Letters in Mathematical Physics

, Volume 99, Issue 1–3, pp 277–297

Review of AdS/CFT Integrability, Chapter III.5: Lüscher Corrections

Open Access
Article

Abstract

In integrable quantum field theories, the large volume spectrum is given by the Bethe Ansatz. The leading corrections, due to virtual particles circulating around the cylinder, are encoded in so-called Lüscher corrections. In order to apply these techniques to the AdS/CFT correspondence, one has to generalize these corrections to the case of generic dispersion relations and to multiparticle states. We review these various generalizations and the applications of Lüscher’s corrections to the study of the worldsheet QFT of the superstring in AdS5 × S5 and, consequently, to anomalous dimensions of operators in \({\mathcal{N}=4}\) SYM theory.

Mathematics Subject Classification (2010)

81T30 81T40 81U15 81U20 

Keywords

AdS/CFT correspondence integrability integrable quantum field theory. 

References

  1. 1.
    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:1012.3990Google Scholar
  2. 2.
    Ahn, C., Nepomechie, R.: Review of AdS/CFT integrability, Chapter III.2: exact world-sheet S-matrix. Lett. Math. Phys. Published in this volume. arxiv:1012.3991Google Scholar
  3. 3.
    Luscher M.: Volume dependence of the energy spectrum in massive quantum field theories. 1. Stable particle states. Commun. Math. Phys. 104, 177 (1986)MathSciNetADSCrossRefGoogle Scholar
  4. 4.
    Rej A., Serban D., Staudacher M.: Planar \({\mathcal{N} \,{=} \,4}\) gauge theory and the Hubbard model. JHEP 0603, 018 (2006)MathSciNetADSCrossRefGoogle Scholar
  5. 5.
    Zamolodchikov A.B.: Thermodynamic Bethe Ansatz in relativistic models. Scaling three state potts and Lee-Yang models. Nucl. Phys. B 342, 695 (1990)MathSciNetADSCrossRefGoogle Scholar
  6. 6.
    Destri C., De Vega H.J.: Unified approach to thermodynamic Bethe Ansatz and finite size corrections for lattice models and field theories. Nucl. Phys. B 438, 413 (1995) hep-th/9407117MathSciNetADSMATHCrossRefGoogle Scholar
  7. 7.
    Fioravanti D., Mariottini A., Quattrini E., Ravanini F.: Excited state Destri-De Vega equation for sine-Gordon and restricted sine-Gordon models. Phys. Lett. B 390, 243 (1997) hep-th/9608091MathSciNetADSCrossRefGoogle Scholar
  8. 8.
    Bajnok, Z.: Review of AdS/CFT integrability, Chapter III.6: thermodynamic Bethe Ansatz. Lett. Math. Phys. Published in this volume. arxiv:1012.3995Google Scholar
  9. 9.
    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: 1012.3996Google Scholar
  10. 10.
    Sieg, C.: Review of AdS/CFT integrability, chapter I.2: the spectrum from perturbative gauge theory. Lett. Math. Phys. Published in this volume. arxiv:1012.3984Google Scholar
  11. 11.
    Klassen T.R., Melzer E.: On the relation between scattering amplitudes and finite size mass corrections in QFT. Nucl. Phys. B 362, 329 (1991)MathSciNetADSCrossRefGoogle Scholar
  12. 12.
    Arutyunov G., Frolov S.: Foundations of the AdS 5 × S 5 superstring. Part I. J. Phys. A 42, 254003 (2009) arxiv:0901.4937MathSciNetADSGoogle Scholar
  13. 13.
    Janik R.A., Lukowski T.: Wrapping interactions at strong coupling – the giant magnon. Phys. Rev. D 76, 126008 (2007) arxiv:0708.2208ADSCrossRefGoogle Scholar
  14. 14.
    Dorey P., Tateo R.: Excited states by analytic continuation of TBA equations. Nucl. Phys. B 482, 639 (1996) hep-th/9607167MathSciNetADSMATHCrossRefGoogle Scholar
