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

本文旨在探讨洛伦兹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界基本定律和力的核心。当20世纪60年代发现无限维Kac-Moody代数时，描述规范理论的工具集得到了扩大。仿射代数是前者的子类，在弦理论和共形场论的理解和描述中起着至关重要的作用。双曲型Kac-Moody代数和洛伦兹Kac-Moody代数是最近才发展起来的，人们已经确定，它们表征了弦理论的更现代版本，即m理论的对称性。特别地，特殊李代数的E10和En扩展起着核心作用45。对洛伦兹Kac-Moody代数的基础数学的理解是相当新颖的，部分仍然不完整。虽然有限维或仿射型的简单李代数已经被很好地研究和完全分类，但洛伦兹Kac-Moody代数仍在研究中。分类方案是基于对无限连通的Dynkin图或等价的它们的根系或Cartan矩阵的研究。一种特殊类型的Kac-Moody代数已经被详细研究过，通常被称为“双曲”代数。它们的Dynkin图以这样一种方式连接起来：删除任何一个节点，留下一组（可能是不相连的）连接的Dynkin图，每个Dynkin图都是有限型的，除了最多一个是仿射型的。双曲型Kac-Moody代数已被分类，具有不超过10个节点和一个洛伦兹的Cartan矩阵，即非奇异且恰好具有一个负特征值。与此同时，作为对上述理解的一部分，一维空间和一维时间的可积量子场论也正是利用这些有限维和无限维李代数的数学工具发展起来的。它们被用在形式因子自提法中，这种方法使人们能够在耦合常数的微扰理论中构造所有阶的散射矩阵。根据n粒子形式因子的后续扩展允许在耦合中以非微扰方式计算量子相关函数。以n粒子形式因子表示的膨胀已知收敛得非常快。到目前为止，还没有基于洛伦兹Kac-Moody al-gebras的这类理论。本提案的目的是填补这一空白，并研究它们的性质。这一过程的完成甚至部分完成不仅会扩大可积量子场论的范围，丰富对它们的理解，而且还有望为正在进行的m理论研究提供新的思路。即使扩展模型破坏了可积性，后者仍然成立。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论背景下开发的工具，即s矩阵自举方法和形状因子方法。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），主要是洛伦兹Kac-Moody代数。我将从基于后一种代数的经典模型的研究开始，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，但我必须熟悉一些量子可积版本的更高级的技术，特别是关于洛伦兹Kac-Moody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.