Advertisement

Y-system, TBA and Quasi-Classical Strings in AdS 4 × CP3

  • Nikolay Gromov
  • Fedor Levkovich-Maslyuk
Open Access
Article

Abstract

We study the exact spectrum of the AdS4/CFT3 duality put forward by Aharony, Bergman, Jafferis and Maldacena (ABJM). We derive thermodynamic Bethe ansatz (TBA) equations for the planar ABJM theory, starting from “mirror” asymptotic Bethe equations which we conjecture. We also propose generalization of the TBA equations for excited states. The recently proposed Y-system is completely consistent with the TBA equations for a large subsector of the theory, but should be modified in general. We find the general asymptotic infinite length solution of the Y-system, and also several solutions to all wrapping orders in the strong coupling scaling limit. To make a comparison with results obtained from string theory, we assume that the all-loop Bethe ansatz of N.G. and P. Vieira is the valid worldsheet theory description in the asymptotic regime. In this case we find complete agreement, to all orders in wrappings, between the solution of our Y-system and generic quasi-classical string spectrum in AdS3 × S1.

Keywords

AdS-CFT Correspondence Integrable Field Theories 

References

  1. [1]
    J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231 [Int. J. Theor. Phys. 38 (1999) 1113] [hep-th/9711200] [SPIRES].MATHMathSciNetADSGoogle Scholar
  2. [2]
    O. Aharony, O. Bergman, D.L. Jafferis and J. Maldacena, N=6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals, JHEP 10 (2008) 091 [arXiv:0806.1218] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  3. [3]
    O. Aharony, O. Bergman and D.L. Jafferis, Fractional M2-branes, JHEP 11 (2008) 043 [arXiv:0807.4924] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  4. [4]
    J.A. Minahan and K. Zarembo, The Bethe-ansatz for N = 4 super Yang-Mills, JHEP 03 (2003) 013 [hep-th/0212208] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  5. [5]
    J.A. Minahan and K. Zarembo, The Bethe ansatz for superconformal Chern-Simons, JHEP 09 (2008) 040 [arXiv:0806.3951] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  6. [6]
    D. Gaiotto, S. Giombi and X. Yin, Spin chains in N = 6 superconformal Chern-Simons-matter theory, JHEP 04 (2009) 066 [arXiv:0806.4589] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  7. [7]
    D. Bak and S.-J. Rey, Integrable spin chain in superconformal Chern-Simons theory, JHEP 10 (2008) 053 [arXiv:0807.2063] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  8. [8]
    C. Kristjansen, M. Orselli and K. Zoubos, Non-planar ABJM theory and integrability, JHEP 03 (2009) 037 [arXiv:0811.2150] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  9. [9]
    B.I. Zwiebel, Two-loop integrability of planar N = 6 superconformal Chern-Simons theory, J. Phys. A 42 (2009) 495402 [arXiv:0901.0411] [SPIRES].MathSciNetGoogle Scholar
  10. [10]
    J.A. Minahan, W. Schulgin and K. Zarembo, Two loop integrability for Chern-Simons theories with N = 6 supersymmetry, JHEP 03 (2009) 057 [arXiv:0901.1142] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  11. [11]
    D. Bak, H. Min and S.-J. Rey, Generalized dynamical spin chain and 4-loop integrability in N = 6 superconformal Chern-Simons theory, Nucl. Phys. B 827 (2010) 381 [arXiv:0904.4677] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  12. [12]
    B. Chen and J.-B. Wu, Semi-classical strings in AdS 4CP 3, JHEP 09 (2008) 096 [arXiv:0807.0802] [SPIRES].ADSCrossRefGoogle Scholar
  13. [13]
    J.A. Minahan, O. Ohlsson Sax and C. Sieg, Magnon dispersion to four loops in the ABJM and ABJ models, arXiv:0908.2463 [SPIRES].
  14. [14]
    J.A. Minahan, O. Ohlsson Sax and C. Sieg, Anomalous dimensions at four loops in N = 6 superconformal Chern-Simons theories, arXiv:0912.3460 [SPIRES].
