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Review of AdS/CFT Integrability: An Overview

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This is the introductory chapter of a review collection on integrability in the context of the AdS/CFT correspondence. In the collection, we present an overview of the achievements and the status of this subject as of the year 2010.

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References

  1. Maldacena J.M.: The large N limit of superconformal field theories and supergravity. Adv. Theor. Math. Phys. 2, 231 (1998) hep-th/9711200

    MathSciNet  ADS  MATH  Google Scholar 

  2. Gubser S.S., Klebanov I.R., Polyakov A.M.: Gauge theory correlators from non-critical string theory. Phys. Lett. B. 428, 105 (1998). doi:10.1016/S0370-2693(98)00377-3 (hep-th/9802109)

    Article  MathSciNet  ADS  Google Scholar 

  3. Witten E.: Anti-de Sitter space and holography. Adv. Theor. Math. Phys. 2, 253 (1998) hep-th/9802150

    MathSciNet  MATH  Google Scholar 

  4. Kovacs, S.: N=4 supersymmetric Yang-Mills theory and the AdS/SCFT correspondence. hep-th/9908171

  5. Aharony O., Gubser S.S., Maldacena J.M., Ooguri H., Oz Y.: Large N field theories, string theory and gravity. Phys. Rept. 323, 183 (2000). doi:10.1016/S0370-1573(99)00083-6 (hep-th/9905111)

    Article  MathSciNet  ADS  Google Scholar 

  6. D’Hoker, E., Freedman, D.Z.: Supersymmetric gauge theories and the AdS/CFT correspondence. hep-th/0201253

  7. Maldacena, J.M.: Lectures on AdS/CFT. hep-th/0309246

  8. Nastase, H.: Introduction to AdS-CFT. arxiv:0712.0689

  9. Polchinski, J.: Introduction to Gauge/Gravity Duality. arxiv:1010.6134

  10. Benna, M.K., Klebanov, I.R.: Gauge-String Dualities and Some Applications. arxiv: 0803.1315

  11. Klebanov I.R., Maldacena J.M.: Solving quantum field theories via curved spacetimes. Phys. Today 62, 28 (2009). doi:10.1063/1.3074260

    Article  Google Scholar 

  12. ’t Hooft, G.: A planar diagram theory for strong interactions. Nucl. Phys. B. 72, 461 (1974). doi:10.1016/0550-3213(74)90154-0

  13. Horowitz, G.T., Polchinski, J.: Gauge/gravity duality. gr-qc/0602037

  14. Kawai H., Lewellen D.C., Tye S.-H.H.: A relation between tree amplitudes of closed and open strings. Nucl. Phys. B. 269, 1 (1986). doi:10.1016/0550-3213(86)90362-7

    Article  MathSciNet  ADS  Google Scholar 

  15. Bern Z., Carrasco J.J.M., Johansson H.: New Relations for Gauge-Theory Ampli- tudes. Phys. Rev. D. 78, 085011 (2008). doi:10.1103/PhysRevD.78.085011 (arxiv: 0805.3993)

    Article  MathSciNet  ADS  Google Scholar 

  16. Bern Z., Carrasco J.J.M., Johansson H.: Perturbative quantum gravity as a double copy of gauge theory. Phys. Rev. Lett. 105, 061602 (2010). doi:10.1103/PhysRevLett.105.061602 (arxiv:1004.0476)

    Article  MathSciNet  ADS  Google Scholar 

  17. Ioffe, B.L., Fadin, V.S., Lipatov, L.N. (eds): Quantum Chromodynamics: Perturbative and Nonperturbative Aspects. Cambridge University Press, Cambridge (2010)

    MATH  Google Scholar 

  18. Seiberg, N., Witten, E.: Monopole condensation, and confinement in \({\mathcal{N}}\) = 2 supersymmetric Yang–Mills theory. Nucl. Phys. B. 426, 19 (1994). doi:10.1016/0550-3213(94)90124-4 (hep-th/9407087)

  19. Seiberg, N., Witten, E.: Monopoles, duality and chiral symmetry breaking in \({\mathcal{N}}\) = 2 supersymmetric QCD. Nucl. Phys. B. 431, 484 (1994). doi:10.1016/0550-3213(94)90214-3 (hep-th/9408099)

