Extensions of integrable quantum field theories based on Lorentzian Kac-Moody algebras

本文旨在研究洛伦兹Kac-Moody代数在可积量子场论中的作用。自20世纪30年代以来，有限维李代数一直是描述自然力和基本定律的核心。这个描述规范理论的工具集在1960年无限维Kac-Moody代数被发现时被扩大了。仿射代数是前者的子类，现在在理解和描述弦理论和共形场理论中起着重要的作用。双曲Kac-Moody代数和Lorentzian Kac-Moody代数是最近才发展起来的，它们刻画了对称性的一种更现代的弦论形式，即M-理论。特别地，例外李代数的E10和En扩张起着中心作用。关于Lorentzian Kac-Moody代数的基本数学理解是相当新颖的，并且部分仍然是不完整的。虽然有限维单李代数或仿射型单李代数已经得到了很好的研究和完全分类，但Lorentzian Kac-Moody代数仍在研究中。这种分类模式是基于对inite连通动态图或等价于它们的根系或Cartan矩阵的研究。一类已被详细研究过的Kac-Moody代数通常被称为“双曲”。他们的动态图以这样一种方式连接，即任何一个节点的删除都会留下一组(可能是不连通的)连通的动态图，其中每个图都是有限类型的，除了至多一个仿射类型。给出了双曲Kac-Moody代数的分类，它具有不超过10个结点和一个洛伦兹Cartan矩阵，即非奇异且恰好有一个负本征值。同时，作为上述理解的一部分，利用有限和无限维李代数的这些数学工具，已经精确地发展了一空间和一时间维的可积量子场理论。它们被用在形状因子Bootstrap方法中，使人们能够在耦合常数中构造微扰理论中的所有阶散射矩阵。随后n粒子形式因子的展开允许在耦合中以非微扰方式计算量子关联函数。已知n粒子形式因子的展开式收敛非常快。到目前为止，还没有基于Lorentzian Kac-Moody al-gebras的这类理论。这项提议的目的是填补这一空白，并研究它们的性质。这项工作的完成甚至部分完成，不仅将扩大可积量子场论的范围，丰富其理解，而且有望为正在进行的M理论研究带来新的曙光。即使扩展的模型破坏了可积性，后者也是成立的。方法本项目中使用的方法将部分来自标准量子力学，但特别是在可积量子场理论背景下开发的工具，即S矩阵自助法和形状因子法。数学工具将是有限维李代数、无限维李代数(特别是Kac-Moody代数)和主要是洛伦兹Kac-Moody代数。我将继续学习基于这些代数的经典模型，并使用在经典可积系统的背景下发展起来的技术。鉴于我的背景，我已经熟悉了量子场论的一般原理，但我必须熟悉它们的量子可积版本中的一些更高级的技术，特别是关于洛伦兹卡克-穆迪代数的数学。

This proposal aims to investigate the role played by Lorentzian Kac-Moody algebras in integrable quantum field theories. Finite dimensional Lie algebras are central in the de-scription of the fundamental laws and forces of nature since the 1930s. This tool set to describe gauge theories was enlarged when infinite dimensional Kac-Moody algebras were discovered in the 1960. Affine algebras, which are subclasses of the former, play now a vi-tal role in the understanding and description of string theory and conformal field theory1 . Hyperbolic Kac-Moody algebras and Lorentzian Kac-Moody algebras have only been de-veloped fairly recently2 and it is established that they characterize the symmetries a more modern versions of string theory, that is M-theory3 . In particular, the E10 and En exten-sions of the exceptional Lie algebras play a central role45 . The understanding of the underlying mathematics regarding Lorentzian Kac-Moody algebras is fairly novel and in parts still incomplete. While simple Lie algebras of finite dimensional or affine type are well studied and fully classified, Lorentzian Kac-Moody algebras are still under investigation. The classification scheme is based on the study offinite connected Dynkin diagrams or equivalently their root systems or Cartan matrices. A particular type of Kac-Moody algebras that has been studied in some detail are usually referred as 'hyperbolic'. Their Dynkin diagrams are connected in such a way such that deletion of any one node leaves a (possibly disconnected) set of connected Dynkin diagrams each of which is of finite type except for at most one of affine type. The hyperbolic Kac-Moody algebras have been classified, possess no more than ten nodes and a Cartan matrix that is Lorentzian, that is, nonsingular and endowed with exactly one negative eigenvalue. In parallel and as part of the understanding of the above, integrable quantum field theories in one space and one time dimensions have been developed using precisely these mathematical tools of finite and infinite dimensional Lie algebras. They were employed in the form factor bootstrap approach6 that enables one to construct scattering matrices7 t9 all orders in perturbation theory in the coupling constants. A subsequent expansion in terms of n-particle form factors allows to compute quantum correlation functions in non-perturbative fashion in the coupling. The expansion in terms of n-particle form factors is known to converge very rapidly. So far no theories of this type have been developed based on Lorentzian Kac-Moody al-gebras. The aim of this proposal is to fill this gap and study their properties. A completion or even a partial completion of this will not only enlarge the set of integrable quantum field theories and enrich their understanding, but it is also expected to shed new light on the ongoing investigations in M-theory. The latter will also hold even if the extended models turn out to break the integrability. MethodologyThe methods to be used in this project will be in part those from standard quantum mechanics, but especially the tool developed in the context of integrable quantum field theories, that is the S-matrix bootstrap method and the form factor approach. The math-ematical tools will be finite dimensional Lie algebras, infinite dimensional Lie algebras (in particular Kac-Moody algebra) and mainly Lorentzian Kac-Moody algebras. I will com-mence with the study of classical models based on these latter algebras and employ also techniques developed in the context of classical integrable systems. Given my background, I am already familiar with the general principals of quantum field theory, but I will have to familiarize myself with some of the more advanced techniques of their quantum integrable versions and especially with the mathematics around Lorentzian Kac-Moody algebras.