  15. 15.
    Dorey P., Tateo R.: Excited states in some simple perturbed conformal field theories. Nucl. Phys. B 515, 575 (1998) hep-th/9706140MathSciNetADSMATHCrossRefGoogle Scholar
  16. 16.
    Heller M.P., Janik R.A., Lukowski T.: A new derivation of Lüscher F-term and fluctuations around the giant magnon. JHEP 0806, 036 (2008) arxiv:0801.4463MathSciNetADSCrossRefGoogle Scholar
  17. 17.
    Ambjorn J., Janik R.A., Kristjansen C.: Wrapping interactions and a new source of corrections to the spin-chain/string duality. Nucl. Phys. B 736, 288 (2006) hep-th/0510171MathSciNetADSCrossRefGoogle Scholar
  18. 18.
    Arutyunov G., Frolov S.: On string S-matrix, bound states and TBA. JHEP 0712, 024 (2007) arxiv:0710.1568MathSciNetADSCrossRefGoogle Scholar
  19. 19.
    Schafer-Nameki S.: Exact expressions for quantum corrections to spinning strings. Phys. Lett. B 639, 571 (2006) hep-th/0602214MathSciNetADSCrossRefGoogle Scholar
  20. 20.
    Schafer-Nameki S., Zamaklar M., Zarembo K.: How accurate is the quantum string Bethe ansatz?. JHEP 0612, 020 (2006) hep-th/0610250MathSciNetADSCrossRefGoogle Scholar
  21. 21.
    Beisert N., Dippel V., Staudacher M.: A novel long range spin chain and planar \({\mathcal{N} \,{=} \,4}\) super Yang-Mills. JHEP 0407, 075 (2004) hep-th/0405001MathSciNetADSCrossRefGoogle Scholar
  22. 22.
    Sieg C., Torrielli A.: Wrapping interactions and the genus expansion of the 2-point function of composite operators. Nucl. Phys. B 723, 3 (2005) hep-th/0505071MathSciNetADSMATHCrossRefGoogle Scholar
  23. 23.
    Hofman D.M., Maldacena J.M.: Giant magnons. J. Phys. A 39, 13095 (2006) hep-th/0604135MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Arutyunov G., Frolov S., Zamaklar M.: Finite-size effects from giant magnons. Nucl. Phys. B 778, 1 (2007) hep-th/0606126MathSciNetADSMATHCrossRefGoogle Scholar
  25. 25.
    Astolfi D., Forini V., . Grignani G., Semenoff G.W.: Gauge invariant finite size spectrum of the giant magnon. Phys. Lett. B 651, 329 (2007) hep-th/0702043MathSciNetADSCrossRefGoogle Scholar
  26. 26.
    Vieira, P., Volin, D.: Review of AdS/CFT integrability, Chapter III.3: the dressing factor. Lett. Math. Phys. Published in this volume. arxiv:1012.3992Google Scholar
  27. 27.
    Hatsuda Y., Suzuki R.: Finite-size effects for dyonic giant magnons. Nucl. Phys. B 800, 349 (2008) arxiv:0801.0747ADSMATHCrossRefGoogle Scholar
  28. 28.
    Gromov N., Schafer-Nameki S., Vieira P.: Quantum wrapped giant magnon. Phys. Rev. D 78, 026006 (2008) arxiv:0801.3671MathSciNetADSCrossRefGoogle Scholar
  29. 29.
    Grignani G., Harmark T., Orselli M., Semenoff G.W.: Finite size giant magnons in the string dual of N = 6 superconformal Chern-Simons theory. JHEP 0812, 008 (2008) arXiv:0807.0205MathSciNetGoogle Scholar
  30. 30.
    Bombardelli D., Fioravanti D.: Finite-size corrections of the CP3 giant magnons: the Lüscher terms. JHEP 0907, 034 (2009) arxiv:0810.0704MathSciNetADSCrossRefGoogle Scholar
  31. 31.
    Lukowski T., Sax O.O.: Finite size giant magnons in the SU(2) × SU(2) sector of AdS 4 × CP 3. JHEP 0812, 073 (2008) arxiv:0810.1246ADSCrossRefGoogle Scholar
  32. 32.
    Ahn C., Bozhilov P.: Finite-size effect of the dyonic giant magnons in \({\mathcal{N} \,{=} \,6}\) super Chern-Simons theory. Phys. Rev. D 79, 046008 (2009) arxiv:0810.2079ADSCrossRefGoogle Scholar