  15. [15]
    I. Bena, J. Polchinski and R. Roiban, Hidden symmetries of the AdS 5 × S 5 superstring, Phys. Rev. D 69 (2004) 046002 [hep-th/0305116] [SPIRES].MathSciNetADSGoogle Scholar
  16. [16]
    B. Stefanski, jr, Green-Schwarz action for type IIA strings on AdS 4 × CP 3, Nucl. Phys. B 808 (2009) 80 [arXiv:0806.4948] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  17. [17]
    G. Arutyunov and S. Frolov, Superstrings on AdS 4 × CP 3 as a coset σ-model, JHEP 09 (2008) 129 [arXiv:0806.4940] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  18. [18]
    J. Gomis, D. Sorokin and L. Wulff, The complete AdS 4 × CP 3 superspace for the type IIA superstring and D-branes, JHEP 03 (2009) 015 [arXiv:0811.1566] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  19. [19]
    D. Astolfi, V.G.M. Puletti, G. Grignani, T. Harmark and M. Orselli, Finite-size corrections in the SU(2) x SU(2) sector of type IIA string theory on AdS 4 xCP 3, Nucl. Phys. B 810 (2009) 150 [arXiv:0807.1527] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  20. [20]
    P. Sundin, The AdS 4 × CP 3 string and its Bethe equations in the near plane wave limit, JHEP 02 (2009) 046 [arXiv:0811.2775] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  21. [21]
    P. Sundin, On the worldsheet theory of the type IIA AdS 4 × CP 3 superstring, JHEP 04 (2010) 014 [arXiv:0909.0697] [SPIRES].CrossRefGoogle Scholar
  22. [22]
    K. Zarembo, Worldsheet spectrum in AdS 4/CFT 3 correspondence, JHEP 04 (2009) 135 [arXiv:0903.1747] [SPIRES].ADSCrossRefGoogle Scholar
  23. [23]
    C. Kalousios, C. Vergu and A. Volovich, Factorized tree-level scattering in AdS 4 xCP 3, JHEP 09 (2009) 049 [arXiv:0905.4702] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  24. [24]
    M. Staudacher, The factorized S-matrix of CFT/AdS, JHEP 05 (2005) 054 [hep-th/0412188] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  25. [25]
    N. Beisert, The SU(2|2) dynamic S-matrix, Adv. Theor. Math. Phys. 12 (2008) 945 [hep-th/0511082] [SPIRES].MathSciNetGoogle Scholar
  26. [26]
    N. Beisert, The analytic Bethe Ansatz for a chain with centrally extended SU(2—2) symmetry, J. Stat. Mech. 0701 (2007) P017 [nlin/0610017] [SPIRES].Google Scholar
  27. [27]
    R.A. Janik, The AdS 5 × S 5 superstring worldsheet S-matrix and crossing symmetry, Phys. Rev. D 73 (2006) 086006 [hep-th/0603038] [SPIRES].MathSciNetADSGoogle Scholar
  28. [28]
    N. Beisert, R. Hernandez and E. Lopez, A crossing-symmetric phase for AdS 5 × S 5 strings, JHEP 11 (2006) 070 [hep-th/0609044] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  29. [29]
    V.A. Kazakov, A. Marshakov, J.A. Minahan and K. Zarembo, Classical/quantum integrability in AdS/CFT, JHEP 05 (2004) 024 [hep-th/0402207] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  30. [30]
    V.A. Kazakov and K. Zarembo, Classical/quantum integrability in non-compact sector of AdS/CFT, JHEP 10 (2004) 060 [hep-th/0410105] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  31. [31]
    N. Beisert, V.A. Kazakov and K. Sakai, Algebraic curve for the SO(6) sector of AdS/CFT, Commun. Math. Phys. 263 (2006) 611 [hep-th/0410253] [SPIRES].MATHMathSciNetADSCrossRefGoogle Scholar
  32. [32]
    S. Schäfer-Nameki, The algebraic curve of 1-loop planar N = 4 SYM, Nucl. Phys. B 714 (2005) 3 [hep-th/0412254] [SPIRES].ADSCrossRefGoogle Scholar
  33. [33]
    N. Beisert, V.A. Kazakov, K. Sakai and K. Zarembo, The algebraic curve of classical superstrings on AdS 5 × S 5, Commun. Math. Phys. 263 (2006) 659 [hep-th/0502226] [SPIRES].MATHMathSciNetADSCrossRefGoogle Scholar
  34. [34]
    N. Beisert and M. Staudacher, Long-range PSU(2, 2|4) Bethe Ansaetze for gauge theory and strings, Nucl. Phys. B 727 (2005) 1 [hep-th/0504190] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  35. [35]
