The Classical r-matrix of AdS/CFT and its Lie Bialgebra Structure

Abstract

In this paper we investigate the algebraic structure of AdS/CFT in the strong-coupling limit. We propose an expression for the classical r-matrix with (deformed) u(2|2) symmetry, which leads to a quasi-triangular Lie bialgebra as the underlying symmetry algebra. On the fundamental representation our r-matrix coincides with the classical limit of the quantum R-matrix.

31 pages, v2: added comment on classical double structure in 4.5. new section 5 on relation to other algebras (from old appendix and new results). fixed typos and mathematical inaccuracies, added references, v3: improved mathematical presentation, to appear in CMP

References in corpus (51)

• Strings in flat space and pp waves from $\mathcal{N}=4$ Super Yang Mills
• The Bethe-Ansatz for N=4 Super Yang-Mills
• Hidden Symmetries of the AdS_5 x S^5 Superstring
• Transcendentality and Crossing
• The N=4 SYM Integrable Super Spin Chain
• The Dilatation Operator of Conformal N=4 Super Yang-Mills Theory
• Long-Range PSU(2,2|4) Bethe Ansaetze for Gauge Theory and Strings
• Bethe Ansatz for Quantum Strings
• Classical/quantum integrability in AdS/CFT
• Giant Magnons
• Multi-spin string solutions in AdS_5 x S^5
• The Factorized S-Matrix of CFT/AdS
• The AdS_5xS^5 superstring worldsheet S-matrix and crossing symmetry
• The Dilatation Operator of N=4 Super Yang-Mills Theory and Integrability
• A Crossing-Symmetric Phase for AdS_5 x S^5 Strings
• Spin chains and string theory
• Stringing Spins and Spinning Strings
• The Algebraic Curve of Classical Superstrings on AdS_5xS^5
• The Zamolodchikov-Faddeev Algebra for AdS_5 x S^5 Superstring
• Planar N=4 gauge theory and the Inozemtsev long range spin chain
• Quantum corrections to the string Bethe ansatz
• Quantizing String Theory in AdS_5 X S^5: Beyond the pp-Wave
• A Relation Between Approaches to Integrability in Superconformal Yang-Mills Theory
• The Off-shell Symmetry Algebra of the Light-cone AdS_5 x S^5 Superstring
• On AdS_5 x S^5 String S-matrix
• On Quantum Corrections to Spinning Strings and Bethe Equations
• Quantum Deformations of the One-Dimensional Hubbard Model
• Worldsheet Scattering in AdS_5 x S^5
• The AdS5xS5 Superstring in Light-Cone Gauge and its Bethe Equations
• On the Hopf algebra structure of the AdS/CFT S-matrix
• Connecting giant magnons to the pp-wave: An interpolating limit of $Ad{S}_5\times S^5$
• Strings in the near plane wave background and AdS/CFT
• The Bethe ansatz approach for factorizable centrally extended S-matrices
• The magnon kinematics of the AdS/CFT correspondence
• The AdS(5)xS(5) superstring quantum spectrum from the algebraic curve
• World-sheet scattering in AdS_5 x S^5 at two loops
• Classical r-matrix of the su(2|2) SYM spin-chain
• A Secret Symmetry of the AdS/CFT S-matrix
• Yangian Symmetries of Matrix Models and Spin Chains: The Dilatation Operator of $\mathcal{N}$$=4$ SYM
• A Yangian Double for the AdS/CFT Classical r-matrix
• The Asymptotic Spectrum of the N=4 Super Yang-Mills Spin Chain
• On Hidden Symmetries of a Super Gauge Theory and Twistor String Theory
• Magnon Bound-state Scattering in Gauge and String Theory
• On Symmetry Enhancement in the psu(1,1|2) Sector of N=4 SYM
• Yangian Symmetry in D=4 Superconformal Yang-Mills Theory
• q-Deformed Supersymmetry and Dynamic Magnon Representations
• Drinfeld second realization of the quantum affine superalgebras of $D^{(1)}(2,1;x)$ via the Weyl groupoid
• On the Scattering Phase for AdS_5 x S^5 Strings
• Quantum deformed magnon kinematics
• Hidden Symmetries and Integrable Hierarchy of the N=4 Supersymmetric Yang-Mills Equations
• A central extension of U_q sl(2,2)^{(1)} and R-matrices with a new parameter

