A deformation of quantum affine algebra in squashed WZNW models

We proceed to study infinite-dimensional symmetries in two-dimensional squashed Wess-Zumino-Novikov-Witten (WZNW) models at the classical level. The target space is given by squashed S^3 and the isometry is SU(2)_L x U(1)_R. It is known that SU(2)_L is enhanced to a couple of Yangians. We reveal here that an infinite-dimensional extension of U(1)_R is a deformation of quantum affine algebra, where a new deformation parameter is provided with the coefficient of the Wess-Zumino term. Then we consider the relation between the deformed quantum affine algebra and the pair of Yangians from the viewpoint of the left-right duality of monodromy matrices. The integrable structure is also discussed by computing the r/s-matrices that satisfy the extended classical Yang-Baxter equation. Finally two degenerate limits are discussed.

💡 Research Summary

The paper investigates the infinite‑dimensional symmetry structures of two‑dimensional squashed Wess‑Zumino‑Novikov‑Witten (WZNW) models at the classical level. The target space is a squashed three‑sphere S³ characterized by a squashing parameter C, and the isometry group is SU(2)ₗ × U(1)ᵣ. The authors first review the left‑hand (SU(2)ₗ) description, constructing improved flat currents jₗ^{± μ} that incorporate the deformation from the Wess‑Zumino term (parameter K = nλ²/(8π)). Using these currents they build two Lax pairs L_{L}^{±}(x;λ_{L}^{±}) and the associated monodromy matrices U_{L}^{±}(λ_{L}^{±}). Expanding the monodromy matrices around λ_{L}^{±}=∞ yields an infinite tower of conserved charges Q_{L}^{±}(n). The Poisson algebra of the first few charges reproduces the defining relations of a Yangian Y(su(2)ₗ), confirming that the left sector is enhanced from the finite SU(2)ₗ symmetry to a Yangian. The authors also compute the Maillet r‑ and s‑matrices, which depend separately on the spectral parameters λ and μ through a scalar function h_{L}^{C,K}(λ) =