Gauge Fields, Geometry, and Strings

规范字段、几何图形和字符串

• 批准号: 1620637
• 负责人: Christopher Beasley
• 金额: $ 18万
• 依托单位: Northeastern University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-09-01 至 2020-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1620637&HistoricalAwards=false
• 关键词: Gauge; Fields; Geometry; Strings

This award funds the research activities of Professor Christopher Beasley at Northeastern University.According to modern particle physics, three of the four fundamental forces in Nature are expressed mathematically within the framework of certain types of theories known as gauge theories. These three forces are the electromagnetic force, the weak nuclear force (which is responsible for radioactive nuclear decays), and the strong nuclear force (which holds atomic nuclei together). The fourth fundamental force, gravity, stands as the notable exception. The discovery of gauge theories is undoubtably one of the great scientific achievements of the 20th century, crucial not only for our modern understanding of elementary particle physics but also for vast swaths of modern geometry and topology. Yet many foundational questions about the dynamics of gauge theory remain poorly understood. These questions have bearing on such topics as the structure of the proton, the character of dark matter and dark energy, or even the mechanism for high-temperature superconductivity. Professor Beasley will conduct research to develop new mathematical methods and tools, inspired by string theory, for analyzing a special class of gauge theories defined in two spatial dimensions. These gauge theories preserve supersymmetry, which relates particles of different spins to each other and is among the possible extensions of the Standard Model now being tested in experiments at the Large Hadron Collider (LHC). Research in this area thus advances the national interest by promoting the progress of science in one of its most fundamental directions: the discovery and understanding of new physical laws. The project will also have significant broader impacts. Substantial portions of the project involve joint work between Professor Beasley and his graduate students. The project thereby serves to train the next generation of mathematical physicists in the methods of quantum field theory and string theory, as well as to disseminate those methods more broadly throughout the mathematical community. Professor Beasley will also give a series of popular scientific talks for first-year undergraduates, will supervise junior and senior mathematics majors in a Research Capstone, and will videotape his lectures in a graduate course on quantum field theory for mathematicians.More technically, Professor Beasley will study perturbative and non-perturbative properties of supersymmetric Wilson loop operators associated to Legendrian knots in contact three-manifolds. He will investigate a relation between these operators and infinite-dimensional symmetry algebras known to occur in two-dimensional quantum field theory, as well as a new localization method for Feynman integrals. Professor Beasley will also construct a novel, time-dependent topological supersymmetry on three-manifolds which fiber over the circle. These ingredients will be applied to perform a variety of exact computations in gauge theory. The project invokes deep notions from contact geometry and topology, and so will establish further interdisciplinary connections between theoretical physics and mathematics.

该奖项资助东北大学克里斯托弗·比斯利教授的研究活动。根据现代粒子物理学，自然界四种基本力中的三种在称为规范理论的某些类型的理论框架内以数学方式表达。 这三种力是电磁力、弱核力（导致放射性核衰变）和强核力（将原子核结合在一起）。 第四种基本力，即重力，是一个值得注意的例外。 规范理论的发现无疑是 20 世纪最伟大的科学成就之一，它不仅对于我们对基本粒子物理的现代理解至关重要，而且对于现代几何学和拓扑学的广泛领域也至关重要。 然而，关于规范理论动力学的许多基本问题仍然知之甚少。 这些问题涉及质子的结构、暗物质和暗能量的特性，甚至高温超导的机制等。 比斯利教授将受弦理论的启发进行研究，开发新的数学方法和工具，用于分析在二维空间维度定义的特殊类别的规范理论。这些规范理论保留了超对称性，它将不同自旋的粒子相互联系起来，并且是标准模型的可能扩展之一，目前正在大型强子对撞机（LHC）的实验中进行测试。 因此，这一领域的研究通过促进科学在其最基本方向之一的进步（发现和理解新的物理定律）来促进国家利益。 该项目还将产生更广泛的影响。 该项目的很大一部分涉及比斯利教授和他的研究生之间的共同工作。 因此，该项目旨在培训下一代数学物理学家使用量子场论和弦理论方法，并在整个数学界更广泛地传播这些方法。 比斯利教授还将为一年级本科生举办一系列受欢迎的科学讲座，指导大三和大四数学专业的研究顶峰，并将在面向数学家的量子场论研究生课程中录制他的讲座。将研究与接触三流形中勒让德结相关的超对称威尔逊环算子的微扰和非微扰特性。 他将研究这些算子与二维量子场论中已知的无限维对称代数之间的关系，以及费曼积分的新定位方法。 比斯利教授还将在三个流形上构造一个新颖的、时间相关的拓扑超对称性，该流形在圆上形成纤维。 这些成分将用于执行规范理论中的各种精确计算。 该项目引用了接触几何和拓扑的深层概念，因此将在理论物理和数学之间建立进一步的跨学科联系。