  In the approach recently proposed by K. Costello and M. Yamazaki, which is
based on a four-dimensional variant of Chern-Simons theory, we derive a simple
and unifying two-dimensional form for the action of many integrable
$\sigma$-models which are known to admit descriptions as affine Gaudin models.
This includes both the Yang-Baxter deformation and the $\lambda$-deformation of
the principal chiral model. We also give an interpretation of Poisson-Lie
$T$-duality in this setting and derive the action of the $\mathsf{E}$-model.
