Gauge Fields, Geometry, and Strings

Northeastern University, Boston MA

Investigators

Abstract

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.