Abstract
We explicitly derive Lax pairs for string theories on Yang-Baxter deformed backgrounds, 1) gravity duals for noncommutative gauge theories, 2) γ-deformations of S5, 3) Schrödinger spacetimes and 4) abelian twists of the global AdS5. Then we can find out a concise derivation of Lax pairs based on simple replacement rules. Furthermore, each of the above deformations can be reinterpreted as twisted boundary conditions with the undeformed background by using the rules. As another derivation, the Lax pair for gravity duals for noncommutative gauge theories is reproduced from the one for a q-deformed AdS5×S5 by taking a scaling limit.
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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].
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].
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].
F. Delduc, M. Magro and B. Vicedo, On classical q-deformations of integrable σ-models, JHEP 11 (2013) 192 [arXiv:1308.3581] [INSPIRE].
I. Kawaguchi and K. Yoshida, Hybrid classical integrability in squashed σ-models, Phys. Lett. B 705 (2014) 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 and K. Yoshida, Hidden Yangian symmetry in σ-model on squashed sphere, JHEP 11 (2010) 032 [arXiv:1008.0776] [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, 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, 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].
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].
F. Delduc, M. Magro and B. Vicedo, Derivation of the action and symmetries of the q-deformed AdS 5 × S 5 superstring, JHEP 10 (2014) 132 [arXiv:1406.6286] [INSPIRE].
I. Kawaguchi, T. Matsumoto and K. Yoshida, Jordanian deformations of the AdS 5 × S 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].
T. Matsumoto and K. Yoshida, Lunin-Maldacena backgrounds from the classical Yang-Baxter equation — towards the gravity/CYBE correspondence, JHEP 06 (2014) 135 [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].
T. Matsumoto and K. Yoshida, Yang-Baxter deformations and string dualities, JHEP 03 (2015) 137 [arXiv:1412.3658] [INSPIRE].
T. Matsumoto and K. Yoshida, Yang-Baxter σ-models based on the CYBE, Nucl. Phys. B 893 (2015) 287 [arXiv:1501.03665] [INSPIRE].
T. Matsumoto and K. Yoshida, Schrödinger geometries arising from Yang-Baxter deformations, JHEP 04 (2015) 180 [arXiv:1502.00740] [INSPIRE].
S.J. van Tongeren, On classical Yang-Baxter based deformations of the AdS 5 × S 5 superstring, JHEP 06 (2015) 048 [arXiv:1504.05516] [INSPIRE].
S.J. van Tongeren, Yang-Baxter deformations, AdS/CFT and twist-noncommutative gauge theory, arXiv:1506.01023 [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].
C.P. Herzog, M. Rangamani and S.F. Ross, Heating up Galilean holography, JHEP 11 (2008) 080 [arXiv:0807.1099] [INSPIRE].
J. Maldacena, D. Martelli and Y. Tachikawa, Comments on string theory backgrounds with non-relativistic conformal symmetry, JHEP 10 (2008) 072 [arXiv:0807.1100] [INSPIRE].
A. Adams, K. Balasubramanian and J. McGreevy, Hot Spacetimes for Cold Atoms, JHEP 11 (2008) 059 [arXiv:0807.1111] [INSPIRE].
T. Matsumoto and K. Yoshida, Integrable deformations of the AdS 5 × S 5 superstring and the classical Yang-Baxter equation — Towards the gravity/CYBE correspondence, J. Phys. Conf. Ser. 563 (2014) 012020 [arXiv:1410.0575] [INSPIRE].
H. Lin, O. Lunin and J.M. Maldacena, Bubbling AdS space and 1/2 BPS geometries, JHEP 10 (2004) 025 [hep-th/0409174] [INSPIRE].
T. Matsumoto, D. Orlando, S. Reffert, J.-i. Sakamoto and K. Yoshida, Yang-Baxter deformations of Minkowski spacetime, arXiv:1505.04553 [INSPIRE].
S. Zakrzewski, Poisson structures on the Poincaré group, Commun. Math. Phys. 185 (1997) 285 [q-alg/9602001]
J. Lukierski and M. Woronowicz, New Lie-algebraic and quadratic deformations of Minkowski space from twisted Poincaré symmetries, Phys. Lett. B 633 (2006) 116 [hep-th/0508083] [INSPIRE].
A. Borowiec, J. Lukierski, M.n. Mozrzymas and V.N. Tolstoy, N = 1/2 Deformations of Chiral Superspaces from New Twisted Poincaré and Euclidean Superalgebras, JHEP 06 (2012) 154 [arXiv:1112.1936] [INSPIRE].