    N. Beisert, B. Eden and M. Staudacher, Transcendentality and crossing, J. Stat. Mech. (2007) P01021 [hep-th/0610251] [SPIRES].
  36. [36]
    N. Gromov and P. Vieira, The AdS4/CFT3 algebraic curve, JHEP 02 (2009) 040 [arXiv:0807.0437] [SPIRES]MathSciNetADSCrossRefGoogle Scholar
  37. [37]
    N. Gromov and P. Vieira, The all loop AdS4/CFT3 Bethe Ansatz, JHEP 01 (2009) 016 [arXiv:0807.0777] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  38. [38]
    C. Ahn and R.I. Nepomechie, N=6 super Chern-Simons theory S-matrix and all-loop Bethe ansatz equations, JHEP 09 (2008) 010 [arXiv:0807.1924] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  39. [39]
    A. Babichenko, B. Stefanski, Jr. and K. Zarembo, Integrability and the AdS 3/CFT 2 correspondence, JHEP 03 (2010) 058 [arXiv:0912.1723] [SPIRES].CrossRefGoogle Scholar
  40. [40]
    J. Ambjorn, R.A. Janik and C. Kristjansen, Wrapping interactions and a new source of corrections to the spin-chain/string duality, Nucl. Phys. B 736 (2006) 288 [hep-th/0510171] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  41. [41]
    R.A. Janik and T. Lukowski, Wrapping interactions at strong coupling – the giant magnon, Phys. Rev. D 76 (2007) 126008 [arXiv:0708.2208] [SPIRES].ADSGoogle Scholar
  42. [42]
    M.P. Heller, R.A. Janik and T. Lukowski, A new derivation of Lüscher F-term and fluctuations around the giant magnon, JHEP 06 (2008) 036 [arXiv:0801.4463] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  43. [43]
    Z. Bajnok and R.A. Janik, Four-loop perturbative Konishi from strings and finite size effects for multiparticle states, Nucl. Phys. B 807 (2009) 625 [arXiv:0807.0399] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  44. [44]
    C.-N. Yang and C.P. Yang, One-dimensional chain of anisotropic spin-spin interactions. I: proof of Bethe’s hypothesis for ground state in a finite system, Phys. Rev. 150 (1966) 321 [SPIRES].ADSCrossRefGoogle Scholar
  45. [45]
    A.B. Zamolodchikov, On the thermodynamic Bethe ansatz equations for reflectionless ADE scattering theories, Phys. Lett. B 253 (1991) 391 [SPIRES].MathSciNetADSGoogle Scholar
  46. [46]
    N. Dorey, Magnon bound states and the AdS/CFT correspondence, J. Phys. A 39 (2006) 13119 [hep-th/0604175] [SPIRES].MathSciNetGoogle Scholar
  47. [47]
    M. Takahashi, Thermodynamics of one-dimensional solvable models, Cambridge University Press, Cambridge U.K. (1999).CrossRefGoogle Scholar
  48. [48]
    F.H.L. Essler, H.Frahm, F.Göhmann, A. Klümper and V. Korepin, The one-dimensional Hubbard model, Cambridge University Press, Cambridge U.K. (2005).MATHCrossRefGoogle Scholar
  49. [49]
    V.V. Bazhanov, S.L. Lukyanov and A.B. Zamolodchikov, Quantum field theories in finite volume: Excited state energies, Nucl. Phys. B 489 (1997) 487 [hep-th/9607099] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  50. [50]
    P. Dorey and R. Tateo, Excited states by analytic continuation of TBA equations, Nucl. Phys. B 482 (1996) 639 [hep-th/9607167] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  51. [51]
    D. Fioravanti, A. Mariottini, E. Quattrini and F. Ravanini, Excited state Destri-De Vega equation for sine-Gordon and restricted sine-Gordon models, Phys. Lett. B 390 (1997) 243 [hep-th/9608091] [SPIRES].MathSciNetADSGoogle Scholar
  52. [52]
    A.G. Bytsko and J. Teschner, Quantization of models with non-compact quantum group symmetry: modular XXZ magnet and lattice sinh-Gordon model, J. Phys. A 39 (2006) 12927 [hep-th/0602093] [SPIRES].MathSciNetGoogle Scholar
  53. [53]
    N. Gromov, V. Kazakov and P. Vieira, Finite volume spectrum of 2D field theories from hirota dynamics, JHEP 12 (2009) 060 [arXiv:0812.5091] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  54. [54]
