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，自然界中四种基本力中的三种在被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（将原子核保持在一起）。 第四种基本力，重力，是一个显著的例外。 规范理论的发现无疑是20世纪世纪最伟大的科学成就之一，不仅对我们现代理解基本粒子物理至关重要，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。 教授比斯利将进行研究，开发新的数学方法和工具，灵感来自弦理论，用于分析一类特殊的规范理论定义在两个空间维度。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的方向之一-发现和理解新的物理定律-上的进步，促进了国家利益。 该项目还将产生重大的广泛影响。 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联合工作。 因此，该项目旨在培养下一代数学物理学家掌握量子场论和弦论的方法，并在整个数学界更广泛地传播这些方法。 此外，比斯利教授亦会为大学一年级学生作一系列科普讲座，并会在一个研究中心指导大三及大四数学专业的学生，以及为数学家在研究生课程中作有关量子场论的讲座录像。此外，比斯利教授亦会研究与接触三维流形中的勒让德结有关的超对称威尔逊环算子的微扰及非微扰性质。 他将调查这些运营商和无限维对称代数已知发生在二维量子场论之间的关系，以及一个新的本地化方法费曼积分。 教授比斯利还将构建一个新的，时间相关的拓扑超对称性的三个流形的纤维在圆。 这些成分将应用于规范理论中的各种精确计算。 该项目从接触几何和拓扑学中调用了深刻的概念，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.