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

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 $s{l}_2$, the ${\mathbb{Z}}_2^2$-graded affine $\widehat{s{l}_2}$ algebra with two central extensions, the ${\mathbb{Z}}_2^2$-graded Virasoro algebra obtained from a Hamiltonian reduction.