Quantum transfer-matrices for the sausage model

  • Vladimir V. Bazhanov
  • Gleb A. Kotousov
  • Sergei L. Lukyanov
Open Access
Regular Article - Theoretical Physics


In this work we revisit the problem of the quantization of the two-dimensional O(3) non-linear sigma model and its one-parameter integrable deformation — the sausage model. Our consideration is based on the so-called ODE/IQFT correspondence, a variant of the Quantum Inverse Scattering Method. The approach allowed us to explore the integrable structures underlying the quantum O(3)/sausage model. Among the obtained results is a system of non-linear integral equations for the computation of the vacuum eigenvalues of the quantum transfer-matrices.


Field Theories in Lower Dimensions Integrable Field Theories Lattice Integrable Models Sigma Models 


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.


  1. [1]
    L.D. Faddeev, E.K. Sklyanin and L.A. Takhtajan, The quantum inverse problem method. 1, Theor. Math. Phys. 40 (1980) 688 [Teor. Mat. Fiz. 40 (1979) 194] [INSPIRE].
  2. [2]
    R.J. Baxter, Partition function of the eight vertex lattice model, Annals Phys. 70 (1972) 193 [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    R.J. Baxter, Exactly solved models in statistical mechanics, Academic Press, London U.K., (1982) [INSPIRE].zbMATHGoogle Scholar
  4. [4]
    C. Destri and H.J. de Vega, Light cone lattice approach to fermionic theories in 2D: the massive Thirring model, Nucl. Phys. B 290 (1987) 363 [INSPIRE].CrossRefGoogle Scholar
  5. [5]
    K. Pohlmeyer, Integrable Hamiltonian systems and interactions through quadratic constraints, Commun. Math. Phys. 46 (1976) 207 [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    V.E. Zakharov and A.V. Mikhailov, Relativistically invariant two-dimensional models in field theory integrable by the inverse problem technique (in Russian), Sov. Phys. JETP 47 (1978) 1017 [Zh. Eksp. Teor. Fiz. 74 (1978) 1953] [INSPIRE].
  7. [7]
    A.M. Polyakov and P.B. Wiegmann, Theory of non-Abelian Goldstone bosons, Phys. Lett. B 131 (1983) 121 [INSPIRE].CrossRefGoogle Scholar
  8. [8]
    L.D. Faddeev and N. Yu. Reshetikhin, Integrability of the principal chiral field model in (1 + 1)-dimension, Annals Phys. 167 (1986) 227 [INSPIRE].MathSciNetCrossRefGoogle Scholar
  9. [9]
    A.B. Zamolodchikov and A.B. Zamolodchikov, Factorized S matrices in two-dimensions as the exact solutions of certain relativistic quantum field models, Annals Phys. 120 (1979) 253 [INSPIRE].MathSciNetCrossRefGoogle Scholar
  10. [10]
    A.G. Bytsko, The zero curvature representation for nonlinear O(3) σ-model, J. Math. Sci. 85 (1994) 1619 [hep-th/9403101] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  11. [11]
    V.V. Bazhanov, S.L. Lukyanov and A.B. Zamolodchikov, Integrable structure of conformal field theory, quantum KdV theory and thermodynamic Bethe ansatz, Commun. Math. Phys. 177 (1996) 381 [hep-th/9412229] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    D. Fioravanti and M. Rossi, A braided Yang-Baxter algebra in a theory of two coupled lattice quantum KdV: algebraic properties and ABA representations, J. Phys. A 35 (2002) 3647 [hep-th/0104002] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  13. [13]
    D. Fioravanti and M. Rossi, Exact conserved quantities on the cylinder 1: conformal case, JHEP 07 (2003) 031 [hep-th/0211094] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  14. [14]
    V.A. Fateev, E. Onofri and A.B. Zamolodchikov, The sausage model (integrable deformations of O(3) σ-model), Nucl. Phys. B 406 (1993) 521 [INSPIRE].CrossRefzbMATHGoogle Scholar
  15. [15]
    S.L. Lukyanov, The integrable harmonic map problem versus Ricci flow, Nucl. Phys. B 865 (2012) 308 [arXiv:1205.3201] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    E.K. Sklyanin, On the complete integrability of the Landau-Lifshitz equation, LOMI-E-79-3, Russia, (1980).
