Integrability and Conformal Symmetry in Four Dimensions

可积模型为研究支撑我们物理现实的理论框架提供了一个肥沃的舞台。它们在两个维度上有一个自然的家，并以更高维度系统的优雅极限为特色。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对应为四维量子场理论提供了场所，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，但也有其他例子，如量子色动力学中的高能散射。本提案的目的是加深对四个时空维可积场理论的基本原理的理解，并扩充相关的可积性工具箱，以便于显式计算。更明确地说，典型的ADS/CFT对偶属于可积模型的一类，其对称性为所谓的杨安。这个非局域量子群是著名的量子杨-巴克斯特方程的有理解的基础。它是在四维最大超对称杨-米尔斯理论或其在AdS5xS5背景上的对偶弦理论中的几个观察点上被确认的。虽然杨年可以自然地在两个维度上定义，但要理解它是如何在4D量子场论中实现的是一个挑战。此外，众所周知，在二维中，量子群对称性会对散射矩阵施加强大的约束。然而，对于上述四维的Yang-Mills理论，类似的约束条件还远远没有被理解。正如我们最近发现的那样，ADS/CFT系统具有进一步的非局域对称性，它充当了所谓的谱参数的生成器。具有这一性质的对称称为主对称，出现在二维的许多可积模型中。然而，了解它们在ADS/CFT通信中的作用才刚刚开始。一般而言，谱参数对可积模型起着至关重要的作用。当研究ADS/CFT的可积性时，得到了强大的工具和新的见解，但在各种与这种对偶性没有直接关系的情况下，发现了四维可积性。例如QCD中的散射过程的Regge极限或某些超对称性较小的场论。后一种情况显示了与ADS/CFT系统相似的结构，但其研究的一般框架尚不清楚。该提议涉及以下几点：1)确定ADS/CFT的新的主对称性的作用；2)将杨安识别为各种可观测的可积工具；3)在四维场论中建立可积结构的统一图景。

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.