On q-deformed symmetries as Poisson-Lie symmetries and application to Yang-Baxter type models

Abstract

Yang-Baxter type models are integrable deformations of integrable field theories, such as the principal chiral model on a Lie group $G$ or $\sigma$-models on (semi-)symmetric spaces $G/F$. The deformation has the effect of breaking the global $G$-symmetry of the original model, replacing the associated set of conserved charges by ones whose Poisson brackets are those of the $q$-deformed Poisson-Hopf algebra ${\mathcal{U}}_q(\mathfrak{g})$. Working at the Hamiltonian level, we show how this $q$-deformed Poisson algebra originates from a Poisson-Lie $G$-symmetry. The theory of Poisson-Lie groups and their actions on Poisson manifolds, in particular the formalism of the non-abelian moment map, is reviewed. For a coboundary Poisson-Lie group $G$, this non-abelian moment map must obey the Semenov-Tian-Shansky bracket on the dual group $G^{\ast }$, up to terms involving central quantities. When the latter vanish, we develop a general procedure linking this Poisson bracket to the defining relations of the Poisson-Hopf algebra ${\mathcal{U}}_q(\mathfrak{g})$, including the $q$-Poisson-Serre relations. We consider reality conditions leading to $q$ being either real or a phase. We determine the non-abelian moment map for Yang-Baxter type models. This enables to compute the corresponding action of $G$ on the fields parametrising the phase space of these models.

45 pages

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• Abelian Yang-Baxter Deformations and TsT transformations
• Assembling integrable sigma-models as affine Gaudin models
• Poisson-Lie duals of the eta deformed symmetric space sigma model
• Homogeneous Yang-Baxter deformations as undeformed yet twisted models
• Integrable supersymmetric deformations of $\mathrm{Ad}{\mathrm{S}}_3\times {\mathrm{S}}^3\times {\mathrm{T}}^4$
• Exploring The Lambda Model Of The Hybrid Superstring
• O($D$,$D$)-covariant two-loop $\beta$-functions and Poisson-Lie T-duality
• Probing analytical and numerical integrability: The curious case of $(Ad{S}_5\times S^5{)}_{\eta }$
• Constrained affine Gaudin models and diagonal Yang-Baxter deformations
• Quantum Groups as Global Symmetries
• Field theory aspects of $\eta$-deformed superstring background