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 $F_0$ is the bosonic diagonal subgroup of the product supergroup $\widehat{F}=\widehat{G} \times \widehat{G}$. They include the AdS$_3$ $\times$ 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.