  33. 33.
    Ahn C., Kim M., Lee B.H.: Quantum finite-size effects for dyonic magnons in the AdS 4 × CP 3. JHEP 1009, 062 (2010) arxiv:1007.1598ADSCrossRefGoogle Scholar
  34. 34.
    Bajnok Z., Palla L., Takacs G.: Finite size effects in quantum field theories with boundary from scattering data. Nucl. Phys. B 716, 519 (2005) hep-th/0412192MathSciNetADSMATHCrossRefGoogle Scholar
  35. 35.
    Correa D.H., Young C.A.S.: Finite size corrections for open strings/open chains in planar AdS/CFT. JHEP 0908, 097 (2009) arxiv:0905.1700MathSciNetADSCrossRefGoogle Scholar
  36. 36.
    Bajnok, Z., Palla, L.: Boundary finite size corrections for multiparticle states and planar AdS/CFT. arxiv:1010.5617Google Scholar
  37. 37.
    Gromov N., Schafer-Nameki S., Vieira P.: Efficient precision quantization in AdS/CFT. JHEP 0812, 013 (2008) arxiv:0807.4752MathSciNetADSCrossRefGoogle Scholar
  38. 38.
    Dorey N.: Magnon bound states and the AdS/CFT correspondence. J. Phys. A 39, 13119 (2006) hep-th/0604175MathSciNetMATHCrossRefGoogle Scholar
  39. 39.
    Bajnok Z., Janik R.A.: Four-loop perturbative Konishi from strings and finite size effects for multiparticle states. Nucl. Phys. B 807, 625 (2009) arxiv:0807.0399MathSciNetADSMATHCrossRefGoogle Scholar
  40. 40.
    Fiamberti F., Santambrogio A., Sieg C., Zanon D.: Wrapping at four loops in \({\mathcal{N} \,{=} \,4}\) SYM. Phys. Lett. B 666, 100 (2008) arxiv:0712.3522MathSciNetADSCrossRefGoogle Scholar
  41. 41.
    Fiamberti F., Santambrogio A., Sieg C., Zanon D.: Anomalous dimension with wrapping at four loops in \({\mathcal{N} \,{=} \,4}\) SYM. Nucl. Phys. B 805, 231 (2008) arxiv:0806.2095MathSciNetADSMATHCrossRefGoogle Scholar
  42. 42.
    Velizhanin V.N.: The four-loop anomalous dimension of the Konishi operator in N = 4 supersymmetric Yang-Mills theory. JETP Lett. 89, 6 (2009) arxiv:0808.3832ADSCrossRefGoogle Scholar
  43. 43.
    Lipatov, L.N.: Reggeization of the vector meson and the vacuum singularity in nonabelian gauge theories. Sov. J. Nucl. Phys. 23, 338 (1976) [Yad. Fiz. 23, 642 (1976)]Google Scholar
  44. 44.
    Kuraev, E.A., Lipatov, L.N., Fadin, V.S.: The Pomeranchuk singularity in nonabelian gauge theories. Sov. Phys. JETP 45, 199 (1977) [Zh. Eksp. Teor. Fiz. 72, 377 (1977)]Google Scholar
  45. 45.
    Balitsky, I.I., Lipatov, L.N.: The Pomeranchuk singularity in quantum chromodynamics. Sov. J. Nucl. Phys. 28, 822 (1978) [Yad. Fiz. 28, 1597 (1978)]Google Scholar
  46. 46.
    Kotikov A.V., Lipatov L.N., Rej A., Staudacher M., Velizhanin V.N.: Dressing and wrapping. J. Stat. Mech. 0710, P10003 (2007) arxiv:0704.3586CrossRefGoogle Scholar
  47. 47.
    Bajnok Z., Janik R.A., Lukowski T.: Four loop twist two, BFKL, wrapping and strings. Nucl. Phys. B 816, 376 (2009) arxiv:0811.4448MathSciNetADSMATHCrossRefGoogle Scholar
  48. 48.
    Beccaria M., Forini V., Lukowski T., Zieme S.: Twist-three at five loops, Bethe Ansatz and wrapping. JHEP 0903, 129 (2009) arxiv:0901.4864ADSCrossRefGoogle Scholar
  49. 49.
    Beccaria M., De Angelis G.F.: On the wrapping correction to single magnon energy in twisted N = 4 SYM. Int. J. Mod. Phys. A 24, 5803 (2009) arxiv:0903.0778MathSciNetADSMATHCrossRefGoogle Scholar
  50. 50.
    Ahn C., Bajnok Z., Bombardelli D., Nepomechie R.I.: Finite-size effect for four-loop Konishi of the beta-deformed N = 4 SYM. Phys. Lett. B 693, 380 (2010) arxiv:1006.2209MathSciNetADSCrossRefGoogle Scholar
