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

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. Methodology
The 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.