Journal of High Energy Physics

, 2011:69 | Cite as

Revisiting the Y = 0 open spin chain at one loop



In 2005, Berenstein and Vázquez determined an open spin chain Hamiltonian describing the one-loop anomalous dimensions of determinant-like operators corresponding to open strings attached to Y = 0 maximal giant gravitons. We construct the transfer ma- trix (generating functional of conserved quantities) containing this Hamiltonian, thereby directly proving its integrability. We find the eigenvalues of this transfer matrix and the corresponding Bethe equations, which we compare with proposed all-loop Bethe equations. We note that the Bethe ansatz solution has a certain “gauge” freedom, and is not com- pletely unique.


Lattice Integrable Models Bethe Ansatz AdS-CFT Correspondence Boundary Quantum Field Theory 


  1. [1]
    N. Beisert et al., Review of AdS/CFT Integrability: An Overview, arXiv:1012.3982 [INSPIRE].
  2. [2]
    J. Minahan and K. Zarembo, The Bethe ansatz for N = 4 super Yang-Mills, JHEP 03 (2003) 013 [hep-th/0212208] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  3. [3]
    C. Sieg, Review of AdS/CFT Integrability, Chapter I.2: The spectrum from perturbative gauge theory, arXiv:1012.3984 [INSPIRE].
  4. [4]
    R.A. Janik, Review of AdS/CFT Integrability, Chapter III.5: Lüscher Corrections, arXiv:1012.3994 [INSPIRE].
  5. [5]
    Z. Bajnok, Review of AdS/CFT Integrability, Chapter III.6: Thermodynamic Bethe Ansatz, arXiv:1012.3995 [INSPIRE].
  6. [6]
    D. Bombardelli, D. Fioravanti and R. Tateo, Thermodynamic Bethe Ansatz for planar AdS/CFT: A Proposal, J. Phys. A 42 (2009) 375401 [arXiv:0902.3930] [INSPIRE].MathSciNetGoogle Scholar
  7. [7]
    N. Gromov, V. Kazakov, A. Kozak and P. Vieira, Exact Spectrum of Anomalous Dimensions of Planar N = 4 Supersymmetric Yang-Mills Theory: TBA and excited states, Lett. Math. Phys. 91 (2010) 265 [arXiv:0902.4458] [INSPIRE].MathSciNetADSMATHCrossRefGoogle Scholar
  8. [8]
    G. Arutyunov and S. Frolov, Thermodynamic Bethe Ansatz for the AdS 5× S 5 Mirror Model, JHEP 05 (2009) 068 [arXiv:0903.0141] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  9. [9]
    K. Zoubos, Review of AdS/CFT Integrability, Chapter IV.2: Deformations, Orbifolds and Open Boundaries, arXiv:1012.3998 [INSPIRE].
  10. [10]
    D.M. Hofman and J.M. Maldacena, Reflecting magnons, JHEP 11 (2007) 063 [arXiv:0708.2272] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  11. [11]
    D. Berenstein and S.E. Vazquez, Integrable open spin chains from giant gravitons, JHEP 06 (2005) 059 [hep-th/0501078] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  12. [12]
    I. Cherednik, Factorizing Particles on a Half Line and Root Systems, Theor. Math. Phys. 61 (1984) 977 [INSPIRE].MathSciNetMATHCrossRefGoogle Scholar
  13. [13]
    E. Sklyanin, Boundary Conditions for Integrable Quantum Systems, J. Phys. A 21 (1988) 2375 [INSPIRE].MathSciNetADSGoogle Scholar
  14. [14]
    S. Ghoshal and A.B. Zamolodchikov, Boundary S matrix and boundary state in two-dimensional integrable quantum field theory, Int. J. Mod. Phys. A 9 (1994) 3841 [hep-th/9306002] [INSPIRE].MathSciNetADSGoogle Scholar
  15. [15]
    A. Agarwal, Open spin chains in super Yang-Mills at higher loops: Some potential problems with integrability, JHEP 08 (2006) 027 [hep-th/0603067] [INSPIRE].ADSCrossRefGoogle Scholar
  16. [16]
    K. Okamura and K. Yoshida, Higher Loop Bethe Ansatz for Open Spin-Chains in AdS/CFT, JHEP 09 (2006) 081 [hep-th/0604100] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  17. [17]
    C. Ahn and R.I. Nepomechie, The Zamolodchikov-Faddeev algebra for open strings attached to giant gravitons, JHEP 05 (2008) 059 [arXiv:0804.4036] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  18. [18]
