Higher-point functions and integrability

在四维量子场论中，计算通常只能通过耦合常数的微扰理论来完成。在量子电动力学中，这就是电荷。它是一个非常小的参数，因此微扰展开的主导项产生实验数据的良好近似。在其他的物理模型中，例如低能的量子色动力学（QCD），耦合并不小，因此级数展开几乎没有帮助。另一方面，许多具有耦合常数的二维系统是可积的，即可以在一般耦合下精确求解。可积系统的理论在数学上是非常丰富的。然而，直到最近才被证明很难将相关的概念引入到四维物理学中，这个议程的目的是在四维QFT中利用可积性发展非微扰方法。我们提出研究四维（N=4 SYM）的最大超对称规范理论。虽然这个模型本身与唯象学无关，但它与物理理论有一些共同的特征。例如，模型的微扰理论再现了QCD的一部分。特别地，在QCD中高能粒子散射的几率幅中最复杂的部分可以从N=4模型中得到，由于它的高度对称性，N=4 SYM具有许多有趣的性质。最显著的是，在强耦合极限下，它可以用特定弯曲空间中的弦理论来描述。这就是著名的AdS/CFT对偶猜想，通常被称为“AdS/CFT对应”。这个规范/弦理论系统的谱问题，即弦能级的计算，或者等价地，场论中的反常维数的计算，是由一个可积模型描述的，该模型在强耦合和弱耦合的相反区域之间有力地插值。该领域目前的工作关注的是控制更高点物体的可积结构，其中功能运动学依赖起着重要的作用。本建议的主要动机是发展的想法，平铺n点相关函数的基本补丁固定对称。我们的目标是在这个新的竞技场的成功的可积性为基础的方法，并最终再次在政权的强，甚至intermediate coupling.The项目提供了一个新的视角微扰场论。在可积系统领域也有可能产生副产品，因为所研究的数量和所采用的方法都是新的。

In quantum field theory (QFT) in four dimensions, calculations can often only be done by perturbation theory in the coupling constant. In the case of quantum electrodynamics, this is the electric charge. It is a very small parameter, and hence the leading terms of the perturbative expansion yield a good approximation of experimental data. In other physical models - e.g. quantum chromodynamics (QCD) at low energy - the coupling is not small, and thus the series expansion is hardly helpful. A non-perturbative treatment is needed.On the other hand, many two-dimensional systems possessing a coupling constant are integrable, i.e. can be solved exactly at generic coupling. The theory of integrable systems is mathematically very rich. However, until recently it has proven hard to import the relevant concepts into four-dimensional physics.This agenda is aimed at the development of non-perturbative methods in four-dimensional QFT utilising integrability. We propose studying the maximally supersymmetric gauge theory in four dimensions (N=4 SYM). Although this model is not itself phenomenologically relevant, it shares some generic features with physical theories. For instance, the perturbation theory of the model reproduces a part of that of QCD. In particular, the most complicated part of the probability amplitudes for high energy particle scattering in QCD may be obtained from the N=4 model.Due to its high symmetry, N=4 SYM has a number of interesting properties. Most prominently, in the strong coupling limit it is described by a string theory in a certain curved space. This is the famous AdS/CFT duality conjecture, frequently called ``AdS/CFT correspondence''. The spectral problem of this gauge/string theory system, i.e. the calculation of the string energy levels or, equivalently, of the anomalous dimensions in field theory, is described by an integrable model that powerfully interpolates between the opposite regimes of strong and weak coupling.Current work in the field is concerned with integrable structures controlling higher-point objects, where the functional kinematical dependence plays an important role. The main incentive of the present proposal is to develop the idea of tiling n-point correlation functions by elementary patches fixed by symmetry. We aim at the success of integrability-based methods in this new arena, and ultimately once more at the regime of strong and even intermediate coupling.The project offers a fresh perspective on perturbative field theory. Potentially there will be spin-offs also in the field of integrable systems, because the quantities studied and the methods employed are new.