The Faddeev Knots and the Skyrme Solitons

摘要奖：DMS-0406446主要研究人员：杨益松这个项目是为Faddeev量子场论模型中出现的纽结孤子发展一个普遍的存在理论。问题的复杂结构需要创新的技术和想法，而不是成熟的工具，如集中-紧凑法。最近，PI与林芳华合作研究了这些Faddev结的存在性，并得到了一系列重要的结果。他们研究的另一个特点是，尽管紧致性可能会失败，但存在可能是两个关键不等式的结果：第一个不等式被他们称为实体不等式，确保了至少一个拓扑极小值的存在；第二个不等式给出了Faddeev能量的最优次线性上增长估计，根据相关的拓扑电荷，意味着存在无穷族的拓扑极小值，因此存在Faddeev结。这一进展为更高水平地理解Faddeev结的各个未知方面开辟了新的方向，包括低电荷非结(环状孤子)的存在，高电荷结的存在及其稳定性，以及鉴于其拓扑和能量特性的结的转变图。Faddeev能量和Skyrme能量之间的相似性可以用来解决Skyrme孤子难以存在的问题。事实上，F·H·林和PI的工作的副产品将是单位电荷Skyrme孤子存在的证明，这纠正了1986年发表的M.Esteban有缺陷(但众所周知)的证明。这项研究也可能扩展到在凝聚态物质物理和宇宙学中有应用的两维Skyrme模型。本课题还为其他重要相关领域中具有拓扑特征的全局能量最小化子的存在性问题的研究提供了新的方向。例如，开尔文勋爵首次探索了使用纽结来模拟原子和分子的想法。最近，纽结的概念启发了许多基本领域，包括粒子和凝聚态物理、统计力学、宇宙学、分子生物学和合成化学。直到最近，数学工作一直专注于纽结的拓扑和组合分类，而不是存在。1995年，Faddeev和Niemi开始了一项开创性的研究，使用高功率的计算机模拟，将纽结的存在作为称为Faddeev模型的量子场理论模型的孤子解。这项拟议的工作发展了这种结的数学存在理论。这项工作将对纽结的数学结构及其计算机实现提供新的见解。

AbstractAward: DMS-0406446Principal Investigator: Yisong YangThis project is to develop a general existence theory for theknotted solitons arising in Faddeev's quantum field theorymodel. The difficult structure of the problem requires innovativetechniques and ideas beyond the well developed tools such as themethod of concentration-compactness. Recently, the PI hascollaborated with F. H. Lin on the existence of these Faddeevknots and a series of important results have been obtained. Anotable feature in their study is that, although compactness mayfail, existence may be achieved as a consequence of two keyinequalities: the first inequality, called by them theSubstantial Inequality, ensures the existence of at least onetopological minimizer; the second inequality, giving an optimalsublinear upper growth estimate for the Faddeev energy in termsof the associated topological charge, implies the existence of aninfinite family of topological minimizers, hence the existence ofthe Faddeev knots. This progress opens new directions forachieving a higher level of understanding of various unsettledaspects of the Faddeev knots, including the existence ofunit-charge unknots (ring-like solitons), existence ofhigh-charge knots and their stability, and transition pictures ofknots in view of their topological and energeticalcharacteristics. The similarity between the Faddeev energy andthe Skyrme energy may be exploited to tackle the difficultexistence problem of the Skyrme solitons. Indeed, a by-product ofthe work of F. H. Lin and the PI will be a proof of theexistence of unit-charge Skyrme solitons which corrects theflawed (but well-known) proof of M. Esteban published in 1986.This study may also be extended to the Skyrme model in twospatial dimensions which has applications in condensed matterphysics and cosmology. This project also suggests new directionsfor the study of the existence problems of global energyminimizers with topological characteristics in other importantrelated areas.The concept of knots is of foundational importance inscience. For example, Lord Kelvin first explored the idea ofusing knots to model atoms and molecules. More recently, theconcept of knots has inspired various fundamental areas,including particle and condensed-matter physics, statisticalmechanics, cosmology, molecular biology, and synthetic chemistry.Up until recently, mathematical work had been focused ontopological and combinatorial classifications of knots, but noton existence. In 1995, Faddeev and Niemi started a seminal studyon the existence of knots as the soliton solutions of a quantumfield theory model known as the Faddeev model using high-powercomputer simulations. The proposed work develops a mathematicalexistence theory of such knots. This work will give new insightinto the mathematical structure of knots and their computerrealization.