Gauge/string theory duality and integrable systems

在四维量子场论（QFT）中，计算常常只能用微扰理论来进行。在电磁学的情况下，这是一个非常小的参数，使得微扰展开的主导项产生实验数据的良好近似。在其他的现实模型中--例如低能的量子色动力学（QCD）--耦合常数并不小，因此级数展开几乎没有帮助;需要非微扰处理。另一方面，许多二维系统是可积的，也就是说它们可以精确求解。可积系统的理论在数学上非常丰富，但迄今为止，将相关概念引入四维物理中已被证明是困难的。本项目旨在发展四维QFT的非微扰方法。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，但它具有物理理论的一些一般特征;例如，这个模型的微扰理论再现了QCD本身的一部分。例如，在QCD中粒子散射的概率幅的计算可以部分地从N=4模型中得到，而其余部分则是结构上更简单的部分。最突出的是，模型的强耦合极限是由弦理论在一定的弯曲空间中描述的。弦理论系统，最近已经构造了一个可积模型，它足够强大，可以在强耦合和弱耦合的相反区域之间插值。我们希望进一步发展这一图景，从而同时了解中强耦合下的QFT、弯曲空间上弦理论的量子化以及与四维物理学有关的可积系统。

In quantum field theory (QFT) in four dimensions calculations can often only be done by perturbation theory w.r.t the coupling constant. In the case of electromagnetism this is the electric charge which is a very small parameter so that the leading terms of the perturbative expansion yield a good approximation of experimental data. In other realistic models - e.g.quantum chromodynamics (QCD) at low energy - the coupling constant is not small and thus the series expansion is hardly helpful; a non-perturbative treatment would be needed.On the other hand, many two-dimensional systems are integrable, i.e. they can be solved exactly. The theory of integrable systems is mathematically very rich, but so far it has proven hard to import the relevant concepts into four-dimensional physics.This project is aimed at the development of non-perturbative methods in four-dimensional QFT. 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 of physical theories; e.g. the perturbation theory of the model reproduces a part of that of QCD itself. For instance, the calculation of probability amplitudes for particle scattering in QCD may be partially obtained from the N=4 model, while the remainder is the structurally simpler part.Due to its high symmetry, the N=4 SYM theory has a number of interesting properties; most prominently the strong coupling limit of the model is described by a string theory in a certain curved space.For an important set of quantities in this gauge/string theory system, an integrable model has recently been constructed which is powerful enough to interpolate between the opposite regimes of strong and weak coupling. We wish to further develop this picture, and thereby to learn simultaneously about QFT at intermediate and strong coupling, the quantisation of string theory on curved spaces, and integrable systems related to four-dimensional physics.