Deformed integrable $\sigma$-models, classical $R$-matrices and classical exchange algebra on Drinfel'd doubles

Abstract

We describe a unifying framework for the systematic construction of integrable deformations of integrable $\sigma$-models within the Hamiltonian formalism. It applies equally to both the `Yang-Baxter' type as well as `gauged WZW' type deformations which were considered recently in the literature. As a byproduct, these two families of integrable deformations are shown to be Poisson-Lie T-dual of one another.

33 pages, 4 figures

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Cited by in corpus (90)

• Scale invariance of the eta-deformed AdS5 x S5 superstring, T-duality and modified type II equations
• On integrable deformations of superstring sigma models related to AdS_n x S^n supercosets
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• Yang-Baxter deformations, AdS/CFT, and twist-noncommutative gauge theory
• Generalised integrable $\lambda$- and $\eta$-deformations and their relation
• $\eta$ and $\lambda$ deformations as $\mathcal{E}$-models
• Abelian Yang-Baxter Deformations and TsT transformations
• Doubled aspects of generalised dualities and integrable deformations
• Homogeneous Yang-Baxter deformations as non-abelian duals of the AdS_5 sigma-model
• Beta Function of k Deformed AdS_5 x S^5 String Theory
• Integrable Deformations of Sigma Models
• Poisson-Lie T-duals of the bi-Yang-Baxter models
• $T$-folds from Yang-Baxter deformations
• A unifying 2d action for integrable $\sigma$-models from 4d Chern-Simons theory
• Integrable flows between exact CFTs
• Integrable 2d sigma models: quantum corrections to geometry from RG flow
• Generalized IIB supergravity from exceptional field theory
• Quantum deformations of the flat space superstring
• Yang-Baxter Deformations Beyond Coset Spaces (a slick way to do TsT)
• Assembling integrable sigma-models as affine Gaudin models
• On jordanian deformations of AdS_5 and supergravity
• Yang-Baxter $\sigma$-model with WZNW term as $\mathcal{E}$-model
• Generalized type IIB supergravity equations and non-Abelian classical r-matrices
• Type II DFT solutions from Poisson-Lie T-duality/plurality
• $\lambda$-deformations of left-right asymmetric CFTs
• Integrable coupled sigma-models
• 4-dimensional Chern-Simons theory and integrable field theories
• S-Matrices and Quantum Group Symmetry of k-Deformed Sigma Models
• All-loop anomalous dimensions in integrable $\lambda$-deformed $\sigma$-models
• All-loop correlators of integrable $\lambda$-deformed $\sigma$-models
• Double and cyclic $\lambda$-deformations and their canonical equivalents
• Deformations of the Almheiri-Polchinski model
• Yang-Baxter deformations of the Principal Chiral Model plus Wess-Zumino term
• Type IIB supergravity solution for the T-dual of the eta-deformed AdS_5 x S^5 superstring
• Non-split and split deformations of AdS_5
• Novel all loop actions of interacting CFTs: Construction, integrability and RG flows
• Quantum deformations of D=4 Euclidean, Lorentz, Kleinian and quaternionic o^*(4) symmetries in unified o(4;C) setting
• Comments on $\eta$-deformed principal chiral model from 4D Chern-Simons theory
• Integrable $\mathcal{E}$-Models, 4d Chern-Simons Theory and Affine Gaudin Models. I. Lagrangian Aspects
• Poisson-Lie duals of the eta deformed symmetric space sigma model
• Spinning strings in the $\eta$-deformed Neumann-Rosochatius system
• RG flows of integrable $\sigma$-models and the twist function
• Supercoset construction of Yang-Baxter deformed AdS${}_5\times$S${}^5$ backgrounds
• On q-deformed symmetries as Poisson-Lie symmetries and application to Yang-Baxter type models
• Homogeneous Yang-Baxter deformations as undeformed yet twisted models
• Integrable Lambda Models And Chern-Simons Theories
• Yang-Baxter deformations and generalized supergravity -- A short summary
• Quantum Inverse Scattering and the Lambda Deformed Principal Chiral Model
• Holomorphic Chern-Simons theory and affine Gaudin models
• On Yang-Baxter models, twist operators, and boundary conditions
• $\lambda$-deformed $Ad{S}_5\times S^5$ superstring from 4D Chern-Simons theory
• Quantum Anisotropic Sigma and Lambda Models as Spin Chains
• Semiclassical spectrum of a Jordanian deformation of $Ad{S}_5\times S^5$
• Towards the gravity/CYBE correspondence beyond integrability -- Yang-Baxter deformations of $T^{1,1}$
• Strong integrability of $\lambda$-deformed models
• AdS solutions and $\lambda$-deformations
• O($D$,$D$)-covariant two-loop $\beta$-functions and Poisson-Lie T-duality
• Weyl anomaly and the $C$-function in $\lambda$-deformed CFTs
• Exploring The Lambda Model Of The Hybrid Superstring
• Yang-Baxter deformations of Minkowski spacetime
• Integrable Deformations from Twistor Space
• Exact results from the geometry of couplings and the effective action
• Generalised T-duality and Integrable Deformations
• Constrained affine Gaudin models and diagonal Yang-Baxter deformations
• Yang-Baxter deformations of $W_{2,4}\times T^{1,1}$ and the associated T-dual models
• All Jordanian deformations of the $Ad{S}_5\times S^5$ superstring
• Integrable deformation of ${\mathbb{CP}}^n$ and generalised Kaehler geometry
• Symmetric space $\lambda$-model exchange algebra from 4d holomorphic Chern-Simons theory
• $\lambda$-Deformation Of The $Ad{S}_5\times S^5$ Pure Spinor Superstring
• Field theory and $\lambda$-deformations: Expanding around the identity
• Novel integrable interpolations
• T-dualization of Gödel string cosmologies via Poisson-Lie T-duality approach
• Generalized quark-antiquark potentials from a $q$-deformed AdS${}_5\times$S${}^5$ background
• Asymptotics in an Asymptotic CFT
• Webs of integrable theories
• Basic quantizations of $D=4$ Euclidean, Lorentz, Kleinian and quaternionic ${\mathfrak{o}}^{\star }(4)$ symmetries
• Lax pairs on Yang-Baxter deformed backgrounds
• Integrability of the $\lambda$-deformation of the PCM with spectators
• $\lambda$-deformations in the upper-half plane
• Poisson-Lie T-duality defects and target space fusion
• Yang-Baxter deformations of the flat space string
• Type-II backgrounds from deformed coset CFTs
• Contact 4d Chern-Simons theory: Generalities
• Jordanian spin chains for twisted strings in $Ad{S}_5\times S^5$
• On exactly solvable Yang-Baxter models and enhanced symmetries
• Integrable models based on non-semi-simple groups and plane wave target spacetimes
• Scale without conformal invariance from integrable deformations of coset CFTs
• Kerr-Schild perturbations of coset CFTs as scale invariant integrable $\sigma$-models
• Dynamically restoring conformal invariance in (integrable) $\sigma$-models
• Equivariant Nijenhuis Lie Algebras: Extensions to Classical Lie-Theoretic Structures