Integrable $\mathbb{Z}_2^2$-graded Extensions of the Liouville and Sinh-Gordon Theories

TL;DR

This paper develops a framework for constructing integrable extensions of classical 2D Toda and affine Toda theories using $Z_2^2$-graded color Lie algebras, leading to new models like the extended Liouville and Sinh-Gordon theories.

Contribution

It introduces a novel $Z_2^2$-graded covariant extension of the Lax pair formalism and applies it to define new integrable models based on graded color Lie algebras.

Findings

Defined $Z_2^2$-graded extended Liouville and Sinh-Gordon models.

Constructed these models using $Z_2^2$-graded affine ${ m sl}_2$ and Virasoro algebras.

Demonstrated the models' integrability through the extended Lax pair formalism.

Abstract

In this paper we present a general framework to construct integrable $\mathbb{Z}_2^2$-graded extensions of classical, two-dimensional Toda and conformal affine Toda theories. The scheme is applied to define the extended Liouville and Sinh-Gordon models; they are based on $\mathbb{Z}_2^2$-graded color Lie algebras and their fields satisfy a parabosonic statististics. The mathematical tools here introduced are the $\mathbb{Z}_2^2$-graded covariant extensions of the Lax pair formalism and of the Polyakov's soldering procedure. The $\mathbb{Z}_2^2$-graded Sinh-Gordon model is derived from an affine $\mathbb{Z}_2^2$-graded color Lie algebra, mimicking a procedure originally introduced by Babelon-Bonora to derive the ordinary Sinh-Gordon model. The color Lie algebras under considerations are: the $6$-generator $\mathbb{Z}_2^2$-graded $sl_2$, the $\mathbb{Z}_2^2$-graded affine ${\widehat{sl_2}}$ algebra with two central extensions, the $\mathbb{Z}_2^2$-graded Virasoro algebra obtained from a Hamiltonian reduction.