  20. Nekrasov, N.A., Shatashvili, S.L.: Supersymmetric vacua and Bethe ansatz. Nucl. Phys. Proc. Suppl. 192–193, 91 (2009). doi:10.1016/j.nuclphysbps.2009.07.047 (arxiv:0901.4744)

  21. Nekrasov N.A., Shatashvili S.L.: Quantum integrability and supersymmetric vacua. Prog. Theor. Phys. Suppl. 177, 105 (2009). doi:10.1143/PTPS.177.105 (arxiv:0901.4748)

    Article  ADS  MATH  Google Scholar 

  22. Lotter, H.: Phenomenology of the BFKL pomeron and unitarity corrections at low x. hep-ph/9705288

  23. Forshaw J.R., Ross D.A.: Quantum Chromodynamics and the Pomeron. Cambridge University Press, Cambridge (1997)

    Book  Google Scholar 

  24. Donnachie S., Dosch G., Landshoff P., Nachtmann O.: Pomeron Physics and QCD. Cambridge University Press, Cambridge (2002)

    Book  MATH  Google Scholar 

  25. Essler F.H.L., Frahm H., Göhmann F., Klümper A., Korepin V.E.: The one-dimensional Hubbard model. Cambridge University Press, Cambridge (2005)

    Book  MATH  Google Scholar 

  26. Plefka J.: Spinning strings and integrable spin chains in the AdS/CFT correspondence. Living Rev. Rel. 8, 9 (2005) hep-th/0507136

    Google Scholar 

  27. Minahan, J.A.: A brief introduction to the Bethe ansatz in \({\mathcal{N}}\) = 4 super-Yang–Mills. J. Phys. A. 39, 12657 (2006). doi:10.1088/0305-4470/39/41/S02

  28. Dorey N.: Notes on integrability in gauge theory and string theory. J. Phys. A. 42, 254001 (2009). doi:10.1088/1751-8113/42/25/254001

    Article  MathSciNet  ADS  Google Scholar 

  29. Arutyunov, G., Frolov, S.: Foundations of the AdS5 × S5 Superstring. Part I. J. Phys. A. 42, 254003 (2009). doi:10.1088/1751-8113/42/25/254003 (arxiv:0901.4937)

    Google Scholar 

  30. Basso, B., Korchemsky, G.P.: Nonperturbative scales in AdS/CFT. J. Phys. A. 42, 254005 (2009). doi:10.1088/1751-8113/42/25/254005 (arxiv:0901.4945)

    Google Scholar 

  31. Alday L.F.: Scattering amplitudes and the AdS/CFT correspondence. J. Phys. A. 42, 254006 (2009). doi:10.1088/1751-8113/42/25/254006

    Article  MathSciNet  ADS  Google Scholar 

  32. Serban D.: Integrability and the AdS/CFT correspondence. J. Phys. A. 44, 124001 (2011). doi:10.1088/1751-8113/44/12/124001 (arxiv:1003.4214)

    Article  MathSciNet  ADS  Google Scholar 

  33. Fiamberti, F., Santambrogio, A., Sieg, C.: Superspace methods for the computation of wrapping effects in the standard and beta-deformed \({\mathcal{N}}\) = 4 SYM (arxiv:1006.3475)

  34. Beisert, N.: The dilatation operator of \({\mathcal{N}}\) = 4 super Yang-Mills theory and integrability. Phys. Rept. 405, 1 (2005). doi:10.1016/j.physrep.2004.09.007 (hep-th/0407277)

  35. Swanson, I.: A review of integrable deformations in AdS/CFT. Mod. Phys. Lett. A. 22, 915 (2007). doi:10.1142/S0217732307023614 (arxiv:0705.2844)

  36. Okamura, K. Aspects of Integrability in AdS/CFT Duality (arxiv:0803.3999)

  37. Vicedo B.: Finite-g Strings. J. Phys. A. 44, 124002 (2011). doi:10.1088/1751-8113/44/12/124002 (arxiv:0810.3402)

    Article  MathSciNet  ADS  Google Scholar 

  38. Rej, A.: Integrability and the AdS/CFT correspondence. J. Phys. A. 42, 254002 (2009). doi:10.1088/1751-8113/42/25/254002 (arxiv:0907.3468)

