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Deformed integrable \(\sigma\)-models, classical \(R\)-matrices and classical exchange algebra on Drinfel’d doubles. (English) Zbl 1422.37037

Summary: We describe a unifying framework for the systematic construction of integrable deformations of integrable \(\sigma\)-models within the Hamiltonian formalism. It applies equally to both the ‘Yang-Baxter’ type as well as ‘gauged WZW’ type deformations which were considered recently in the literature. As a byproduct, these two families of integrable deformations are shown to be Poisson-Lie \(T\)-dual of one another.

MSC:

37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
17B62 Lie bialgebras; Lie coalgebras
17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
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[1] Klimčík C 2002 Yang–Baxter sigma models and dS/AdS T duality J. High Energy Phys. JHEP12(2002)051
[2] Klimčík C 2009 On integrability of the Yang–Baxter sigma-model J. Math. Phys.50 043508 · Zbl 1215.81099
[3] Delduc F, Magro M and Vicedo B 2013 On classical q-deformations of integrable &sgr;-models J. High Energy Phys. JHEP11(2013)192 · Zbl 1342.81182
[4] Delduc F, Magro M and Vicedo B 2014 An integrable deformation of the superstring action Phys. Rev. Lett.112 051601 · Zbl 1333.81322
[5] Delduc F, Magro M and Vicedo B 2014 Derivation of the action and symmetries of the q-deformed superstring J. High Energy Phys. JHEP10(2014)132 · Zbl 1333.81322
[6] Arutyunov G, Borsato R and Frolov S 2014 S-matrix for strings on &eegr;-deformed J. High Energy Phys. JHEP04(2014)002 · Zbl 06564633
[7] Hoare B, Roiban R and Tseytlin A A 2014 On deformations of AdSn x Sn supercosets J. High Energy Phys. JHEP06(2014)002 · Zbl 06565136
[8] Arutyunov G, de Leeuw M and van Tongeren S J 2015 The exact spectrum and mirror duality of the (AdS S5) superstring Theor. Math. Phys.182 23 · Zbl 1317.81211
[9] Lunin O, Roiban R and Tseytlin A A 2015 Supergravity backgrounds for deformations of supercoset string models Nucl. Phys. B 891 106 · Zbl 1328.81182
[10] Engelund O T and Roiban R 2015 On the asymptotic states and the quantum S matrix of the &eegr;-deformed S5 superstring J. High Energy Phys. JHEP03(2015)168 · Zbl 06877820
[11] Matsumoto T and Yoshida K 2015 Yang–Baxter sigma models based on the CYBE Nucl. Phys. B 893 287 · Zbl 1348.81379
[12] Kawaguchi I, Matsumoto T and Yoshida K 2014 Jordanian deformations of the superstring J. High Energy Phys. JHEP04(2014)153 · Zbl 06564637
[13] Matsumoto T and Yoshida K 2014 Integrable deformations of the superstring and the classical Yang–Baxter equation–towards the gravity/CYBE correspondence J. Phys. Conf. Ser.563 012020
[14] van Tongeren S J 2015 On classical Yang–Baxter based deformations of the superstring J. High Energy Phys. JHEP1506(2015)048 · Zbl 1388.83332
[15] Sfetsos K 2014 Integrable interpolations: from exact CFTs to non-Abelian T-duals Nucl. Phys. B 880 225 · Zbl 1284.81257
[16] Balog J, Forgacs P, Horvath Z and Palla L 1994 A new family of SU(2) symmetric integrable sigma models Phys. Lett. B 324 403
[17] Hollowood T J, Miramontes J L and Schmidtt D M 2014 Integrable deformations of strings on symmetric spaces J. High Energy Phys. JHEP11(2014)009 · Zbl 1333.81341
[18] Hollowood T J, Miramontes J L and Schmidtt D M 2014 An integrable deformation of the superstring J. Phys. A 47 495402 · Zbl 1305.81120
[19] Demulder S, Sfetsos K and Thompson D C 2015 Integrable {\(\lambda\)}-deformations: squashing coset CFTs and arXiv:1504.02781
[20] Maillet J M 1986 New integrable canonical structures in two-dimensional models Nucl. Phys. B 269 54–76
[21] Itsios G, Sfetsos K, Siampos K and Torrielli A 2014 The classical Yang–Baxter equation and the associated Yangian symmetry of gauged WZW-type theories Nucl. Phys. B 889 64 · Zbl 1326.81180
[22] Sfetsos K and Siampos K 2015 The anisotropic łambda-deformed SU(2) model is integrable Phys. Lett. B 743 160 · Zbl 1343.81131
[23] Semenov-Tian-Shansky M A 1983 What is a classical R-matrix? Funct. Anal. Appl.17 259 · Zbl 0535.58031
