Gauge Fields, Geometry, and Strings

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，自然界的四种基本力中的三种是在某些类型的理论(称为规范理论)的框架内用数学表示的。这三种力是电磁力、弱核力(负责放射性核衰变)和强核力(将原子核保持在一起)。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，不仅对我们对基本粒子物理学的现代理解至关重要，对大量现代几何和拓扑学也是如此。然而，关于规范理论动力学的许多基础性问题仍然知之甚少。这些问题关系到质子的结构、暗物质和暗能量的性质，甚至高温超导的机制。比斯利教授将在弦理论的启发下进行研究，开发新的数学方法和工具，用于分析在两个空间维度定义的一类特殊的规范理论。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最基本的方向之一--发现和理解新的物理定律--来促进国家利益。该项目还将产生重大的更广泛的影响。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合作。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的方法，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做一系列受欢迎的科学演讲，将在研究中心指导初三和高级数学专业的学生，并将为数学家录制他在量子场论研究生课程中的讲座。更严格地说，比斯利教授将研究与接触三流形中的勒让德结有关的超对称威尔逊环算符的微扰和非微扰性质。他将研究这些算符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。比斯利教授还将在三维流形上构建一种新颖的、依赖于时间的拓扑超对称性。这些成分将被应用于规范理论中的各种精确计算。该项目引用了接触几何学和拓扑学中的深层概念，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.