Modave Lectures on Classical Integrability in 2d Field Theories

S. Driezen
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引用次数: 3

Abstract

These lecture notes are based on a blackboard course given at the XVII Modave Summer School in Mathematical Physics held from 13 -- 17 September 2021 in Brussels (Belgium), and aimed at Ph.D. students in High Energy Theoretical Physics. We start with introducing classical integrability in finite-dimensional systems to set the stage for our main purpose: introducing two-dimensional classical field theories which are integrable. We focus on their zero-curvature formulation through the so-called Lax connection, which ensures the existence of an infinite tower of conserved charges. We then move on to their Poisson bracket structure, known as the Sklyanin or Maillet structure, which ensures complete classical integrability. All the concepts that we encounter will be illustrated with the integrable Principal Chiral Model, which is the canonical sigma-model that appears (or its generalisations) on the worldsheet of many string backgrounds. Along the way we briefly comment on the properties of integrability at the quantum level.
二维场论中的经典可积性Modave讲座
这些讲义是根据2021年9月13日至17日在布鲁塞尔(比利时)举行的第十七届Modave数学物理暑期学校的黑板课程编写的,针对的是高能理论物理博士生。我们首先介绍有限维系统中的经典可积性,为我们的主要目的做准备:介绍二维经典场理论,它们是可积的。我们通过所谓的Lax连接来关注它们的零曲率公式,这确保了一个无限的守恒电荷塔的存在。然后我们转向他们的泊松支架结构,称为Sklyanin或Maillet结构,它确保了完全的经典可积性。我们遇到的所有概念都将用可积主手性模型来说明,这是在许多弦背景的世界表上出现(或其推广)的规范sigma模型。在此过程中,我们简要地评论了量子水平上可积性的性质。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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