Abstract
We consider bosonic subsectors of the q-deformed AdS5×S5 superstring action and study the classical integrable structure of anisotropic Landau-Lifshitz sigma models (LLSMs) derived by taking fast-moving string limits. The subsectors are 1) deformed AdS3 × S1 and 2) R × deformed S3. The cases 1) and 2) lead to a time-like warped SL(2) LLSM and a squashed S3 LLSM, respectively. For each of them, we construct an infinite number of non-local conserved charges and show a quantum affine algebra at the classical level. Furthermore, a pp-wave like limit is applied for the case 1). The resulting system is a null-like warped SL(2) LLSM and exhibits a couple of Yangians through non-local gauge transformations associated with Jordanian twists.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J.M. Maldacena, The Large-N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].
N. Beisert et al., Review of AdS/CFT Integrability: An Overview, Lett. Math. Phys. 99 (2012) 3 [arXiv:1012.3982] [INSPIRE].
R.R. Metsaev and A.A. Tseytlin, Type IIB superstring action in AdS 5 × S 5 background, Nucl. Phys. B 533 (1998) 109 [hep-th/9805028] [INSPIRE].
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] [INSPIRE].
M. Lüscher, Quantum Nonlocal Charges and Absence of Particle Production in the Two-Dimensional Nonlinear σ-model, Nucl. Phys. B 135 (1978) 1 [INSPIRE].
M. Lüscher and K. Pohlmeyer, Scattering of Massless Lumps and Nonlocal Charges in the Two-Dimensional Classical Nonlinear σ-model, Nucl. Phys. B 137 (1978) 46 [INSPIRE].
E. Brézin, C. Itzykson, J. Zinn-Justin and J.B. Zuber, Remarks About the Existence of Nonlocal Charges in Two-Dimensional Models, Phys. Lett. B 82 (1979) 442 [INSPIRE].
D. Bernard, Hidden Yangians in 2 − D massive current algebras, Commun. Math. Phys. 137 (1991) 191 [INSPIRE].
N.J. MacKay, On the classical origins of Yangian symmetry in integrable field theory, Phys. Lett. B 281 (1992) 90 [Erratum ibid. B 308 (1993) 444] [INSPIRE].
K. Zarembo, Strings on Semisymmetric Superspaces, JHEP 05 (2010) 002 [arXiv:1003.0465] [INSPIRE].
L. Wulff, Superisometries and integrability of superstrings, arXiv:1402.3122 [INSPIRE].
N. Beisert and P. Koroteev, Quantum Deformations of the One-Dimensional Hubbard Model, J. Phys. A 41 (2008) 255204 [arXiv:0802.0777] [INSPIRE].
N. Beisert, W. Galleas and T. Matsumoto, A Quantum Affine Algebra for the Deformed Hubbard Chain, J. Phys. A 45 (2012) 365206 [arXiv:1102.5700] [INSPIRE].
B. Hoare, T.J. Hollowood and J.L. Miramontes, q-Deformation of the AdS 5 xS 5 Superstring S-matrix and its Relativistic Limit, JHEP 03 (2012) 015 [arXiv:1112.4485] [INSPIRE].
B. Hoare, T.J. Hollowood and J.L. Miramontes, Bound States of the q-Deformed AdS 5 xS 5 Superstring S-matrix, JHEP 10 (2012) 076 [arXiv:1206.0010] [INSPIRE].
B. Hoare, T.J. Hollowood and J.L. Miramontes, Restoring Unitarity in the q-Deformed World-Sheet S-matrix, JHEP 10 (2013) 050 [arXiv:1303.1447] [INSPIRE].
M. de Leeuw, V. Regelskis and A. Torrielli, The Quantum Affine Origin of the AdS/CFT Secret Symmetry, J. Phys. A 45 (2012) 175202 [arXiv:1112.4989] [INSPIRE].
G. Arutyunov, M. de Leeuw and S.J. van Tongeren, The Quantum Deformed Mirror TBA I, JHEP 10 (2012) 090 [arXiv:1208.3478] [INSPIRE].
G. Arutyunov, M. de Leeuw and S.J. van Tongeren, The Quantum Deformed Mirror TBA II, JHEP 02 (2013) 012 [arXiv:1210.8185] [INSPIRE].