  17. [17]
    L.D. Faddeev and L.A. Takhtajan, Hamiltonian methods in the theory of solitons, Springer-Verlag, Berlin Germany, (1987) [INSPIRE].CrossRefzbMATHGoogle Scholar
  18. [18]
    V.V. Bazhanov, S.L. Lukyanov and A.B. Zamolodchikov, Integrable structure of conformal field theory II. Q-operator and DDV equation, Commun. Math. Phys. 190 (1997) 247 [hep-th/9604044] [INSPIRE].
  19. [19]
    V.V. Bazhanov, S.L. Lukyanov and A.B. Zamolodchikov, Integrable structure of conformal field theory III. The Yang-Baxter relation, Commun. Math. Phys. 200 (1999) 297 [hep-th/9805008] [INSPIRE].
  20. [20]
    A. Voros, Spectral zeta functions, Adv. Stud. Pure Math. 21 (1992) 327.MathSciNetzbMATHGoogle Scholar
  21. [21]
    A. Voros, An exact solution method for 1D polynomial Schrödinger equations, J. Phys. A 32 (1999) 5993 [math-ph/9902016].
  22. [22]
    P. Dorey and R. Tateo, Anharmonic oscillators, the thermodynamic Bethe ansatz and nonlinear integral equations, J. Phys. A 32 (1999) L419 [hep-th/9812211] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  23. [23]
    V.V. Bazhanov, S.L. Lukyanov and A.B. Zamolodchikov, Spectral determinants for Schrödinger equation and Q-operators of conformal field theory, J. Statist. Phys. 102 (2001) 567 [hep-th/9812247] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    J. Suzuki, Functional relations in Stokes multipliers and solvable models related to U q(A n(1)), J. Phys. A 33 (2000) 3507 [hep-th/9910215] [INSPIRE].zbMATHGoogle Scholar
  25. [25]
    V.V. Bazhanov, S.L. Lukyanov and A.B. Zamolodchikov, Higher level eigenvalues of Q-operators and Schrödinger equation, Adv. Theor. Math. Phys. 7 (2003) 711 [hep-th/0307108] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    Al.B. Zamolodchikov, Generalized Mathieu equation and Liouville TBA, in Quantum field theories in two dimensions, collected works of Alexei Zamolodchikov, volume 2, A. Belavin, Ya. Pugai and A. Zamolodchikov eds., World Scientific, Singapore, (2012).Google Scholar
  27. [27]
    S.L. Lukyanov and A.B. Zamolodchikov, Quantum sine(h)-Gordon model and classical integrable equations, JHEP 07 (2010) 008 [arXiv:1003.5333] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    P. Dorey, C. Dunning and R. Tateo, The ODE/IM correspondence, J. Phys. A 40 (2007) R205 [hep-th/0703066] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  29. [29]
    P. Dorey, S. Faldella, S. Negro and R. Tateo, The Bethe ansatz and the Tzitzeica-Bullough-Dodd equation, Phil. Trans. Roy. Soc. Lond. A 371 (2013) 20120052 [arXiv:1209.5517] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    P. Adamopoulou and C. Dunning, Bethe ansatz equations for the classical A n(1) affine Toda field theories, J. Phys. A 47 (2014) 205205 [arXiv:1401.1187] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  31. [31]
    K. Ito and C. Locke, ODE/IM correspondence and Bethe ansatz for affine Toda field equations, Nucl. Phys. B 896 (2015) 763 [arXiv:1502.00906] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  32. [32]
    D. Masoero, A. Raimondo and D. Valeri, Bethe ansatz and the spectral theory of affine Lie algebra-valued connections I. The simply-laced case, Commun. Math. Phys. 344 (2016) 719 [arXiv:1501.07421] [INSPIRE].