A. Ballesteros, F.J. Herranz and N.R. Bruno, Quantum (anti)de Sitter algebras and generalizations of the kappa-Minkowski space, hep-th/0409295 [INSPIRE].
I.R. Klebanov and E. Witten, Superconformal field theory on three-branes at a Calabi-Yau singularity, Nucl. Phys. B 536 (1998) 199 [hep-th/9807080] [INSPIRE].
P. Basu and L.A. Pando Zayas, Chaos rules out integrability of strings on AdS 5 × T 1,1, Phys. Lett. B 700 (2014) 243 [arXiv:1103.4107] [INSPIRE].
P. Basu and L.A. Pando Zayas, Analytic Non-integrability in String Theory, Phys. Rev. D 84 (2011) 046006 [arXiv:1105.2540] [INSPIRE].
Y. Asano, D. Kawai, H. Kyono and K. Yoshida, Chaotic strings in a near Penrose limit of AdS 5 × T 1,1, JHEP 08 (2015) 060 [arXiv:1505.07583] [INSPIRE].
A. Catal-Ozer, Lunin-Maldacena deformations with three parameters, JHEP 02 (2006) 026 [hep-th/0512290] [INSPIRE].
P.M. Crichigno, T. Matsumoto and K. Yoshida, Deformations of T 1,1 as Yang-Baxter σ-models, JHEP 12 (2014) 085 [arXiv:1406.2249] [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. McLoughlin and I. Swanson, Integrable twists in AdS/CFT, JHEP 08 (2006) 084 [hep-th/0605018] [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].
G. Arutyunov, R. Borsato and S. Frolov, Puzzles of eta-deformed AdS 5 × S 5, arXiv:1507.04239 [INSPIRE].
V.G. Drinfeld, Hopf algebras and the quantum Yang-Baxter equation, Sov. Math. Dokl. 32 (1985) 254 [INSPIRE].
V.G. Drinfeld, Quantum groups, J. Sov. Math. 41 (1988) 898 [INSPIRE].
M. Jimbo, A q difference analog of U(g) and the Yang-Baxter equation, Lett. Math. Phys. 10 (1985) 63 [INSPIRE].
R.G. Leigh and M.J. Strassler, Exactly marginal operators and duality in four-dimensional N =1 supersymmetric gauge theory, Nucl. Phys. B 447(1995) 95[hep-th/9503121] [INSPIRE].
L.F. Alday, G. Arutyunov and S. Frolov, Green-Schwarz strings in TsT-transformed backgrounds, JHEP 06 (2006) 018 [hep-th/0512253] [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].
A. Borowiec, H. Kyono, J. Lukierski, J.-i. Sakamoto and K. Yoshida, Yang-Baxter σ-models and Lax pairs arising from κ-Poincaré r-matrices, arXiv:1510.03083 [INSPIRE].
B. Hoare, R. Roiban and A.A. Tseytlin, On deformations of AdS n × S n supercosets, JHEP 06 (2014) 002 [arXiv:1403.5517] [INSPIRE].
G. Arutyunov, M. de Leeuw and S.J. van Tongeren, The exact spectrum and mirror duality of the (AdS 5 × S 5) η superstring, Theor. Math. Phys. 182 (2015) 23 [arXiv:1403.6104] [INSPIRE].
G. Arutyunov and S.J. van Tongeren, AdS5 × S5 mirror model as a string σ-model, Phys. Rev. Lett. 113 (2014) 261605 [arXiv:1406.2304] [INSPIRE].
G. Arutyunov and S.J. van Tongeren, Double Wick rotating Green-Schwarz strings, JHEP 05 (2015) 027 [arXiv:1412.5137] [INSPIRE].
T. Kameyama and K. Yoshida, Anisotropic Landau-Lifshitz σ-models from q-deformed AdS 5× S 5 superstrings, JHEP 08 (2014) 110 [arXiv:1405.4467] [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].
M. Khouchen and J. Kluson, Giant Magnon on Deformed AdS 3 × S 3, Phys. Rev. D 90 (2014) 066001 [arXiv:1405.5017] [INSPIRE].
C. Ahn and P. Bozhilov, Finite-size giant magnons on η-deformed AdS 5 × S 5, Phys. Lett. B 737 (2014) 293 [arXiv:1406.0628] [INSPIRE].
A. Banerjee and K.L. Panigrahi, On the rotating and oscillating strings in (AdS 3 × S 3) κ , JHEP 09 (2014) 048 [arXiv:1406.3642] [INSPIRE].