    H.-Y. Chen and D.H. Correa, Comments on the Boundary Scattering Phase, JHEP 02 (2008) 028 [arXiv:0712.1361] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  19. [19]
    W. Galleas, The Bethe Ansatz Equations for Reflecting Magnons, Nucl. Phys. B 820 (2009) 664 [arXiv:0902.1681] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  20. [20]
    N. Beisert, The S-matrix of AdS/CFT and Yangian symmetry, PoS(Solvay)002 [arXiv:0704.0400] [INSPIRE].
  21. [21]
    C. Ahn and R.I. Nepomechie, Yangian symmetry and bound states in AdS/CFT boundary scattering, JHEP 05 (2010) 016 [arXiv:1003.3361] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  22. [22]
    N. MacKay and V. Regelskis, Yangian symmetry of the Y = 0 maximal giant graviton, JHEP 12 (2010) 076 [arXiv:1010.3761] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  23. [23]
    L. Palla, Yangian symmetry of boundary scattering in AdS/CFT and the explicit form of bound state reflection matrices, JHEP 03 (2011) 110 [arXiv:1102.0122] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  24. [24]
    D. Correa and C. Young, Finite size corrections for open strings/open chains in planar AdS/CFT, JHEP 08 (2009) 097 [arXiv:0905.1700] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  25. [25]
    Z. Bajnok and L. Palla, Boundary finite size corrections for multiparticle states and planar AdS/CFT, JHEP 01 (2011) 011 [arXiv:1010.5617] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  26. [26]
    N. Mann and S.E. Vazquez, Classical Open String Integrability, JHEP 04 (2007) 065 [hep-th/0612038] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  27. [27]
    A. Dekel and Y. Oz, Integrability of Green-Schwarz σ-models with Boundaries, JHEP 08 (2011) 004 [arXiv:1106.3446] [INSPIRE].ADSCrossRefGoogle Scholar
  28. [28]
    R.I. Nepomechie, Bethe ansatz equations for open spin chains from giant gravitons, JHEP 05 (2009) 100 [arXiv:0903.1646] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  29. [29]
    R.I. Nepomechie, Nested algebraic Bethe ansatz for open GL(N ) spin chains with projected K-matrices, Nucl. Phys. B 831 (2010) 429 [arXiv:0911.5494] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  30. [30]
    A.B. Zamolodchikov and A.B. Zamolodchikov, Relativistic Factorized S Matrix in Two-Dimensions Having O(N ) Isotopic Symmetry, Nucl. Phys. B 133 (1978) 525 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  31. [31]
    A. Lima-Santos and R. Malara, C n1) , D n1) and A (2n−1) reflection K-matrices, Nucl. Phys. B 675 (2003) 661 [nlin/0307046] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  32. [32]
    E. Abdalla and A. Lima-Santos, Integrable models: From dynamical solutions to string theory, Braz. J. Phys. (2008) [arXiv:0808.3919] [INSPIRE].
  33. [33]
    H. Frahm and N.A. Slavnov, New solutions to the reflection equation and the projecting method, J. Phys. A 32 (1999) 1547 [arXiv:cond-mat/9810312]MathSciNetADSGoogle Scholar
  34. [34]
    N. Beisert, V. Kazakov, K. Sakai and K. Zarembo, Complete spectrum of long operators in N =4 SYM at one loop, JHEP 07(2005)030 [hep-th/0503200] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  35. [35]
    H.J. de Vega and M. Karowski, Exact bethe ansatz solution of 0(2N ) symmetric theories, Nucl. Phys. B 280 (1987) 225.ADSCrossRefGoogle Scholar
  36. [36]
    M. Martins and P. Ramos, The Algebraic Bethe ansatz for rational braid - monoid lattice models, Nucl. Phys. B 500 (1997) 579 [hep-th/9703023] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar

Copyright information

© SISSA, Trieste, Italy 2011

Authors and Affiliations

  1. 1.Physics DepartmentUniversity of MiamiCoral GablesUSA

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