I.V. Cherednik, Relativistically Invariant Quasiclassical Limits of Integrable Two-dimensional Quantum Models, Theor. Math. Phys. 47 (1981) 422 [INSPIRE].
L.D. Faddeev and N.Y. Reshetikhin, Integrability of the Principal Chiral Field Model in (1+1)-dimension, Annals Phys. 167 (1986) 227 [INSPIRE].
J. Balog, P. Forgacs and L. Palla, A Two-dimensional integrable axionic σ-model and T duality, Phys. Lett. B 484 (2000) 367 [hep-th/0004180] [INSPIRE].
S. Schäfer-Nameki, M. Yamazaki and K. Yoshida, Coset Construction for Duals of Non-relativistic CFTs, JHEP 05 (2009) 038 [arXiv:0903.4245] [INSPIRE].
I. Kawaguchi and K. Yoshida, Hidden Yangian symmetry in σ-model on squashed sphere, JHEP 11 (2010) 032 [arXiv:1008.0776] [INSPIRE].
I. Kawaguchi, D. Orlando and K. Yoshida, Yangian symmetry in deformed WZNW models on squashed spheres, Phys. Lett. B 701 (2011) 475 [arXiv:1104.0738] [INSPIRE].
I. Kawaguchi and K. Yoshida, A deformation of quantum affine algebra in squashed Wess-Zumino-Novikov-Witten models, J. Math. Phys. 55 (2014) 062302 [arXiv:1311.4696] [INSPIRE].
I. Kawaguchi and K. Yoshida, Hybrid classical integrability in squashed σ-models, Phys. Lett. B 705 (2011) 251 [arXiv:1107.3662] [INSPIRE].
I. Kawaguchi and K. Yoshida, Hybrid classical integrable structure of squashed σ-models: A Short summary, J. Phys. Conf. Ser. 343 (2012) 012055 [arXiv:1110.6748] [INSPIRE].
I. Kawaguchi, T. Matsumoto and K. Yoshida, The classical origin of quantum affine algebra in squashed σ-models, JHEP 04 (2012) 115 [arXiv:1201.3058] [INSPIRE].
I. Kawaguchi, T. Matsumoto and K. Yoshida, On the classical equivalence of monodromy matrices in squashed σ-model, JHEP 06 (2012) 082 [arXiv:1203.3400] [INSPIRE].
I. Kawaguchi and K. Yoshida, Classical integrability of Schrödinger σ-models and q-deformed Poincaré symmetry, JHEP 11 (2011) 094 [arXiv:1109.0872] [INSPIRE].
I. Kawaguchi and K. Yoshida, Exotic symmetry and monodromy equivalence in Schrödinger σ-models, JHEP 02 (2013) 024 [arXiv:1209.4147] [INSPIRE].
I. Kawaguchi, T. Matsumoto and K. Yoshida, Schroedinger σ-models and Jordanian twists, JHEP 08 (2013) 013 [arXiv:1305.6556] [INSPIRE].
D. Orlando, S. Reffert and L.I. Uruchurtu, Classical Integrability of the Squashed Three-sphere, Warped AdS3 and Schroedinger Spacetime via T-duality, J. Phys. A 44 (2011) 115401 [arXiv:1011.1771] [INSPIRE].
B. Basso and A. Rej, On the integrability of two-dimensional models with U (1) × SU (N ) symmetry, Nucl. Phys. B 866 (2013) 337 [arXiv:1207.0413] [INSPIRE].
D. Orlando and L.I. Uruchurtu, Integrable Superstrings on the Squashed Three-sphere, JHEP 10 (2012) 007 [arXiv:1208.3680] [INSPIRE].
F. Delduc, M. Magro and B. Vicedo, On classical q-deformations of integrable σ-models, JHEP 11 (2013) 192 [arXiv:1308.3581] [INSPIRE].
C. Klimčík, Yang-Baxter σ-models and dS/AdS T duality, JHEP 12 (2002) 051 [hep-th/0210095] [INSPIRE].
C. Klimčík, On integrability of the Yang-Baxter σ-model, J. Math. Phys. 50 (2009) 043508 [arXiv:0802.3518] [INSPIRE].
C. Klimčík, Integrability of the bi-Yang-Baxter σ-model, Lett. Math. Phys. 104 (2014) 1095 [arXiv:1402.2105] [INSPIRE].
R. Squellari, Yang-Baxter σ model: Quantum aspects, Nucl. Phys. B 881 (2014) 502 [arXiv:1401.3197] [INSPIRE].
V.G. Drinfeld, Hopf algebras and the quantum Yang-Baxter equation, Sov. Math. Dokl. 32 (1985) 254 [INSPIRE].