    Google Scholar 

  39. Gromov N.: Integrability in AdS/CFT correspondence: Quasi-classical analysis. J. Phys. A. 42, 254004 (2009). doi:10.1088/1751-8113/42/25/254004

    Article  MathSciNet  ADS  Google Scholar 

  40. Volin D.: Quantum integrability and functional equations. J. Phys. A. 44, 124003 (2011). doi:10.1088/1751-8113/44/12/124003 (arxiv:1003.4725)

    Article  MathSciNet  ADS  Google Scholar 

  41. Puletti, V.G.M.: On string integrability. A journey through the two-dimensional hidden symmetries in the AdS/CFT dualities. Adv. High Energy Phys. 2010, 471238 (2010). doi:10.1155/2010/471238 (arxiv:1006.3494)

  42. de Leeuw, M.: The S-matrix of the AdS5 × S5 superstring. arxiv:1007.4931

  43. Schäfer-Nameki, S.: Strings and super-Yang–Mills theory: The integrable story. J. Stat. Mech. 0612, N001 (2006). doi:10.1088/1742-5468/2006/12/N12001

  44. Nicolai H.: String theory: Back to basics. Nature 449, 797 (2007). doi:10.1038/449797a

    Article  ADS  Google Scholar 

  45. Kristjansen, C., Staudacher, M., Tseytlin, A. (eds.): Gauge-string duality and integrability: Progress and outlook. J. Phys. A. 42, 250301 (2009). doi:10.1088/1751-8121/42/25/250301

  46. Dorey, P., Minahan, J., Tseytlin, A. (eds.): Quantum integrable models and gauge-string duality. J. Phys. A. 44, 120301 (2011). doi:10.1088/1751-8121/44/12/120301

    Google Scholar 

  47. Dorey, P., Dunne, G., Feinberg, J. (eds.): Recent Advances in Low-Dimensional Quantum Field Theories. J. Phys. A. 39(issue 41) (2006) (editorial). doi:10.1088/0305-4470/39/41/E01

  48. Alcaraz, F., Babelon, O., de Gier J., Foda, O. (eds.): The 75th Anniversary of the Bethe Ansatz, topical articles. J. Stat. Mech. http://iopscience.iop.org/1742-5468/focus/extra.topical2

  49. Minahan, J.A.: Review of AdS/CFT Integrability, Chapter I.1: Spin Chains in \({\fancyscript{N}}\) = 4 SYM. Lett. Math. Phys. Published in this volume. arxiv:1012.3983

  50. Sieg, C.: Review of AdS/CFT Integrability, Chapter I.2: The spectrum from perturbative gauge theory. Lett. Math. Phys. Published in this volume. arxiv:1012.3984

  51. Rej, A.: Review of AdS/CFT Integrability, Chapter I.3: Long-range spin chains. Lett. Math. Phys. Published in this volume. arxiv:1012.3985

  52. Tseytlin, A.: Review of AdS/CFT Integrability, Chapter II.1: Classical AdS5 × S5 string solutions. Lett. Math. Phys. Published in this volume. arxiv:1012.3986

  53. McLoughlin, T.: Review of AdS/CFT Integrability, Chapter II.2: Quantum Strings in AdS5 × S5. Lett. Math. Phys. Published in this volume. arxiv:1012.3987

  54. Magro, M.: Review of AdS/CFT Integrability, Chapter II.3: Sigma Model, Gauge Fixing. Lett. Math. Phys. Published in this volume. arxiv:1012.3988

  55. Schäfer-Nameki, S.: Review of AdS/CFT Integrability, Chapter II.4: The Spectral Curve. Lett. Math. Phys. Published in this volume. arxiv:1012.3989

  56. Staudacher, M.: Review of AdS/CFT Integrability, Chapter III.1: Bethe Ansätze and the R-Matrix Formalism. Lett. Math. Phys. Published in this volume. arxiv:1012.3990

  57. Ahn, C., Nepomechie, R.I.: Review of AdS/CFT Integrability, Chapter III.2: Exact world-sheet S-matrix. Lett. Math. Phys. Published in this volume. arxiv:1012.3991

  58. Vieira, P., Volin, D.: Review of AdS/CFT Integrability, Chapter III.3: The dressing factor. Lett. Math. Phys. Published in this volume. arxiv:1012.3992

