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 AdS 5 × S 5 and, consequently, to anomalous dimensions of operators in \({\mathcal{N}=4}\) SYM theory.
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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
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:1012.3991
Luscher M.: Volume dependence of the energy spectrum in massive quantum field theories. 1. Stable particle states. Commun. Math. Phys. 104, 177 (1986)
Rej A., Serban D., Staudacher M.: Planar \({\mathcal{N} \,{=} \,4}\) gauge theory and the Hubbard model. JHEP 0603, 018 (2006)
Zamolodchikov A.B.: Thermodynamic Bethe Ansatz in relativistic models. Scaling three state potts and Lee-Yang models. Nucl. Phys. B 342, 695 (1990)
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/9407117
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/9608091
Bajnok, Z.: Review of AdS/CFT integrability, Chapter III.6: thermodynamic Bethe Ansatz. Lett. Math. Phys. Published in this volume. arxiv:1012.3995
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
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
Klassen T.R., Melzer E.: On the relation between scattering amplitudes and finite size mass corrections in QFT. Nucl. Phys. B 362, 329 (1991)
Arutyunov G., Frolov S.: Foundations of the AdS 5 × S 5 superstring. Part I. J. Phys. A 42, 254003 (2009) arxiv:0901.4937
Janik R.A., Lukowski T.: Wrapping interactions at strong coupling – the giant magnon. Phys. Rev. D 76, 126008 (2007) arxiv:0708.2208
Dorey P., Tateo R.: Excited states by analytic continuation of TBA equations. Nucl. Phys. B 482, 639 (1996) hep-th/9607167
Dorey P., Tateo R.: Excited states in some simple perturbed conformal field theories. Nucl. Phys. B 515, 575 (1998) hep-th/9706140
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:0801.4463
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/0510171
Arutyunov G., Frolov S.: On string S-matrix, bound states and TBA. JHEP 0712, 024 (2007) arxiv:0710.1568
Schafer-Nameki S.: Exact expressions for quantum corrections to spinning strings. Phys. Lett. B 639, 571 (2006) hep-th/0602214
Schafer-Nameki S., Zamaklar M., Zarembo K.: How accurate is the quantum string Bethe ansatz?. JHEP 0612, 020 (2006) hep-th/0610250
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/0405001
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/0505071
Hofman D.M., Maldacena J.M.: Giant magnons. J. Phys. A 39, 13095 (2006) hep-th/0604135
Arutyunov G., Frolov S., Zamaklar M.: Finite-size effects from giant magnons. Nucl. Phys. B 778, 1 (2007) hep-th/0606126
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/0702043
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
Hatsuda Y., Suzuki R.: Finite-size effects for dyonic giant magnons. Nucl. Phys. B 800, 349 (2008) arxiv:0801.0747
Gromov N., Schafer-Nameki S., Vieira P.: Quantum wrapped giant magnon. Phys. Rev. D 78, 026006 (2008) arxiv:0801.3671
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:0807.0205
Bombardelli D., Fioravanti D.: Finite-size corrections of the CP3 giant magnons: the Lüscher terms. JHEP 0907, 034 (2009) arxiv:0810.0704
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:0810.1246
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:0810.2079
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:1007.1598
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/0412192
Correa D.H., Young C.A.S.: Finite size corrections for open strings/open chains in planar AdS/CFT. JHEP 0908, 097 (2009) arxiv:0905.1700
Bajnok, Z., Palla, L.: Boundary finite size corrections for multiparticle states and planar AdS/CFT. arxiv:1010.5617
Gromov N., Schafer-Nameki S., Vieira P.: Efficient precision quantization in AdS/CFT. JHEP 0812, 013 (2008) arxiv:0807.4752
Dorey N.: Magnon bound states and the AdS/CFT correspondence. J. Phys. A 39, 13119 (2006) hep-th/0604175
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:0807.0399
Fiamberti F., Santambrogio A., Sieg C., Zanon D.: Wrapping at four loops in \({\mathcal{N} \,{=} \,4}\) SYM. Phys. Lett. B 666, 100 (2008) arxiv:0712.3522
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:0806.2095
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:0808.3832
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)]
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)]
Balitsky, I.I., Lipatov, L.N.: The Pomeranchuk singularity in quantum chromodynamics. Sov. J. Nucl. Phys. 28, 822 (1978) [Yad. Fiz. 28, 1597 (1978)]
Kotikov A.V., Lipatov L.N., Rej A., Staudacher M., Velizhanin V.N.: Dressing and wrapping. J. Stat. Mech. 0710, P10003 (2007) arxiv:0704.3586
Bajnok Z., Janik R.A., Lukowski T.: Four loop twist two, BFKL, wrapping and strings. Nucl. Phys. B 816, 376 (2009) arxiv:0811.4448
Beccaria M., Forini V., Lukowski T., Zieme S.: Twist-three at five loops, Bethe Ansatz and wrapping. JHEP 0903, 129 (2009) arxiv:0901.4864
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:0903.0778
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:1006.2209
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:0811.4594
Fiamberti F., Santambrogio A., Sieg C.: Five-loop anomalous dimension at critical wrapping order in N = 4 SYM. JHEP 1003, 103 (2010) arxiv:0908.0234
Bombardelli D., Fioravanti D., Tateo R.: Thermodynamic Bethe Ansatz for planar AdS/CFT: a proposal. J. Phys. A 42, 375401 (2009) arxiv:0902.3930
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:0901.3753
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:0902.4458
Arutyunov G., Frolov S.: Thermodynamic Bethe Ansatz for the AdS 5 × S 5 mirror model. JHEP 0905, 068 (2009) arxiv:0903.0141
Arutyunov G., Frolov S.: Simplified TBA equations of the AdS 5 × S 5 mirror model. JHEP 0911, 019 (2009) arxiv:0907.2647
Bajnok Z., Hegedus A., Janik R.A., Lukowski T.: Five loop Konishi from AdS/CFT. Nucl. Phys. B 827, 426 (2010) arxiv:0906.4062
Lukowski T., Rej A., Velizhanin V.N.: Five-loop anomalous dimension of twist-two operators. Nucl. Phys. B 831, 105 (2010) arxiv:0912.1624
Velizhanin V.N.: Six-loop anomalous dimension of twist-three operators in N = 4 SYM. JHEP 1011, 129 (2010) arxiv:1003.4717
Arutyunov G., Frolov S., Suzuki R.: Five-loop Konishi from the mirror TBA. JHEP 1004, 069 (2010) arxiv:1002.1711
Balog J., Hegedus A.: 5-loop Konishi from linearized TBA and the XXX magnet. JHEP 1006, 080 (2010) arxiv:1002.4142
Balog J., Hegedus A.: The Bajnok-Janik formula and wrapping corrections. JHEP 1009, 107 (2010) arxiv:1003.4303
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:1010.5606
Gromov N., Kazakov V., Vieira P.: Finite volume spectrum of 2D field theories from Hirota dynamics. JHEP 0912, 060 (2009) arxiv:0812.5091
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:0907.1759
Bykov D.V., Frolov S.: Giant magnons in TsT-transformed AdS 5 × S 5. JHEP 0807, 071 (2008) arxiv:0805.1070
Acknowledgements
This work was supported by Polish science funds as a research project N N202 105136 (2009–2011).
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Janik, R.A. Review of AdS/CFT Integrability, Chapter III.5: Lüscher Corrections. Lett Math Phys 99, 277–297 (2012). https://doi.org/10.1007/s11005-011-0511-z
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DOI: https://doi.org/10.1007/s11005-011-0511-z