M. Jimbo, A q difference analog of U(g) and the Yang-Baxter equation, Lett. Math. Phys. 10 (1985) 63 [INSPIRE].
V.G. Drinfeld, Quantum groups, J. Sov. Math. 41 (1988) 898 [INSPIRE].
F. Delduc, M. Magro and B. Vicedo, An integrable deformation of the AdS 5 × S 5 superstring action, Phys. Rev. Lett. 112 (2014) 051601 [arXiv:1309.5850] [INSPIRE].
G. Arutyunov, R. Borsato and S. Frolov, S-matrix for strings on η-deformed AdS 5 × S 5, JHEP 04 (2014) 002 [arXiv:1312.3542] [INSPIRE].
B. Hoare, R. Roiban and A.A. Tseytlin, On deformations of AdS n × S n supercosets, arXiv:1403.5517 [INSPIRE].
G. Arutyunov, M. de Leeuw and S.J. van Tongeren, On the exact spectrum and mirror duality of the (AdS 5 xS 5) η superstring, arXiv:1403.6104 [INSPIRE].
N. Reshetikhin, Multiparameter quantum groups and twisted quasitriangular Hopf algebras, Lett. Math. Phys. 20 (1990) 331 [INSPIRE].
A. Stolin and P. P. Kulish, New rational solutions of Yang-Baxter equation and deformed Yangians, Czech. J. Phys. 47 (1997) 123 [q-alg/9608011].
A. Stolin and P. P. Kulish, Deformed Yangians and Integrable Models, Czech. J. Phys. 47 (1997) 1207 [q-alg/9708024].
P.P. Kulish, V.D. Lyakhovsky and A.I. Mudrov, Extended Jordanian twists for Lie algebras, J. Math. Phys. 40 (1999) 4569 [math/9806014] [INSPIRE].
I. Kawaguchi, T. Matsumoto and K. Yoshida, Jordanian deformations of the AdS 5 xS 5 superstring, JHEP 04 (2014) 153 [arXiv:1401.4855] [INSPIRE].
I. Kawaguchi, T. Matsumoto and K. Yoshida, A Jordanian deformation of AdS space in type IIB supergravity, JHEP 06 (2014) 146 [arXiv:1402.6147] [INSPIRE].
O. Lunin and J.M. Maldacena, Deforming field theories with U(1) × U(1) global symmetry and their gravity duals, JHEP 05 (2005) 033 [hep-th/0502086] [INSPIRE].
S. Frolov, Lax pair for strings in Lunin-Maldacena background, JHEP 05 (2005) 069 [hep-th/0503201] [INSPIRE].
A. Hashimoto and N. Itzhaki, Noncommutative Yang-Mills and the AdS/CFT correspondence, Phys. Lett. B 465 (1999) 142 [hep-th/9907166] [INSPIRE].
J.M. Maldacena and J.G. Russo, Large-N limit of noncommutative gauge theories, JHEP 09 (1999) 025 [hep-th/9908134] [INSPIRE].
D. Dhokarh, S.S. Haque and A. Hashimoto, Melvin Twists of global AdS 5 × S 5 and their Non-Commutative Field Theory Dual, JHEP 08 (2008) 084 [arXiv:0801.3812] [INSPIRE].
T. Matsumoto and K. Yoshida, Lunin-Maldacena backgrounds from the classical Yang-Baxter equation - Towards the gravity/CYBE correspondence, arXiv:1404.1838 [INSPIRE].
T. Matsumoto and K. Yoshida, Integrability of classical strings dual for noncommutative gauge theories, JHEP 06 (2014) 163 [arXiv:1404.3657] [INSPIRE].
M. Kruczenski, Spin chains and string theory, Phys. Rev. Lett. 93 (2004) 161602 [hep-th/0311203] [INSPIRE].
R. Hernandez and E. Lopez, The SU(3) spin chain σ-model and string theory, JHEP 04 (2004) 052 [hep-th/0403139] [INSPIRE].