  51. 51.
    Fiamberti F., Santambrogio A., Sieg C., Zanon D.: Single impurity operators at critical wrapping order in the beta-deformed \({\mathcal{N} \,{=} \,4}\) SYM. JHEP 0908, 034 (2009) arxiv:0811.4594ADSCrossRefGoogle Scholar
  52. 52.
    Fiamberti F., Santambrogio A., Sieg C.: Five-loop anomalous dimension at critical wrapping order in N = 4 SYM. JHEP 1003, 103 (2010) arxiv:0908.0234ADSCrossRefGoogle Scholar
  53. 53.
    Bombardelli D., Fioravanti D., Tateo R.: Thermodynamic Bethe Ansatz for planar AdS/CFT: a proposal. J. Phys. A 42, 375401 (2009) arxiv:0902.3930MathSciNetCrossRefGoogle Scholar
  54. 54.
    Gromov N., Kazakov V., Vieira P.: Exact spectrum of anomalous dimensions of planar N = 4 supersymmetric Yang-Mills theory. Phys. Rev. Lett. 103, 131601 (2009) arxiv:0901.3753MathSciNetADSCrossRefGoogle Scholar
  55. 55.
    Gromov N., Kazakov V., Kozak A., Vieira P.: Exact spectrum of anomalous dimensions of planar N = 4 supersymmetric Yang-Mills theory: TBA and excited states. Lett. Math. Phys. 91, 265 (2010) arxiv:0902.4458MathSciNetADSMATHCrossRefGoogle Scholar
  56. 56.
    Arutyunov G., Frolov S.: Thermodynamic Bethe Ansatz for the AdS 5 × S 5 mirror model. JHEP 0905, 068 (2009) arxiv:0903.0141MathSciNetADSCrossRefGoogle Scholar
  57. 57.
    Arutyunov G., Frolov S.: Simplified TBA equations of the AdS 5 × S 5 mirror model. JHEP 0911, 019 (2009) arxiv:0907.2647MathSciNetADSCrossRefGoogle Scholar
  58. 58.
    Bajnok Z., Hegedus A., Janik R.A., Lukowski T.: Five loop Konishi from AdS/CFT. Nucl. Phys. B 827, 426 (2010) arxiv:0906.4062MathSciNetADSMATHCrossRefGoogle Scholar
  59. 59.
    Lukowski T., Rej A., Velizhanin V.N.: Five-loop anomalous dimension of twist-two operators. Nucl. Phys. B 831, 105 (2010) arxiv:0912.1624MathSciNetADSMATHCrossRefGoogle Scholar
  60. 60.
    Velizhanin V.N.: Six-loop anomalous dimension of twist-three operators in N = 4 SYM. JHEP 1011, 129 (2010) arxiv:1003.4717MathSciNetADSCrossRefGoogle Scholar
  61. 61.
    Arutyunov G., Frolov S., Suzuki R.: Five-loop Konishi from the mirror TBA. JHEP 1004, 069 (2010) arxiv:1002.1711ADSCrossRefGoogle Scholar
  62. 62.
    Balog J., Hegedus A.: 5-loop Konishi from linearized TBA and the XXX magnet. JHEP 1006, 080 (2010) arxiv:1002.4142ADSCrossRefGoogle Scholar
  63. 63.
    Balog J., Hegedus A.: The Bajnok-Janik formula and wrapping corrections. JHEP 1009, 107 (2010) arxiv:1003.4303ADSCrossRefGoogle Scholar
  64. 64.
    Bajnok Z., Deeb O.e.: 6-loop anomalous dimension of a single impurity operator from AdS/CFT and multiple zeta values. JHEP 1101, 054 (2011) arxiv:1010.5606ADSCrossRefGoogle Scholar
  65. 65.
    Gromov N., Kazakov V., Vieira P.: Finite volume spectrum of 2D field theories from Hirota dynamics. JHEP 0912, 060 (2009) arxiv:0812.5091MathSciNetADSCrossRefGoogle Scholar
  66. 66.
    Balog J., Hegedus A.: The finite size spectrum of the 2-dimensional O(3) nonlinear sigma-model. Nucl. Phys. B 829, 425–446 (2010) arxiv:0907.1759MathSciNetADSMATHCrossRefGoogle Scholar
  67. 67.
    Bykov D.V., Frolov S.: Giant magnons in TsT-transformed AdS 5 × S 5. JHEP 0807, 071 (2008) arxiv:0805.1070MathSciNetADSCrossRefGoogle Scholar

Copyright information

© The Author(s) 2011

Authors and Affiliations

  1. 1.Institute of PhysicsJagiellonian UniversityKrakówPoland

Personalised recommendations