A Twisted Affine Interpolation of the $\mathbb{Z}_2^2$-graded Integrable Super-Liouville and super-Sinh-Gordon Equations

This paper is a third one in a series devoted to the construction of $2D$, $\mathbb{Z}_2^2$-graded, integrable Toda field theories. The first paper introduced parabosonic models obtained from Rittenberg-Wyler color Lie algebras; the second paper explained how to deal with parafermions (the $\mathbb{Z}_2^2$-graded super-Liouville theory) from finite Rittenberg-Wyler color Lie superalgebras. This new paper extends the construction to the infinite case of (twisted) affine color Lie superalgebras. A $\mathbb{Z}_2^2$-graded extension of ${\widehat{\mathfrak{osp}(2|2)^{(2)}}}$ allows to interpolate the super-Liouville and the super-Sinh-Gordon models, recovered as special limits. The $\mathbb{Z}_2^2$-graded extensions of three frameworks are here discussed: the Polyakov's soldering, the Leznov-Savel'ev Lax pair (zero-curvature) formulation and the Bäcklund transformations. As a byproduct of the construction, the Hamiltonian reduction from the $\mathbb{Z}_2^2$-graded ${\widehat{\mathfrak{osp(2|2)}^{(2)}}}$ WZNW models induces a $\mathbb{Z}_2^2$-graded color Lie superalgebra which admits three inequivalent superVirasoro subalgebras; they are the ${\cal N}=2$ superVirasoro, the $(1,\frac{3}{2},\frac{3}{2},2)$ $\mathbb{Z}_2^2$-graded superVirasoro and the $(\frac{3}{2},\frac{3}{2},2,2)$ $\mathbb{Z}_2^2$-graded superVirasoro.