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On the space of $2d$ integrable models

Lukas W. Lindwasser

TL;DR

The work develops a comprehensive Lie-algebraic framework for 2d integrable models by treating infinite-dimensional symmetry algebras generated by nonlinear, higher-derivative left-moving currents as organizing principles for integrable deformations. It systematically classifies commuting subalgebras for a single scalar, uncovering links to the KdV hierarchy and to known models such as sine-Gordon, Liouville, and Bullough-Dodd, while revealing new sequences of commuting charges. Upon quantization, the classical algebras deform, with Virasoro symmetry emerging for the polynomial sector and nonpolynomial generators obtaining challenging, often nonlocal quantum corrections, yet some first-order $\alpha'$ deformations retain a controllable structure. Collectively, the results map the landscape of potential integrable deformations and suggest pathways to extend the framework to fermionic and WZW theories, as well as to develop intrinsic, algebraic proofs of integrability beyond model-specific techniques.

Abstract

We study infinite dimensional Lie algebras, whose infinite dimensional mutually commuting subalgebras correspond with the symmetry algebra of $2d$ integrable models. These Lie algebras are defined by the set of infinitesimal, nonlinear, and higher derivative symmetry transformations present in theories with a left(right)-moving or (anti)-holomorphic current. We study a large class of such Lagrangian theories. We study the commuting subalgebras of the $2d$ free massless scalar, and find the symmetries of the known integrable models such as sine-Gordon, Liouville, Bullough-Dodd, and Korteweg-de Vries. Along the way, we find several new sequences of commuting charges, which we conjecture are charges of integrable models which are new deformations of a single scalar. After quantizing, the Lie algebra is deformed, and so are their commuting subalgebras.

On the space of $2d$ integrable models

TL;DR

The work develops a comprehensive Lie-algebraic framework for 2d integrable models by treating infinite-dimensional symmetry algebras generated by nonlinear, higher-derivative left-moving currents as organizing principles for integrable deformations. It systematically classifies commuting subalgebras for a single scalar, uncovering links to the KdV hierarchy and to known models such as sine-Gordon, Liouville, and Bullough-Dodd, while revealing new sequences of commuting charges. Upon quantization, the classical algebras deform, with Virasoro symmetry emerging for the polynomial sector and nonpolynomial generators obtaining challenging, often nonlocal quantum corrections, yet some first-order deformations retain a controllable structure. Collectively, the results map the landscape of potential integrable deformations and suggest pathways to extend the framework to fermionic and WZW theories, as well as to develop intrinsic, algebraic proofs of integrability beyond model-specific techniques.

Abstract

We study infinite dimensional Lie algebras, whose infinite dimensional mutually commuting subalgebras correspond with the symmetry algebra of integrable models. These Lie algebras are defined by the set of infinitesimal, nonlinear, and higher derivative symmetry transformations present in theories with a left(right)-moving or (anti)-holomorphic current. We study a large class of such Lagrangian theories. We study the commuting subalgebras of the free massless scalar, and find the symmetries of the known integrable models such as sine-Gordon, Liouville, Bullough-Dodd, and Korteweg-de Vries. Along the way, we find several new sequences of commuting charges, which we conjecture are charges of integrable models which are new deformations of a single scalar. After quantizing, the Lie algebra is deformed, and so are their commuting subalgebras.
Paper Structure (31 sections, 167 equations, 2 figures)

This paper contains 31 sections, 167 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Conventions
  3. Outline
  4. Preliminaries
  5. $2d$ free massless scalars
  6. Lie algebra
  7. Subalgebras
  8. Integrable deformations of a single scalar
  9. $[\ell_0,\ell_0']=0$
  10. $[\ell_0,\ell_n]=0$
  11. $[\ell_1,\ell_2]=0$
  12. The case $\frac{\partial^3\ell_1}{\partial j_1^3}=0$
  13. The case $\frac{\partial^3\ell_1}{\partial j_1^3}\neq 0$
  14. $[\ell_2,\ell_3]=0$
  15. The case $\frac{\partial^3\ell_2}{\partial j_2^3}=0$
  16. ...and 16 more sections

Figures (2)

  • Figure 1: Wave profile of the single wave solution (\ref{['eq:j=0wavesoln']}) to the equation (\ref{['eq:j=0KdV']}).
  • Figure 2: Wave profile of the single wave solution (\ref{['eq:j=-1wavesoln']}) to the equation (\ref{['eq:j=-1KdV']}) with $\sqrt{c/\lambda}=1$.