    N. Gromov, V. Kazakov and P. Vieira, Exact spectrum of anomalous dimensions of planar N = 4 supersymmetric Yang-Mills theory, Phys. Rev. Lett. 103 (2009) 131601 [arXiv:0901.3753] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  55. [55]
    F. Fiamberti, A. Santambrogio, C. Sieg and D. Zanon, Anomalous dimension with wrapping at four loops in N = 4 SYM, Nucl. Phys. B 805 (2008) 231 [arXiv:0806.2095] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  56. [56]
    V.N. Velizhanin, Leading transcedentality contributions to the four-loop universal anomalous dimension in N = 4 SYM, Phys. Lett. B 676 (2009) 112 [arXiv:0811.0607] [SPIRES].MathSciNetADSGoogle Scholar
  57. [57]
    G. Arutyunov and S. Frolov, On string S-matrix, bound states and TBA, JHEP 12 (2007) 024 [arXiv:0710.1568] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  58. [58]
    F. Fiamberti, A. Santambrogio and C. Sieg, Five-loop anomalous dimension at critical wrapping order in N = 4 SYM, JHEP 03 (2010) 103 [arXiv:0908.0234] [SPIRES].CrossRefGoogle Scholar
  59. [59]
    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] [SPIRES].MathSciNetGoogle Scholar
  60. [60]
    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] [SPIRES].MATHMathSciNetADSCrossRefGoogle Scholar
  61. [61]
    G. Arutyunov and S. Frolov, The dressing factor and crossing equations, J. Phys. A 42 (2009) 425401 [arXiv:0904.4575] [SPIRES].MathSciNetADSGoogle Scholar
  62. [62]
    G. Arutyunov and S. Frolov, Thermodynamic Bethe Ansatz for the AdS 5 × S 5 mirror model, JHEP 05 (2009) 068 [arXiv:0903.0141] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  63. [63]
    S. Frolov and R. Suzuki, Temperature quantization from the TBA equations, Phys. Lett. B 679 (2009) 60 [arXiv:0906.0499] [SPIRES].MathSciNetADSGoogle Scholar
  64. [64]
    G. Arutyunov and S. Frolov, String hypothesis for the AdS 5 × S 5 mirror, JHEP 03 (2009) 152 [arXiv:0901.1417] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  65. [65]
    N. Gromov, V. Kazakov and P. Vieira, Exact spectrum of planar \( \mathcal{N} = 4 \) supersymmetric Yang-Mills theory: Konishi dimension at any coupling, Phys. Rev. Lett. 104 (2010) 211601 [arXiv:0906.4240] [SPIRES].ADSCrossRefGoogle Scholar
  66. [66]
    K. Konishi, Anomalous supersymmetry transformation of some composite operators in SQCD, Phys. Lett. B 135 (1984) 439 [SPIRES].ADSGoogle Scholar
  67. [67]
    M. Bianchi, S. Kovacs, G. Rossi and Y.S. Stanev, Properties of the Konishi multiplet in N = 4 SYM theory, JHEP 05 (2001) 042 [hep-th/0104016] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  68. [68]
    B. Eden, C. Jarczak, E. Sokatchev and Y.S. Stanev, Operator mixing in N = 4 SYM: the Konishi anomaly revisited, Nucl. Phys. B 722 (2005) 119 [hep-th/0501077] [SPIRES].MathSciNetADSGoogle Scholar
  69. [69]
    S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from non-critical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [SPIRES].MathSciNetADSGoogle Scholar
  70. [70]
    N. Gromov, Y-system and quasi-classical strings, JHEP 01 (2010) 112 [arXiv:0910.3608] [SPIRES].CrossRefGoogle Scholar
  71. [71]
    G. Arutyunov and S. Frolov, Uniform light-cone gauge for strings in AdS 5 × S 5 : Solving SU(1|1) sector, JHEP 01 (2006) 055 [hep-th/0510208] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  72. [72]
    R. Roiban and A.A. Tseytlin, Quantum strings in AdS 5 × S 5 : strong-coupling corrections to dimension of Konishi operator, JHEP 11 (2009) 013 [arXiv:0906.4294] [SPIRES].ADSCrossRefGoogle Scholar
  73. [73]
    A.A. Tseytlin, Quantum strings in AdS5 x S5 and AdS/CFT duality, Int. J. Mod. Phys. A 25 (2010) 319 [arXiv:0907.3238] [SPIRES].ADSGoogle Scholar
  74. [74]
