Intergratable Quantum Field Theories:New Methods And Applications

在基础物理学（从物质的亚核结构到天体物理学和宇宙学）中有大量不断扩大的问题，这些问题处理的系统的理论描述涉及连续的许多变量，它们之间具有很强的相互联系。量子场论是这类系统的统一数学语言。适当地说，这门语言很难，而且它的“语法规则”一般还没有建立起来。在这种情况下，量子场论的任何实例或“模型”，如果其方程可以一直推导出来，就特别有价值。这样的模型被称为“可积量子场论”。它们主要是作为验证和发展思想和数学方法的试验场，但非常多的可积“模型”直接应用于对复杂材料性质的理论理解。这就是为什么今天可积量子场论构成了数学物理中最重要和最积极发展的领域之一。此外，自从发现“规范/弦二象性”和可积性在这种关系中的作用以来，这一研究领域已经发现自己处于理论高能物理的前沿。不幸的是，在这种情况下出现的可积模型仍然无法完全解决。该研究项目提出了可积量子场论的进一步发展，重点是所谓的可积σ模型。这正是在规/弦二象性背景下出现的一般类型的模型。此外，类似的模型被认为对材料物理中特别复杂的无序系统提供了理论理解。提出了一类新的具有新颖数学结构的可积西格玛模型，并求其全解。这是直接应用于量规/弦二象性系统的中间步骤。有趣的是，为可积量子场论开发的数学方法在数学物理中具有更广泛的意义，并且经常有助于研究更广泛的不可积系统。这个领域也将被探索。该活动由三个项目组成。(1)分析了一类新的目标空间为变形群流形的可积σ模型。对于大多数已知的变形，这种模型的量子可积性需要对标准工具进行重大修改，例如Lax表示、Yang-Baxter代数和Baxter的Q算子及其关系。该项目的目标是开发所有这些修改，并将其作为找到完整解决方案的基础。我们方法的关键将是所谓的ODE/IQFT对应，这是可积量子场论（“IQFT侧”）的新数学工具，它允许将其代数结构“编码”到常微分方程系统（“ODE”侧）中。本课题是当前数学物理研究的主流；它的意义在于与规范/弦二象性的关系，并可能与无序系统的物理学有关。另外两部分的研究是探索可积性方法在非可积系统中的应用。(2)利用1+1 QCD中的t Hooft方程和Baxter的t - q方程之间的关系分析大n展开中的介子衰减和散射振幅，最终目的是获得具有约束相互作用的一般1+1系统的衰减和振幅分析的通用工具，这在凝聚态物理中是非常常见的。(3)第三部分讨论了在不可积量子场论中出现近可积子系统的可能性，这种可能性以前没有被探索过。

There is large and expanding array of problems in fundamental physics (ranging from sub-nuclear structure of matter through astrophysics and cosmology) which deal with systems whose theoretical description involves continuously many variables with strong interconnection between them. Quantum field theory is a unifying mathematical language for such systems. Appropriately, this language is difficult, and in general its "grammatical rules" are not even yet established. In this situation, any instances, or "models", of quantum field theory whose equations can be worked out all the way through are especially valuable. Such models are known as "integrable quantum field theories". Mainly, they serve as the testing ground for verification and development of ideas and mathematical methods, but remarkably many integrable "models" directly apply to theoretical understanding of properties of complex materials. That is why today the integrable quantum field theories constitute one of the most important and actively developed area of mathematical physics. Moreover, since the discovery of "gauge/string duality" and the role of integrability in this relation, this field of research has found itself at the cutting edge of theoretical high-energy physics. Unfortunately, the integrable models emerging in this context still resist full solution. This research project proposes further development of integrable quantum field theories, with emphasis on the so-called integrable sigma models. This is exactly the general class of models which emerge in the context of the gauge/string duality. Also, similar models are believed to provide theoretical understanding of particularly complicated disordered systems in physics of materials. It is proposed to develop a certain new class of integrable sigma-models, with novel mathematical structure, and find full solutions. This is an intermediate step in the approach to the systems of direct application in the gauge/string dualities. Interestingly, mathematical methods developed for integrable quantum field theories turn out to have broader significance in mathematical physics, and often help to investigate a wider class of non-integrable systems. This area will be explored as well.The activity consists of three projects. (1) Analysis of a new class of integrable sigma models whose target spaces are deformed group manifolds. With the most general known deformations, quantum integrability of such models require significant modifications of standard tools such as the Lax representations, the Yang-Baxter algebras, and Baxter's Q operators and their relations. The project's goal is to develop all such modifications, and use that as the basis for finding full solutions. The key to our approach will be the so-called ODE/IQFT correspondence, the new mathematical tool of integrable quantum field theories ("IQFT side") which allows to "encode" its algebraic structures into a system of ordinary differential equations ("ODE" side). This project lays in the mainstream of the current research in mathematical physics; its significance is in the relation to the gauge/string dualities, and possibly to physics of disordered systems. The other two parts the research is to explore applications of the methods of integrability in non-integrable systems. (2) The relation between the 't Hooft's equation in 1+1 QCD and the Baxter's T-Q equation will be used to analyze the meson decays and scattering amplitudes in the large-N expansion, with the ultimate goal of gaining universal tools for analysis of decays and amplitudes in generic 1+1 systems with confining interactions, which are very common in condensed matter physics. (3) The third part addresses possibility of emergence of near-integrable subsystems in otherwise non-integrable quantum field theories, the possibility which was not previously explored.