String Theory and Integrable Systems

TL;DR

This paper reviews key developments in integrable quantum field theories with infinite-dimensional symmetries, highlighting their mathematical structures and relevance across various physics disciplines.

Contribution

It provides an overview of the basic structures and recent achievements in integrable models with quantum group symmetries, emphasizing their applications.

Findings

Analysis of integrable systems of Kadomtsev-Petviashvili type

Hamiltonian approach to Lie-Poisson symmetries

Quantum field theory perspective on 2D relativistic models

Abstract

This is mainly a brief review of some key achievements in a `hot'' area of theoretical and mathematical physics. The principal aim is to outline the basic structures underlying {\em integrable} quantum field theory models with {\em infinite-dimensional} symmetry groups which display a radically new type of {\em quantum group} symmetries. Certain particular aspects are elaborated upon with some detail: integrable systems of Kadomtsev-Petviashvili type and their reductions appearing in matrix models of strings; Hamiltonian approach to Lie-Poisson symmetries; quantum field theory approach to two-dimensional relativistic integrable models with dynamically broken conformal invariance. All field-theoretic models in question are of primary relevance to diverse branches of physics ranging from nonlinear hydrodynamics to string theory of fundamental particle interactions at ultra-high energies.