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• Quantum Deformations of the One-Dimensional Hubbard Model
• The S-matrix of String Bound States
• One-Loop Superconformal and Yangian Symmetries of Scattering Amplitudes in N=4 Super Yang-Mills
• The Bound State S-Matrix for AdS5 x S5 Superstring
• On Drinfeld's second realization of the AdS/CFT su(2|2) Yangian
• The classical R-matrix of AdS/CFT and its Lie dialgebra structure
• Hexagonalization of Correlation Functions II: Two-Particle Contributions
• The Classical Exchange Algebra of AdS5 x S5 String Theory
• Yangians, S-matrices and AdS/CFT
• A Quantum Affine Algebra for the Deformed Hubbard Chain
• Algebra of transfer-matrices and Yang-Baxter equations on the string worldsheet in AdS(5) x S(5)
• Integrable S-matrices, massive and massless modes and the AdS_2 x S^2 superstring
• Secret Symmetries of Type IIB Superstring Theory on AdS3 x S3 x M4
• Bonus Yangian Symmetry for the Planar S-Matrix of N=4 Super Yang-Mills
• The Classical Trigonometric r-Matrix for the Quantum-Deformed Hubbard Chain
• From Scattering Amplitudes to the Dilatation Generator in N=4 SYM
• An Exceptional Algebraic Origin of the AdS/CFT Yangian Symmetry
• Yangian Symmetry of String Theory on $Ad{S}_3\times S^3\times S^3\times S^1$ with Mixed 3-form Flux
• The RTT-Realization for the Deformed gl(2|2) Yangian
• Quantum Symmetries and Marginal Deformations
• The S-matrix algebra of the AdS2 X S2 superstring
• Free Fermions, vertex Hamiltonians, and lower-dimensional AdS/CFT
• Secret Symmetries in AdS/CFT
• Weakly coupled N=4 Super Yang-Mills and N=6 Chern-Simons theories from u(2|2) Yangian symmetry
• The su(2|3) Undynamic Spin Chain
• Universal blocks of the AdS/CFT Scattering Matrix
• Serre Relation and Higher Grade Generators of the AdS/CFT Yangian Symmetry
• Maximally extended sl(2|2), q-deformed d(2,1;epsilon) and 3D kappa-Poincaré
• Review of AdS/CFT Integrability, Chapter II.2: Quantum Strings in AdS5xS5
• Structure of the string R-matrix
• The Bethe Ansatz for AdS5 x S5 Bound States
• Classification of four and six dimensional Drinfel'd superdoubles
• Maximally extended sl(2|2) as a quantum double
• The gl(1|1) Lie superbialgebras
• $q$-Poincaré supersymmetry in $Ad{S}_5/CF{T}_4$
• Yangians in Integrable Field Theories, Spin Chains and Gauge-String Dualities
• Review of AdS/CFT Integrability, Chapter VI.2: Yangian Algebra
• Fast spinning strings on $\eta$ deformed $Ad{S}_5\times S^5$
• Classical Exchange Algebra of the Superstring on S^5 with the AdS-time
• Boost generator in AdS_3 integrable superstrings for general braiding
• $q$-Poincaré invariance of the $Ad{S}_3/CF{T}_2$ $R$-matrix
• The secret symmetries of the AdS/CFT reflection matrices
• Classical r-matrices of two and three dimensional Lie super-bialgebras and their Poisson-Lie supergroups
• The Yangian of sl(n|m) and the universal R-matrix
• On $Ad{S}_2/CF{T}_1$ transfer matrices, Bethe ansatz and scale invariance
• Lie superbialgebra structures on the Lie superalgebra $({\mathcal{C}}^3+\mathcal{A})$ and deformation of related integrable Hamiltonian systems
• The Quantum Affine Origin of the AdS/CFT Secret Symmetry
• Classical Lie Bialgebras for AdS/CFT Integrability by Contraction and Reduction
• PSU(2,2|4) Exchange Algebra of N=4 Superconformal Multiplets
• Spin-chain with PSU(2|2)xU(1)^3 and Non-linear Sigma-model with D(2,1;gamma)
• Constrained WZWN Models on G/{S x U(1)^n} and Exchange Algebra of G-Primaries
• Killing scalar of non-linear sigma models on G/H realizing the classical exchange algebra
• On matrix realizations of the Lie superalgebra D(2, 1 ; α)
• An Algebraic Model for the su(2|2) Light-Cone String Field Theory