    D. Bombardelli and D. Fioravanti, Finite-Size Corrections of the \( \mathbb{C}{\mathbb{P}^3} \) Giant Magnons: the Lúscher terms, JHEP 07 (2009) 034 [arXiv:0810.0704] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  75. [75]
    I. Shenderovich, Giant magnons in AdS 4/CFT 3 : dispersion, quantization and finite–size corrections, arXiv:0807.2861 [SPIRES].
  76. [76]
    N. Berkovits, Super-Poincaré covariant quantization of the superstring, JHEP 04 (2000) 018 [hep-th/0001035] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  77. [77]
    N. Gromov, V. Kazakov and P. Vieira, Finite volume spectrum of 2D field theories from Hirota dynamics, JHEP 12 (2009) 060 [arXiv:0812.5091] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  78. [78]
    O. Bergman and S. Hirano, Anomalous radius shift in AdS 4/CFT 3, JHEP 07 (2009) 016 [arXiv:0902.1743] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  79. [79]
    L. Mazzucato and B.C. Vallilo, On the non-renormalization of the AdS Radius, JHEP 09 (2009) 056 [arXiv:0906.4572] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  80. [80]
    G. Bonelli, P.A. Grassi and H. Safaai, Exploring pure spinor string theory on \( Ad{S_4} \times \mathbb{C}{\mathbb{P}^3} \), JHEP 10 (2008) 085 [arXiv:0808.1051] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  81. [81]
    N. Beisert, J.A. Minahan, M. Staudacher and K. Zarembo, Stringing spins and spinning strings, JHEP 09 (2003) 010 [hep-th/0306139] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  82. [82]
    Z. Tsuboi, Analytic Bethe ansatz and functional equations for Lie superalgebra sl(r + 1|s + 1), J. Phys. A 30 (1997) 7975 [arXiv:0911.5387].MathSciNetADSGoogle Scholar
  83. [83]
    N. Beisert, The analytic Bethe ansatz for a chain with centrally extended su(2|2) symmetry, J. Stat. Mech. 0701 (2007) P017 [nlin/0610017] [SPIRES].Google Scholar
  84. [84]
    V. Kazakov, A.S. Sorin and A. Zabrodin, Supersymmetric Bethe ansatz and Baxter equations from discrete Hirota dynamics, Nucl. Phys. B 790 (2008) 345 [hep-th/0703147] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  85. [85]
    H. Saleur and B. Pozsgay, Scattering and duality in the 2 dimensional OSP(2—2) gross neveu and σ-models, JHEP 02 (2010) 008 [arXiv:0910.0637] [SPIRES].CrossRefGoogle Scholar
  86. [86]
    B. Vicedo, Finite-g Strings, arXiv:0810.3402 [SPIRES].
  87. [87]
    T. Lukowski and O.O. Sax, Finite size giant magnons in the SU(2) x SU(2) sector of AdS 4 xCP 3, JHEP 12 (2008) 073 [arXiv:0810.1246] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  88. [88]
    A.B. Zamolodchikov and A.B. Zamolodchikov, Factorized S-matrices in two dimensions as the exact solutions of certain relativistic quantum field models, Annals Phys. 120 (1979) 253 [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  89. [89]
    P. Dorey, Exact S matrices, hep-th/9810026 [SPIRES].
  90. [90]
    N. Beisert, C. Kristjansen and M. Staudacher, The dilatation operator of N = 4 super Yang-Mills theory, Nucl. Phys. B 664 (2003) 131 [hep-th/0303060] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  91. [91]
    N. Beisert and M. Staudacher, The N = 4 SYM integrable super spin chain, Nucl. Phys. B 670 (2003) 439 [hep-th/0307042] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  92. [92]
    N. Beisert, The SU(2|3) dynamic spin chain, Nucl. Phys. B 682 (2004) 487 [hep-th/0310252] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  93. [93]
    M. Staudacher, The factorized S-matrix of CFT/AdS, JHEP 05 (2005) 054 [hep-th/0412188] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  94. [94]
    N. Beisert, R. Hernandez and E. Lopez, A crossing-symmetric phase for AdS 5 × S 5 strings, JHEP 11 (2006) 070 [hep-th/0609044] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  95. [95]