  59. Freyhult, L.: Review of AdS/CFT Integrability, Chapter III.4: Twist states and the cusp anomalous dimension. Lett. Math. Phys. Published in this volume. arxiv:1012.3993

  60. Janik, R.: Review of AdS/CFT Integrability, Chapter III.5: Lüscher corrections. Lett. Math. Phys. Published in this volume. arxiv:1012.3994

  61. Bajnok, Z.: Review of AdS/CFT Integrability, Chapter III.6: Thermodynamic Bethe Ansatz. Lett. Math. Phys. Published in this volume. arxiv:1012.3995

  62. Kazakov, V., Gromov, N.: Review of AdS/CFT Integrability, Chapter III.7: Hirota Dynamics for Quantum Integrability. Lett. Math. Phys. Published in this volume. arxiv:1012.3996

  63. Kristjansen, C.: Review of AdS/CFT Integrability, Chapter IV.1: Aspects of Non-planarity. Lett. Math. Phys. Published in this volume. arxiv:1012.3997

  64. Zoubos, K.: Review of AdS/CFT Integrability, Chapter IV.2: Deformations, Orbifolds and Open Boundaries. Lett. Math. Phys. Published in this volume. arxiv:1012.3998

  65. Klose, T.: Review of AdS/CFT Integrability, Chapter IV.3: \({\fancyscript{N}}\) = 6 Chern-Simons and Strings on AdS4 × CP3. Lett. Math. Phys. Published in this volume. arxiv:1012.3999

  66. Korchemsky, G.: Review of AdS/CFT Integrability, Chapter IV.4: Integrability in QCD and \({\fancyscript{N}< \psi}\) SYM. Lett. Math. Phys. Published in this volume. arxiv:1012.4000

  67. Dixon L.J.: Gluon scattering in \({\fancyscript{N}}\) = 4 super-Yang-Mills theory from weak to strong coupling. PoS RADCOR 2007, 056 (2007) arxiv:0803.2475

    Google Scholar 

  68. Alday L.F., Roiban R.: Scattering amplitudes, Wilson loops and the string/gauge theory correspondence. Phys. Rept. 468, 153 (2008) arxiv:0807.1889

    Article  MathSciNet  ADS  Google Scholar 

  69. Henn J.M.: Duality between Wilson loops and gluon amplitudes. Fortsch. Phys. 57, 729 (2009) arxiv:0903.0522

    Article  MathSciNet  MATH  ADS  Google Scholar 

  70. Wolf M.: A first course on twistors, integrability and gluon scattering amplitudes. J. Phys. A 43, 393001 (2010) arxiv:1001.3871

    Article  MathSciNet  Google Scholar 

  71. Drummond J. M.: Hidden simplicity of gauge theory amplitudes. Class. Quant. Grav. 27, 214001 (2010) arxiv:1010.2418

    Article  ADS  Google Scholar 

  72. Roiban, R., Spradlin, M., Volovich, A. (eds.): Scattering amplitudes in gauge theories: progress and outlook. J. Phys. A. (to appear)

  73. Roiban, R.: Review of AdS/CFT Integrability, Chapter V.1: Scattering Amplitudes—a Brief Introduction. Lett. Math. Phys. Published in this volume. arxiv:1012.4001

  74. Drummond, J.M.: Review of AdS/CFT Integrability, Chapter V.2: Dual Superconformal Symmetry. Lett. Math. Phys. Published in this volume. arxiv:1012.4002

  75. Alday, L.F.: Review of AdS/CFT Integrability, Chapter V.3: Scattering Amplitudes at Strong Coupling. Lett. Math. Phys. Published in this volume. arxiv:1012.4003

  76. Beisert, N.: Review of AdS/CFT Integrability, Chapter VI.1: Superconformal Algebra. Lett. Math. Phys. Published in this volume. arxiv:1012.4004

  77. Torrielli, A.: Review of AdS/CFT Integrability, Chapter VI.2: Yangian Algebra. Lett. Math. Phys. Published in this volume. arxiv:1012.4005

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Beisert, N., Ahn, C., Alday, L.F. et al. Review of AdS/CFT Integrability: An Overview. Lett Math Phys 99, 3–32 (2012). https://doi.org/10.1007/s11005-011-0529-2

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