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.

23 pages

References in corpus (17)

• Green-Schwarz action for Type IIA strings on $Ad{S}_4\times C{P}^3$
• Abelian Yang-Baxter Deformations and TsT transformations
• Lunin-Maldacena backgrounds from the classical Yang-Baxter equation -- Towards the gravity/CYBE correspondence
• Yang-Baxter sigma models based on the CYBE
• Towards a two-parameter q-deformation of AdS_3 x S^3 x M^4 superstrings
• A Jordanian deformation of AdS space in type IIB supergravity
• Homogeneous Yang-Baxter deformations as non-abelian duals of the AdS_5 sigma-model
• Non-local charges, Zm gradings and coset space actions
• Integrable Deformations of Sigma Models
• Homogeneous Yang-Baxter deformations as generalized diffeomorphisms
• $T$-folds from Yang-Baxter deformations
• Yang-Baxter $\sigma$-model with WZNW term as $\mathcal{E}$-model
• Generalized type IIB supergravity equations and non-Abelian classical r-matrices
• Integrable properties of sigma-models with non-symmetric target spaces
• Yang-Baxter deformations of the Principal Chiral Model plus Wess-Zumino term
• Unimodular jordanian deformations of integrable superstrings
• Flag manifold sigma-models: the 1/N-expansion and the anomaly two-form

Cited by in corpus (1)

• Universal 1-loop divergences for integrable sigma models