    N. Beisert, B. Eden and M. Staudacher, Transcendentality and crossing, J. Stat. Mech. (2007) P01021 [hep-th/0610251] [SPIRES].
  96. [96]
    V.V. Bazhanov, S.L. Lukyanov and A.B. Zamolodchikov, Quantum field theories in finite volume: excited state energies, Nucl. Phys. B 489 (1997) 487 [hep-th/9607099] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  97. [97]
    P. Dorey and R. Tateo, Excited states by analytic continuation of TBA equations, Nucl. Phys. B 482 (1996) 639 [hep-th/9607167] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  98. [98]
    P. Dorey and R. Tateo, Excited states in some simple perturbed conformal field theories, Nucl. Phys. B 515 (1998) 575 [hep-th/9706140] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  99. [99]
    B.I. Zwiebel, Two-loop integrability of planar N = 6 superconformal Chern-Simons theory, J. Phys. A 42 (2009) 495402 [arXiv:0901.0411] [SPIRES].MathSciNetGoogle Scholar
  100. [100]
    J.A. Minahan, W. Schulgin and K. Zarembo, Two loop integrability for Chern-Simons theories with N = 6 supersymmetry, JHEP 03 (2009) 057 [arXiv:0901.1142] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  101. [101]
    C. Ahn and R.I. Nepomechie, Two-loop test of the N = 6 Chern-Simons theory S-matrix, JHEP 03 (2009) 144 [arXiv:0901.3334] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  102. [102]
    A.B. Zamolodchikov, Thermodynamic Bethe Ansatz in relativistic models. scaling three state potts and Lee-Yang models, Nucl. Phys. B 342 (1990) 695 [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  103. [103]
    G. Arutyunov, S. Frolov and R. Suzuki, Exploring the mirror TBA, JHEP 05 (2010) 031 [arXiv:0911.2224] [SPIRES].CrossRefGoogle Scholar
  104. [104]
    N. Gromov and V. Mikhaylov, Comment on the scaling function in AdS 4 × CP 3, JHEP 04 (2009) 083 [arXiv:0807.4897] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  105. [105]
    N. Gromov and P. Vieira, Complete 1-loop test of AdS/CFT, JHEP 04 (2008) 046 [arXiv:0709.3487] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  106. [106]
    T. McLoughlin and R. Roiban, Spinning strings at one-loop in AdS 4 xCP 3, JHEP 12 (2008) 101 [arXiv:0807.3965] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  107. [107]
    L.F. Alday, G. Arutyunov and D. Bykov, Semiclassical quantization of spinning strings in AdS 4 xCP 3, JHEP 11 (2008) 089 [arXiv:0807.4400] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  108. [108]
    C. Krishnan, AdS4/CFT3 at one loop, JHEP 09 (2008) 092 [arXiv:0807.4561] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  109. [109]
    T. McLoughlin, R. Roiban and A.A. Tseytlin, Quantum spinning strings in AdS 4 xCP 3 : testing the Bethe Ansatz proposal, JHEP 11 (2008) 069 [arXiv:0809.4038] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  110. [110]
    V. Mikhaylov, On the fermionic frequencies of circular strings, arXiv:1002.1831 [SPIRES].
  111. [111]
    S. Frolov and A.A. Tseytlin, Semiclassical quantization of rotating superstring in AdS 5 × S 5, JHEP 06 (2002) 007 [hep-th/0204226] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  112. [112]
    D. Bombardelli, D. Fioravanti and R. Tateo, TBA and Y-system for planar AdS 4/CFT 3, Nucl. Phys. B 834 (2010) 543 [arXiv:0912.4715] [SPIRES].ADSGoogle Scholar

Copyright information

© The Author(s) 2010

Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

Authors and Affiliations

  1. 1.Mathematics DepartmentKing’s College LondonLondonU.K.
  2. 2.DESY TheoryHamburgGermany
  3. 3.Institut für Theoretische Physik Universität HamburgHamburgGermany
  4. 4.St. Petersburg INPSt.PetersburgRussia
  5. 5.Physics DepartmentMoscow State UniversityMoscowRussia

Personalised recommendations