RUI: Solitons and Quantum Field Theory

该奖项资助宾夕法尼亚州立大学费耶特校区安迪·罗伊斯顿教授的研究活动。“场”的概念是物理学和日常生活的核心。场（如电场和磁场）存在于整个空间，使重力、电力和磁力等力得以传播。例如，光波和无线电波是电磁场中的涟漪。“量子场论”是理论物理学家用来将基本粒子描述为场中离散涟漪的数学框架。在他的研究中，罗伊斯顿教授的目标是应用以使用被称为“孤子”的物体为中心的新方法来解决量子场论中两个困难且长期存在的问题。孤子是一种特殊类型的粒子，它可以在场自相互作用时存在；我们可以把孤子想象成一个缠结的场。通过研究孤子相互作用和与普通粒子相互作用的数学描述，Royston教授旨在了解某些粒子产生和衰变过程的机制，这些机制超出了传统计算方法的范围。因此，对量子场论数学结构的研究通过在最基础的水平上促进科学的进步来促进国家利益。该项目还将产生更广泛的重大影响。罗伊斯顿教授将让本科生参与他的研究，让他们接触基础科学，并教授他们计算机编码和数值分析等实用技能，这些技能将有利于他们在STEM领域的职业发展。更技术性的是，Royston教授将解决的第一个问题是确定虚拟孤子-反孤子对对涉及微扰粒子的过程的主要贡献。通过量子场论中的交叉对称，这种贡献与孤子发射或吸收高能粒子导致孤子质量数量级的大动量传递有关。罗伊斯顿教授将结合半经典技术和一种新工具——强迫孤子方程——来分析这一过程。罗伊斯顿教授及其合作者在2020年发现的强迫孤子方程是一个波状方程，它描述了一个孤子沿着任意指定的轨迹被驱动。Royston教授将解决的第二个问题是关于某些超对称规范理论中孤子束缚态谱的“过壁”现象，并着重从孤子场构型和模空间的半经典角度来理解过壁的方式。在最近的工作中，Royston教授注意到磁单极子的过壁现象与数学结构之间的密切联系，该结构旨在提供单极子模空间的紧化，作为带角的流形。Royston教授与一位著名数学家合作，旨在通过分析单极子模空间中某个量子力学中零能束缚态的跳跃行为，给出一个完整的过壁描述。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This award funds the research activities of Professor Andy Royston at Penn State Fayette, The Eberly Campus.The concept of a "field" is central to physics and everyday life. Fields (such as electric fields and magnetic fields) exist throughout space and enable the transmission of forces like gravity, electricity, and magnetism. Light and radio waves, for example, are ripples in an electromagnetic field. "Quantum field theory" is the mathematical framework theoretical physicists use to describe fundamental particles as discrete ripples in a field. In his research, Professor Royston aims to apply novel approaches centered on the use of objects called "solitons" to address two difficult and long-standing questions in quantum field theory. A soliton is a special type of particle that can exist when fields self-interact; one can imagine a soliton as a knot of tangled-up field. By studying the mathematical description of solitons interacting with each other and with ordinary particles, Professor Royston aims to understand mechanisms for certain particle creation and decay processes beyond the reach of traditional computational methods. Research on the mathematical structure of quantum field theory thus advances the national interest by promoting the progress of science at its most foundational level. This project will also have significant broader impacts. Professor Royston will involve undergraduates in his research, exposing them to basic science and teaching them practical skills such as computer coding and numerical analysis that will benefit them in STEM careers.More technically, the first question Professor Royston will address is that of determining the leading contribution of virtual soliton-antisoliton pairs to processes involving perturbative particles. Through crossing symmetry in quantum field theory, such contributions are related to a soliton that emits or absorbs a high energy particle resulting in a large momentum transfer of order the mass of the soliton. Professor Royston will combine semiclassical techniques with a new tool --- the forced soliton equation --- to analyze this process. The forced soliton equation, discovered by Professor Royston and collaborators in 2020, is a wavelike equation that describes a soliton being driven along an arbitrarily specifiable trajectory. The second question Professor Royston will address concerns "wall-crossing" phenomena for soliton bound-state spectra in certain supersymmetric gauge theories, and focuses on the manner in which wall-crossing can be understood from the semiclassical perspective of soliton field configurations and moduli spaces. In recent work, Professor Royston has noted a close connection between wall-crossing phenomena for magnetic monopoles and a construction in mathematics that aims to provide a compactification of monopole moduli space as a manifold with corners. Professor Royston, working in collaboration with a leading mathematician, aims to give a complete description of wall-crossing by analyzing the jumping behavior of zero-energy bound states in a certain quantum mechanics on monopole moduli space.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.