  33. [33]
    D. Masoero, A. Raimondo and D. Valeri, Bethe ansatz and the spectral theory of affine Lie algebra-valued connections II. The non simply-laced case, Commun. Math. Phys. 349 (2017) 1063 [arXiv:1511.00895] [INSPIRE].
  34. [34]
    K. Ito and H. Shu, ODE/IM correspondence for modified B 2(1) affine Toda field equation, Nucl. Phys. B 916 (2017) 414 [arXiv:1605.04668] [INSPIRE].CrossRefzbMATHGoogle Scholar
  35. [35]
    C. Babenko and F. Smirnov, Suzuki equations and integrals of motion for supersymmetric CFT, Nucl. Phys. B 924 (2017) 406 [arXiv:1706.03349] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  36. [36]
    R.S. Hamilton, The Ricci flow on surfaces, Contemp. Math. 71, Amer. Math. Soc., Providence RI U.S.A., (1988), pg. 237.Google Scholar
  37. [37]
    S. Elitzur, A. Forge and E. Rabinovici, Some global aspects of string compactifications, Nucl. Phys. B 359 (1991) 581 [INSPIRE].MathSciNetCrossRefGoogle Scholar
  38. [38]
    E. Witten, On string theory and black holes, Phys. Rev. D 44 (1991) 314 [INSPIRE].MathSciNetzbMATHGoogle Scholar
  39. [39]
    Yu. G. Stroganov, A new calculation method for partition functions in some lattice models, Phys. Lett. A 74 (1979) 116 [INSPIRE].MathSciNetCrossRefGoogle Scholar
  40. [40]
    R.J. Baxter and P.A. Pearce, Hard hexagons: interfacial tension and correlation length, J. Phys. A 15 (1982) 897.MathSciNetGoogle Scholar
  41. [41]
    A.N. Kirillov and N.Y. Reshetikhin, Exact solution of the integrable XXZ Heisenberg model with arbitrary spin I. The ground state and the excitation spectrum, J. Phys. A 20 (1987) 1565 [INSPIRE].
  42. [42]
    S.L. Lukyanov, E.S. Vitchev and A.B. Zamolodchikov, Integrable model of boundary interaction: the paperclip, Nucl. Phys. B 683 (2004) 423 [hep-th/0312168] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  43. [43]
    S.L. Lukyanov, A.M. Tsvelik and A.B. Zamolodchikov, Paperclip at θ = π, Nucl. Phys. B 719 (2005) 103 [hep-th/0501155] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  44. [44]
    S.L. Lukyanov and A.B. Zamolodchikov, Integrability in 2D field theory/sigma models, lecture notes for the Les Houches school, to appear, France, (2016).Google Scholar
  45. [45]
    I. Bakas and E. Kiritsis, Beyond the large-N limit: nonlinear W as symmetry of the SL(2, R)/U(1) coset model, Int. J. Mod. Phys. A 7S1A (1992) 55 [hep-th/9109029] [INSPIRE].
  46. [46]
    R. Dijkgraaf, H.L. Verlinde and E.P. Verlinde, String propagation in a black hole geometry, Nucl. Phys. B 371 (1992) 269 [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  47. [47]
    A.B. Zamolodchikov and Al.B. Zamolodchikov, unpublished notes, (1995).Google Scholar
  48. [48]
    A. Gerasimov, A. Marshakov and A. Morozov, Free field representation of parafermions and related coset models, Nucl. Phys. B 328 (1989) 664 [INSPIRE].MathSciNetCrossRefGoogle Scholar
  49. [49]
    V.A. Fateev and A.B. Zamolodchikov, Parafermionic currents in the two-dimensional conformal quantum field theory and selfdual critical points in Z N invariant statistical systems, Sov. Phys. JETP 62 (1985) 215 [Zh. Eksp. Teor. Fiz. 89 (1985) 380] [INSPIRE].
