Research in Integrable Quantum Field Theory

该奖项资助新泽西州立大学罗格斯大学的谢尔盖·卢基亚诺夫教授的研究活动。自1920年S创立以来，量子场论已成为现代理论物理中不可或缺的工具。除了粒子物理学-开发这个框架的最初动机-它在描述相变物理以及在多体系统、湍流、宇宙学等方面的大量应用方面取得了显著的成功。随着主题列表的不断扩大，我们被迫不断修改我们对量子力学的基本理解，并采用新的强大的数学技术来处理日益复杂的问题。要做到这一点，一个完美的理论实验室已经被证明涉及到低维的“可积”模型和共形场论。这些理论拥有一组大的，常常是无限的对称性，这些对称性一起允许使用强大的精确方法来求解。反过来，这种更高的控制程度已经-并仍在引导-对许多重要的物理概念有深刻的见解。因此，可积量子场论(IQFT)领域处于理论高能物理的前沿。Lukyanov教授的研究旨在为IQFT的进一步发展奠定基础。成功的实现将使人们能够克服当传统方法应用于主流理论高能物理背景下出现的许多理论时出现的臭名昭著的技术困难。因此，这项研究符合国家利益，因为它将对美国国内基础科学的发展和理解产生重要影响。卢基亚诺夫教授还打算让研究生和博士后参与他的研究，并通过讲座和工作坊将结果传播给更广泛的年轻科学家社区。更严格地说，这项研究专注于可积量子自旋链和非线性西格玛模型-正是规范/弦对偶性背景下出现的理论类型。本研究分为三个项目。第一个是研究一类处于临界点的多参数量子自旋链。这种方法的关键将是所谓的ODE/IQFT对应，这是一种新的可积量子场论(“IQFT”侧)的数学工具，它允许人们将其代数结构“编码”成一个常微分方程组(“ODE”侧)。这部分与仪表/弦的二元性直接相关。第二个项目致力于研究量子色动力学中散射幅度的高能渐近性。第三个项目的最终目标是将ODE/IQFT方法扩展到可积的非线性Sigma模型。这个项目所考虑的新方法，一旦得到充分发展，应该会产生广泛的影响。第一个和第三个项目可能会带来全新的方向和发现，包括在描述凝聚态物理中无序系统方面的潜在应用。相比之下，第二个项目与粒子物理中的一个现实问题有关。这三个项目都为学生提供了极好的QFT和统计力学方面的培训机会，从本科水平的问题到最抽象的分析技术的高级培训。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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 Sergei Lukyanov at Rutgers, The State University of New Jersey.Since its inception in the 1920's, Quantum Field Theory (QFT) has become an indispensable tool in modern theoretical physics. Apart from particle physics --- the original motivation for developing this framework --- it has found remarkable success in describing the physics of phase transitions as well as numerous applications in many-body systems, turbulence, cosmology, etc. As the list of topics continues to expand, we are forced to constantly revise our fundamental understanding of QFT and incorporate new powerful mathematical techniques that can handle problems of increasing complexity. A perfect theoretical laboratory for doing this has turned out to involve low-dimensional "integrable" models and conformal field theories. These are theories that possess a large, oftentimes infinite, set of symmetries that together allow the use of powerful exact methods for their solution. In turn this elevated degree of control has led --- and is still leading --- to profound insights into many important physical concepts. As a result, the field of Integrable Quantum Field Theory (IQFT) sits at the cutting edge of theoretical high-energy physics. The research of Professor Lukyanov is aimed at the further development of IQFTs. A successful realization would allow one to overcome the notorious technical difficulties that emerge when traditional methods are applied to many of the theories that appear within the context of mainstream theoretical high-energy physics. This research is therefore in the national interest because it will have an important impact on the development and understanding of fundamental science within the United States. Professor Lukyanov also intends to involve graduate students and postdocs in his research and disseminate the results to the broader community of young scientists via lectures and workshops.More technically, this research is focused on integrable quantum spin chains and non-linear sigma models --- exactly the types of theories that emerge within the context of gauge/string duality. This research is organized into three projects. The first is the study of a certain multiparametric quantum spin chain at criticality. The key to the approach will be the so-called ODE/IQFT correspondence, a new mathematical tool of Integrable Quantum Field Theories ("IQFT" side) which allows one to "encode" its algebraic structures into a system of Ordinary Differential Equations ("ODE" side). This part is directly related to the gauge/string duality. The second project is devoted to the study of the high-energy asymptotics of scattering amplitudes in quantum chromodynamics. The ultimate objective of the third project is to extend the ODE/IQFT approach to integrable non-linear sigma models. The new methods considered in this project, when fully developed, should be of broad impact. The first and third projects could lead to entirely new directions and discoveries, including potential applications to the description of disordered systems in condensed-matter physics. By contrast, the second project has to do with a realistic problem in particle physics. All three projects provide excellent training opportunities for students in QFT and statistical mechanics, from undergraduate-level problems to advanced training in the most abstract analytic techniques.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.