Three-parameter integrable deformation of Z_4 permutation supercosets

A three-parameter integrable deformation of ${\mathbb{Z}}_4$ permutation supercosets is constructed. These supercosets are of the form $ {\widehat{F}} / {F_0}$where$F_0$isthebosonicdiagonalsubgroupoftheproductsupergroup$\widehat{F}=\widehat{G} \times \widehat{G}$\mathrm{.}TheyincludetheAdS$_3$ $\times$ S${}^3$ and AdS${}_3$ $\times$ S${}^3$ $\times$ S${}^3$ supercosets. This deformation encompasses both the bi-Yang-Baxter deformation of the semi-symmetric space $\sigma$-model on ${\mathbb{Z}}_4$ permutation supercosets and the mixed flux model. Truncating the action at the bosonic level, we show that one recovers the bi-Yang-Baxter deformation of the principal chiral model plus Wess-Zumino term.