Integrable Z₂²-graded Extensions of the Liouville and Sinh-Gordon Theories

In this paper we present a general framework to construct integrable Z₂²-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 Z₂²-graded color Lie algebras and their fields satisfy a parabosonic statististics. The mathematical tools here introduced are the Z₂²-graded covariant extensions of the Lax pair formalism and of the Polyakov's soldering procedure. The Z₂²-graded Sinh-Gordon model is derived from an affine Z₂²-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 Z₂²-graded sl₂, the Z₂²-graded affine sl₂ algebra with two central extensions, the Z₂²-graded Virasoro algebra obtained from a Hamiltonian reduction.