Lax pairs for new $\mathbb{Z}_N$-symmetric coset $\sigma$-models and their Yang-Baxter deformations

Two-dimensional $\sigma$-models with $\mathbb{Z}_N$-symmetric homogeneous target spaces have been shown to be classically integrable when introducing WZ-terms in a particular way. This article continues the search for new models of this type now allowing some kinetic terms to be absent, analogously to the Green-Schwarz superstring $\sigma$-model on $\mathbb{Z}_4$-symmetric homogeneous spaces. A list of such integrable $\mathbb{Z}_N$-symmetric (super)coset $\sigma$-models for $N \leq 6$ and their Lax pairs is presented. For arbitrary $N$, a big class of integrable models is constructed that includes both the known pure spinor and Green-Schwarz superstring on $\mathbb{Z}_4$-symmetric cosets. Integrable Yang-Baxter deformations of this class of $\mathbb{Z}_N$-symmetric (super)coset $\sigma$-models can be constructed in same way as in the known $\mathbb{Z}_2$- or $\mathbb{Z}_4$-cases. Deformations based on solutions of the modified classical Yang-Baxter equation, the so-called $\eta$-deformation, require deformation of the constants defining the Lagrangian and the corresponding Lax pair. Homogeneous Yang-Baxter deformations (i.e. those based on solutions to the classical Yang-Baxter equation) leave the equations of motion and consequently the Lax pair invariant and are expected to be classically equivalent to the undeformed model. As an example, the relationship between $\mathbb{Z}_3$-symmetric homogeneous spaces and nearly (para-)K\"ahler geometries is revisited. Confirming existing literature it is shown that the integrable choice of WZ-term in the $\mathbb{Z}_3$-symmetric coset $\sigma$-model associated to a nearly K\"ahler background gives an imaginary contribution to the action.