Integrability and Conformal Symmetry in Four Dimensions

四维可积性和共形对称性

• 批准号: 363895012
• 负责人: Dr. Florian Loebbert
• 金额: --
• 依托单位: Bethe Zentrum für Theoretische Physik (BCTP)
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2017
• 资助国家: 德国
• 起止时间: 2016-12-31 至 2022-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/363895012?language=en
• 关键词: Integrability; Conformal; Symmetry; Four; Dimensions

Integrable models furnish a fertile arena for studying the theoretical framework that underlies our physical reality. They have a natural home in two dimensions and feature in elegant limits of higher dimensional systems. Notably, integrable models have played a major role throughout the history of physics since Kepler. The planar AdS/CFT correspondence provides the venue for a four-dimensional quantum field theory that is believed to be completely integrable. While this case represents the most popular occurrence of integrability in four dimensional QFT, other examples exist, such as the high energy scattering in quantum chromodynamics. It is the aim of the present proposal to develop a solid understanding of the principles that underly integrable field theories in four spacetime dimensions and to enlarge the associated integrability toolbox in order to facilitate explicit calculations.To be more explicit, the prototypical AdS/CFT duality belongs to the class of integrable models whose symmetry is the so-called Yangian. This nonlocal quantum group underlies rational solutions to the famous quantum Yang-Baxter equation. It was identified on several observables within the maximally supersymmetric Yang-Mills theory in four dimensions or its dual string theory on the background AdS5 x S5. While the Yangian can naturally be defined in two dimensions, it is a challenge to understand how it is realized in a 4d quantum field theory. Moreover, in two dimensions quantum group symmetries are known to impose powerful constraints, e.g. on the scattering matrix. For the above Yang-Mills theory in four dimensions, however, similar constraints are far from being understood.As we recently discovered, the AdS/CFT system has a further nonlocal symmetry which acts as a generator of the so-called spectral parameter. Symmetries with this property are termed master symmetries and appear in many integrable models in 2d. Understanding their role for the AdS/CFT correspondence, however, is just at its beginning. Generically, the spectral parameter plays a crucial role for integrable models. It allows to package conservation laws in compact form and to make important statements using the theorems of complex analysis.While studying the integrability of AdS/CFT results in powerful tools and novel insights, four-dimensional integrability is found in various situations not directly related to this duality. Examples are the Regge limit of scattering processes in QCD or certain field theories with less supersymmetry. The latter cases show similar structures as the AdS/CFT system but a general framework for their study is not known.The following points are addressed by this proposal: 1) To establish the role of the novel master symmetry of AdS/CFT, 2) to identify the Yangian as an integrability tool on various observables and 3) to develop a unified picture of integrable structures in four-dimensional field theories.

可积模型为研究物理现实背后的理论框架提供了广阔的舞台。它们在二维空间中拥有自然的家园，并具有高维系统的优雅限制。值得注意的是，自开普勒以来，可积模型在整个物理学史上发挥了重要作用。平面 AdS/CFT 对应关系为四维量子场论提供了场所，该理论被认为是完全可积的。虽然这种情况代表了四维 QFT 中最常见的可积性，但还存在其他示例，例如量子色动力学中的高能量散射。本提案的目的是加深对四个时空维度可积场论的原理的理解，并扩大相关的可积性工具箱，以便于显式计算。更明确地说，原型 AdS/CFT 对偶性属于可积模型类别，其对称性是所谓的 Yangian。这个非局域量子群是著名的量子杨-巴克斯特方程有理解的基础。它是在四维最大超对称杨-米尔斯理论或其背景 AdS5 x S5 上的双弦理论中的几个可观测量上确定的。虽然杨量自然可以在二维中定义，但理解它如何在 4d 量子场论中实现是一个挑战。此外，在二维中，量子群对称性会施加强大的约束，例如在散射矩阵上。然而，对于上述四维杨-米尔斯理论，类似的约束还远未被理解。正如我们最近发现的，AdS/CFT 系统具有进一步的非局域对称性，它充当所谓谱参数的生成器。具有此属性的对称性称为主对称性，出现在许多二维可积模型中。然而，了解它们在 AdS/CFT 通信中的作用才刚刚开始。一般来说，光谱参数对于可积模型起着至关重要的作用。它允许以紧凑的形式封装守恒定律，并使用复分析定理做出重要的陈述。在研究 AdS/CFT 的可积性时，会产生强大的工具和新颖的见解，在与这种二元性不直接相关的各种情况下发现了四维可积性。例如 QCD 中散射过程的 Regge 极限或某些具有较少超对称性的场论。后一种情况显示了与 AdS/CFT 系统类似的结构，但其研究的一般框架尚不清楚。该提案解决了以下几点：1）建立 AdS/CFT 新颖的主对称性的作用，2）将 Yangian 确定为各种可观测量的可积性工具，3）开发四维场论中可积结构的统一图景。