Gauge Fields, Geometry, and Strings

该奖项资助了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，自然界的四种基本力中有三种是在某种被称为规范理论的理论框架内用数学方法表达的。这三种力分别是电磁力、弱核力（导致放射性核衰变）和强核力（将原子核聚集在一起）。第四种基本力，重力，是一个明显的例外。规范理论的发现无疑是20世纪最伟大的科学成就之一，它不仅对我们对基本粒子物理学的现代理解至关重要，而且对现代几何学和拓扑学的广大领域也至关重要。然而，关于规范理论动力学的许多基本问题仍然知之甚少。这些问题涉及到质子的结构、暗物质和暗能量的性质，甚至高温超导的机理等问题。Beasley教授将在弦理论的启发下开展研究，开发新的数学方法和工具，用于分析在两个空间维度上定义的特殊规范理论。这些规范理论保持了超对称性，它将不同自旋的粒子相互联系起来，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的方向之一的进步来促进国家利益：发现和理解新的物理定律。该项目还将产生更广泛的重大影响。这个项目的很大一部分涉及比斯利教授和他的研究生的共同工作。因此，该项目有助于培养下一代量子场论和弦理论方法的数学物理学家，并在整个数学界更广泛地传播这些方法。Beasley教授还将为一年级的本科生做一系列流行的科学讲座，指导三年级和四年级的数学专业学生进行一项研究，并将他在一门面向数学家的量子场论研究生课程上的讲座录下来。更技术性的是，Beasley教授将研究接触三流形中与Legendrian结相关的超对称Wilson环算子的微扰和非微扰性质。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，以及费曼积分的一种新的局部化方法。Beasley教授还将在三流形上构建一种新颖的、与时间相关的拓扑超对称结构。这些成分将应用于规范理论中的各种精确计算。该项目涉及接触几何和拓扑学的深层概念，因此将在理论物理和数学之间建立进一步的跨学科联系。

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.