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

Abstract

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 $N\le 6$ 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-)Kähler 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 Kähler background gives an imaginary contribution to the action.

arXiv comment23 pages

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