P. Bozhilov, Some three-point correlation functions in the eta-deformed AdS 5 × S 5, arXiv:1502.00610 [INSPIRE].
A. Banerjee, S. Bhattacharya and K.L. Panigrahi, Spiky strings in κ-deformed AdS, JHEP 06 (2015) 057 [arXiv:1503.07447] [INSPIRE].
M. Khouchen and J. Klusoň, D-brane on deformed AdS 3 × S 3, JHEP 08 (2015) 046 [arXiv:1505.04946] [INSPIRE].
G. Arutyunov and D. Medina-Rincon, Deformed Neumann model from spinning strings on (AdS 5 × S 5) η , JHEP 10 (2014) 50 [arXiv:1406.2536] [INSPIRE].
T. Kameyama and K. Yoshida, A new coordinate system for q-deformed AdS 5 × S 5 and classical string solutions, J. Phys. A 48 (2015) 075401 [arXiv:1408.2189] [INSPIRE].
T. Kameyama and K. Yoshida, Minimal surfaces in q-deformed AdS 5 × S 5 with Poincaré coordinates, J. Phys. A 48 (2015) 245401 [arXiv:1410.5544] [INSPIRE].
N. Bai, H.-H. Chen and J.-B. Wu, Holographic Cusped Wilson loops in q-deformed AdS 5 × S 5 Spacetime, Chin. Phys. C 39 (2015) 103102 [arXiv:1412.8156] [INSPIRE].
B. Hoare, Towards a two-parameter q-deformation of AdS 3 × S 3 × M 4 superstrings, Nucl. Phys. B 891 (2015) 259 [arXiv:1411.1266] [INSPIRE].
O. Lunin, R. Roiban and A.A. Tseytlin, Supergravity backgrounds for deformations of AdS n × S n supercoset string models, Nucl. Phys. B 891 (2015) 106 [arXiv:1411.1066] [INSPIRE].
B. Hoare and A.A. Tseytlin, Type IIB supergravity solution for the T-dual of the η-deformed AdS 5 × S 5 superstring, JHEP 10 (2015) 060 [arXiv:1508.01150] [INSPIRE].
K. Sfetsos, Integrable interpolations: From exact CFTs to non-Abelian T-duals, Nucl. Phys. B 880 (2014) 225 [arXiv:1312.4560] [INSPIRE].
K. Sfetsos and D.C. Thompson, Spacetimes for λ-deformations, JHEP 12 (2014) 164 [arXiv:1410.1886] [INSPIRE].
S. Demulder, K. Sfetsos and D.C. Thompson, Integrable λ-deformations: Squashing Coset CFTs and AdS 5 × S 5, JHEP 07 (2015) 019 [arXiv:1504.02781] [INSPIRE].
K. Sfetsos, K. Siampos and D.C. Thompson, Generalised integrable λ- and η-deformations and their relation, Nucl. Phys. B 899 (2015) 489 [arXiv:1506.05784] [INSPIRE].
T.J. Hollowood, J.L. Miramontes and D.M. Schmidtt, Integrable Deformations of Strings on Symmetric Spaces, JHEP 11 (2014) 009 [arXiv:1407.2840] [INSPIRE].
T.J. Hollowood, J.L. Miramontes and D.M. Schmidtt, An Integrable Deformation of the AdS 5 × S 5 Superstring, J. Phys. A 47 (2014) 495402 [arXiv:1409.1538] [INSPIRE].
T.J. Hollowood, J.L. Miramontes and D.M. Schmidtt, S-Matrices and Quantum Group Symmetry of k-Deformed σ-models, arXiv:1506.06601 [INSPIRE].
C. Appadu and T.J. Hollowood, β-function of k Deformed AdS 5 × S 5 String Theory, arXiv:1507.05420 [INSPIRE].
B. Vicedo, Deformed integrable σ-models, classical R-matrices and classical exchange algebra on Drinfel’d doubles, J. Phys. A 48 (2015) 355203 [arXiv:1504.06303] [INSPIRE].
B. Hoare and A.A. Tseytlin, On integrable deformations of superstring σ-models related to AdS n × S n supercosets, Nucl. Phys. B 897 (2015) 448 [arXiv:1504.07213] [INSPIRE].
C. Klimčík, η and λ deformations as ℰ-models, Nucl. Phys. B 900 (2015) 259 [arXiv:1508.05832] [INSPIRE].
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Kameyama, T., Kyono, H., Sakamoto, Ji. et al. Lax pairs on Yang-Baxter deformed backgrounds. J. High Energ. Phys. 2015, 43 (2015). https://doi.org/10.1007/JHEP11(2015)043
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DOI: https://doi.org/10.1007/JHEP11(2015)043