想结合的众多努力之一。 我们对理论X的追求将不可避免地阐明更广泛的思想圈子，并有助于对当代物理学的数学理解。 过去几代人在数学 - 物理界面上的工作增添了现代世界：我们的计算机，GPS系统，运输，复杂的医疗工具，更多的归功于它们的存在归功于该领域的基础研究，这些研究可以追溯到一个世纪以上。 尽管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。