Workshop on Large-N at Swansea

1974年特胡夫特提出用SU（N）规范理论来近似QCD，其中色数N取为无穷大。特胡夫特表明，在这一限制的动态规范理论是控制平面图。这个想法被称为“特胡夫特大N极限”。特胡夫特还提出，在这一限制规范理论可能有一个描述方面的树水平弦理论。特胡夫特大氮极限导致了几个现象学上的成功，其中包括对茨威格规则和维滕-韦内齐亚诺公式的解释。在1998年，Maldacena证明了大N N=4 SYM在AdS 5 x S5空间上有一个IIB型超引力的对偶描述，在四维空间中给出了特胡夫特思想的具体实现。这种联系被称为AdS/CFT对偶。由于这种对偶性，通过对弱耦合对偶描述应用一组可控变形，一个全新的具有非平凡动力学的强耦合场论类可以用分析方法来处理。这个例子说明了可解场论及其性质的分类是多么强大的工具，通过允许探索新的大类模型，这些模型不能用标准的微扰技术来研究。自从Maldacena的最初工作以来，弦规范对偶性已经发展到包括与QCD和超越标准模型现象学直接相关的理论和描述凝聚态系统的理论的例子。此外，激烈的理论活动规范弦对偶刺激了一个社区的形成，致力于研究理论在大N极限与晶格技术，这提供了第一原理的结果，使我们能够连接理论在大N和模型在有限N.我们建议组织一个为期四天的研讨会在斯旺西关于大N规范理论的各个方面：正式的，唯象和晶格模拟。这次会议的重点是双重的。在形式方面，我们要强调的重要性，寻找非平凡的系统，可以完全解决和理解，在一个令人满意的和严格的方式。在应用方面，我们要总结的进展，该计划可能走向的动力系统表现出非平凡的功能，如共形不动点和相变的理解，无论是分析和数值晶格研究。特别注意也将给予唯象的影响，如QCD的研究，大N规范理论与通用表示的李代数的基本对称性和动力学电弱对称性破缺。

In 1974 't Hooft proposed to approximate QCD by an SU(N) gauge theory, where the number of colours, N, is taken to infinity. 't Hooft showed that in this limit the dynamics of the gauge theory is controlled by planar graphs. This idea is known as the ``'t Hooft large-N limit''. 't Hooft also proposed that in this limit the gauge theory may have a description in terms of a tree level string theory. The 't Hooft large-N limit has led to several phenomenological successes, among them an explanation of the Zweig rule and the Witten-Veneziano formula for the eta' mass. In 1998 Maldacena conjectured that large-N N=4 SYM has a dual description in terms of type IIB supergravity on the space AdS5 x S5 , giving a concrete realization of 't Hooft's idea in a four-dimensional setup. This connection is known as the AdS/CFT duality. Thanks to this duality, a completely new class of strongly coupled field theories with non-trivial dynamics became accessible to analytical methods, by applying a set of controllable deformations to the weakly-coupled dual description.This example illustrates how powerful a tool the classification of solvable field theories and of their properties is, by allowing to explore new large classes of models which cannot be studied with standard perturbative techniques. Since the original work by Maldacena, string-gauge duality has evolved to include examples of theories of immediate relevance for QCD and Beyond the Standard Model phenomenology and theories that describe condensed matter system. Besides, the intense theoretical activity on gauge-string duality has stimulated the formation of a community dedicated to studies of theories in the large-N limit with lattice techniques, which provide first-principle results enabling us to connect theories at large-N and models at finite N.We propose to organize a four day workshop at Swansea about various aspects of large-N gauge theories: formal, phenomenological and lattice simulations. The focus of this conference is twofold. On the formal side, we want to stress the importance of searching for non-trivial systems that can be fully solved and understood in a satisfactory and rigorous way. On the application side, we want to summarise the progress that this program made possible towards the understanding of dynamical systems exhibiting non-trivial features such as conformal fixed points and phase transitions, both analytically and with numerical lattice studies. Special attention will be given also to the phenomenological implications such as the study of QCD, of large-N gauge theories with generic representations of the Lie algebra of the underlying symmetries and of Dynamical Electro-Weak Symmetry Breaking.