[24] Klimčík C and Severa P 1995 Dual non-Abelian duality and the Drinfeld double Phys. Lett. B 351 455
[25] Klimčík C and Severa P 1996 Poisson–Lie T duality and loop groups of Drinfeld doubles Phys. Lett. B 372 65 · Zbl 1037.81576
[26] Rajeev S G 1989 Nonabelian bosonization without Wess–Zumino terms: 1. New current algebra Phys. Lett. B 217 123
[27] Delduc F, Magro M and Vicedo B 2015 Integrable double deformation of the principal chiral model Nucl. Phys. B 891 312 · Zbl 1328.81200
[28] Balog J, Dabrowski L and Fehér L 1990 Classical R-matrix and exchange algebra in WZNW and Toda theories Phys. Lett. B 244 227–34 · Zbl 0724.58069
[29] Alekseev A and Shatashvili S 1990 Quantum groups and WZNW models Commun. Math. Phys.133 353–68 · Zbl 0748.17009
[30] Faddeev L D 1990 On the exchange matrix for WZNW model Commun. Math. Phys.132 131–8 · Zbl 0731.17018
[31] Gawedzki K 1991 Classical origin of quantum group symmetries in Wess–Zumino–Witten conformal field theory Commun. Math. Phys.139 201–13 · Zbl 0721.58045
[32] Falceto F and Gawedzki K 1993 Lattice Wess–Zumino–Witten model and quantum groups J. Geom. Phys.11 251 · Zbl 0785.17013
[33] Frolov S 2005 Lax pair for strings in Lunin–Maldacena background J. High Energy Phys. JHEP05(2005)069
[34] Alday L F, Arutyunov G and Frolov S 2006 Green–Schwarz strings in TsT-transformed backgrounds J. High Energy Phys. JHEP06(2006)018
[35] Matsumoto T and Yoshida K 2014 Lunin–Maldacena backgrounds from the classical Yang–Baxter equation–towards the gravity/CYBE correspondence J. High Energy Phys. JHEP06(2014)135 · Zbl 1333.83196
[36] Semenov-Tian-Shansky M A 2008 Integrable systems: the R-matrix approach RIMS-1650
[37] Yu A, Alekseev and Malkin A Z 1994 Symplectic structures associated to Lie–Poisson groups Commun. Math. Phys.162 147–73 · Zbl 0797.58020
[38] Majid S 1990 Matched pairs of Lie groups associated to solutions of the Yang–Baxter equations Pac. J. Math.141 311–32 · Zbl 0735.17017
[39] Wolf J A 1969 The action of a real semisimple Lie group on a complex flag manifold: I. Orbit structure and holomorphic arc components Bull. Am. Math. Soc.75 1121–237 · Zbl 0183.50901
[40] Aomoto K 1966 On some double coset decompositions of complex semi-simple Lie groups J. Math. Soc. Japan18 1 · Zbl 0142.26701
[41] Harnad J and Kupershmidt B A 1995 Symplectic geometries on , Hamiltonian group actions and integrable systems J. Geom. Phys.16 168–206 · Zbl 0829.53027
[42] Sevostyanov A 1996 The classical R-matrix method for nonlinear sigma-model Int. J. Mod. Phys. A 11 4241–54 · Zbl 1044.37541
[43] Vicedo B 2011 The classical R-matrix of AdS/CFT and its Lie dialgebra structure Lett. Math. Phys.95 249 · Zbl 1214.37047
[44] Vicedo B 2010 Hamiltonian dynamics and the hidden symmetries of the superstring J. High Energy Phys. JHEP01(2010)102 · Zbl 1269.81152
[45] Kawaguchi I, Matsumoto T and Yoshida K 2013 Schroedinger sigma models and Jordanian twists J. High Energy Phys. JHEP08(2013)013 · Zbl 1342.83108
[46] Kawaguchi I and Yoshida K 2013 Exotic symmetry and monodromy equivalence in Schrödinger sigma models J. High Energy Phys. JHEP02(2013)024 · Zbl 1342.81240
[47] Belavin A A and Drinfel’d V G 1984 Triangle Equations and Simple Lie Algebras(Classical Reviews in Mathematics and Mathematical Physics vol 1) (New York: Harwood Academic)
[48] Cahen M, Gutt S and Rawnsley J 1994 Some remarks on the classification of Poisson–Lie groups Contemp. Math.179 1 · Zbl 0820.58018
[49] Balog J, Fehér L and Palla L 1999 The chiral WZNW phase space and its Poisson–Lie groupoid Phys. Lett. B 463 83–92 · Zbl 0987.81095
[50] Balog J, Fehér L and Palla L 2000 Chiral extensions of the WZNW phase space, Poisson–Lie symmetries and groupoids Nucl. Phys. B 568 503–42 · Zbl 0961.81098
[51] Feher L and Marshall I 2002 On a Poisson–Lie analogue of the classical dynamical Yang–Baxter equation for self-dual Lie algebras Lett. Math. Phys.62 51–62 · Zbl 1030.53081
[52] Klimčík C 2014 Integrability of the bi-Yang–Baxter sigma-model Lett. Math. Phys.104 1095 · Zbl 1359.70102
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