The Faddeev Knots and the Skyrme Solitons

项目负责人：杨一松本项目旨在建立Faddeev量子场理论模型中产生的结孤子的一般存在性理论。这个问题的复杂结构需要创新的技术和想法，而不仅仅是像集中-紧致法这样的成熟工具。最近，PI与F. H. Lin合作研究了这些Faddeevknots的存在性，并获得了一系列重要的结果。他们研究的另一个特点是，尽管紧性可能会失败，但存在性可以通过两个关键不等式来实现：第一个不等式，被他们称为实质不等式，确保至少存在一个拓扑最小值；第二个不等式给出了Faddeev能量在相关拓扑电荷方面的最优次线性上增长估计，这意味着存在无穷一族的拓扑极小值，因此存在Faddeev结。这一进展为实现对Faddeev结的各种不稳定方面的更高水平的理解开辟了新的方向，包括单位电荷解结（环状孤子）的存在，高电荷结的存在及其稳定性，以及基于其拓扑和能量特征的结的跃迁图。利用Faddeev能量和Skyrme能量之间的相似性，可以解决Skyrme孤子的难存在问题。事实上，F. H. Lin和PI的工作的一个副产品将是证明单位电荷Skyrme孤子的存在，这将纠正1986年发表的M. Esteban有缺陷（但众所周知）的证明。本研究也可以推广到二维空间的Skyrme模型，该模型在凝聚态物理和宇宙学中具有应用。本项目也为其他重要相关领域中具有拓扑特征的全局能量极小器存在性问题的研究指明了新的方向。结的概念在科学上具有重要的基础意义。例如，开尔文勋爵首先探索了用结来模拟原子和分子的想法。最近，结的概念启发了许多基础领域，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，数学工作一直集中在结点的本体论和组合分类上，而不是存在性。1995年，Faddeev和Niemi开始了一项开创性的研究，研究结的存在性，作为量子场理论模型（即Faddeev模型）的孤子解。提出的工作发展了这种结的数学存在理论。这项工作将为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.