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.