B. Stefanski Jr. and A.A. Tseytlin, Large spin limits of AdS/CFT and generalized Landau-Lifshitz equations, JHEP 05 (2004) 042 [hep-th/0404133] [INSPIRE].
S. Bellucci, P.-Y. Casteill, J.F. Morales and C. Sochichiu, SL(2) spin chain and spinning strings on AdS 5 × S 5, Nucl. Phys. B 707 (2005) 303 [hep-th/0409086] [INSPIRE].
T. Kameyama and K. Yoshida, String theories on warped AdS backgrounds and integrable deformations of spin chains, JHEP 05 (2013) 146 [arXiv:1304.1286] [INSPIRE].
W.-Y. Wen, Spin chain from marginally deformed AdS 3 × S 3, Phys. Rev. D 75 (2007) 067901 [hep-th/0610147] [INSPIRE].
G. Arutyunov and S. Frolov, Foundations of the AdS 5 × S 5 Superstring. Part I, J. Phys. A 42 (2009) 254003 [arXiv:0901.4937] [INSPIRE].
L.D. Faddeev and L.A. Takhtajan, Hamiltonian Methods in the Theory of Solitons, Springer Series in Soviet Mathematics, Springer, New York U.S.A. (1987).
J. Lukierski, H. Ruegg, A. Nowicki and V.N. Tolstoi, Q deformation of Poincaré algebra, Phys. Lett. B 264 (1991) 331 [INSPIRE].
Ch. Ohn, A ∗-product on SL(2) and the corresponding nonstandard quantum-U(sl(2)), Lett. Math. Phys. 25 (1992) 85.
M. Kruczenski, A.V. Ryzhov and A.A. Tseytlin, Large spin limit of AdS 5 × S 5 string theory and low-energy expansion of ferromagnetic spin chains, Nucl. Phys. B 692 (2004) 3 [hep-th/0403120] [INSPIRE].
M. Kruczenski and A.A. Tseytlin, Semiclassical relativistic strings in S 5 and long coherent operators in N = 4 SYM theory, JHEP 09 (2004) 038 [hep-th/0406189] [INSPIRE].
J.A. Minahan, A. Tirziu and A.A. Tseytlin, 1/J 2 corrections to BMN energies from the quantum long range Landau-Lifshitz model, JHEP 11 (2005) 031 [hep-th/0510080] [INSPIRE].
N. Beisert, L. Fiévet, M. de Leeuw and F. Loebbert, Integrable Deformations of the XXZ Spin Chain, J. Stat. Mech. 2013 (2013) P09028 [arXiv:1308.1584] [INSPIRE].
R. Hernandez and E. Lopez, Spin chain σ-models with fermions, JHEP 11 (2004) 079 [hep-th/0410022] [INSPIRE].
B. Stefanski Jr., Landau-Lifshitz σ-models, fermions and the AdS/CFT correspondence, JHEP 07 (2007) 009 [arXiv:0704.1460] [INSPIRE].
R. Roiban, A. Tirziu and A.A. Tseytlin, Asymptotic Bethe ansatz S-matrix and Landau-Lifshitz type effective 2 − D actions, J. Phys. A 39 (2006) 13129 [hep-th/0604199] [INSPIRE].
A. Tirziu, Quantum Landau-Lifshitz model at four loops: 1/J and 1/J 2 corrections to BMN energies, Phys. Rev. D 73 (2006) 106001 [hep-th/0601139] [INSPIRE].
A. Melikyan, A. Pinzul, V.O. Rivelles and G. Weber, On S-matrix factorization of the Landau-Lifshitz model, JHEP 10 (2008) 002 [arXiv:0808.2489] [INSPIRE].
A. Melikyan and A. Pinzul, On quantum integrability of the Landau-Lifshitz model, J. Math. Phys. 50 (2009) 103518 [arXiv:0812.0188] [INSPIRE].
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1405.4467
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Kameyama, T., Yoshida, K. Anisotropic Landau-Lifshitz sigma models from q-deformed AdS5×S5 superstrings. J. High Energ. Phys. 2014, 110 (2014). https://doi.org/10.1007/JHEP08(2014)110
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP08(2014)110