  50. [50]
    V.A. Fateev and A.B. Zamolodchikov, Selfdual solutions of the star triangle relations in Z N models, Phys. Lett. A 92 (1982) 37 [INSPIRE].CrossRefGoogle Scholar
  51. [51]
    G. Felder, BRST approach to minimal models, Nucl. Phys. B 317 (1989) 215 [Erratum ibid. B 324 (1989) 548] [INSPIRE].
  52. [52]
    A.G. Izergin and V.E. Korepin, The lattice quantum sine-Gordon model, Lett. Math. Phys. 5 (1981) 199 [INSPIRE].MathSciNetCrossRefGoogle Scholar
  53. [53]
    E.K. Sklyanin, Some algebraic structures connected with the Yang-Baxter equation. Representations of quantum algebras, Funct. Anal. Appl. 17 (1983) 273 [Funkt. Anal. Pril. 17 (1983) 34] [INSPIRE].
  54. [54]
    V.V. Bazhanov and Yu. G. Stroganov, Chiral Potts model as a descendant of the six vertex model, J. Statist. Phys. 59 (1990) 799 [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  55. [55]
    R.J. Baxter, V.V. Bazhanov and J.H.H. Perk, Functional relations for transfer matrices of the chiral Potts model, Int. J. Mod. Phys. B 4 (1990) 803 [INSPIRE].MathSciNetCrossRefGoogle Scholar
  56. [56]
    E.K. Sklyanin, Separation of variables — new trends, Prog. Theor. Phys. Suppl. 118 (1995) 35 [solv-int/9504001] [INSPIRE].
  57. [57]
    S.L. Lukyanov, ODE/IM correspondence for the Fateev model, JHEP 12 (2013) 012 [arXiv:1303.2566] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  58. [58]
    V.V. Bazhanov and S.L. Lukyanov, Integrable structure of quantum field theory: classical flat connections versus quantum stationary states, JHEP 09 (2014) 147 [arXiv:1310.4390] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  59. [59]
    D. Fioravanti, Geometrical loci and CFTs via the Virasoro symmetry of the mKdV-SG hierarchy: an excursus, Phys. Lett. B 609 (2005) 173 [hep-th/0408079] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  60. [60]
    B. Feigin and E. Frenkel, Quantization of soliton systems and Langlands duality, Adv. Stud. Pure Math. 61 (2011) 185 [arXiv:0705.2486] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  61. [61]
    A. Klumper, M.T. Batchelor and P.A. Pearce, Central charges of the 6- and 19-vertex models with twisted boundary conditions, J. Phys. A 24 (1991) 3111 [INSPIRE].MathSciNetzbMATHGoogle Scholar
  62. [62]
    C. Destri and H.J. de Vega, New approach to thermal Bethe ansatz, Phys. Rev. Lett. 69 (1992) 2313 [hep-th/9203064] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  63. [63]
    C. Destri and H.J. De Vega, Unified approach to thermodynamic Bethe ansatz and finite size corrections for lattice models and field theories, Nucl. Phys. B 438 (1995) 413 [hep-th/9407117] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  64. [64]
    A.B. Zamolodchikov and A.B. Zamolodchikov, Structure constants and conformal bootstrap in Liouville field theory, Nucl. Phys. B 477 (1996) 577 [hep-th/9506136] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  65. [65]
    V.A. Fateev, I.V. Frolov and A.S. Shvarts, Quantum fluctuations of instantons in the nonlinear σ-model, Nucl. Phys. B 154 (1979) 1 [INSPIRE].CrossRefGoogle Scholar
  66. [66]
    A.P. Bukhvostov and L.N. Lipatov, Instanton-anti-instanton interaction in the O(3) nonlinear σ model and an exactly soluble fermion theory, Nucl. Phys. B 180 (1981) 116 [Pisma Zh. Eksp. Teor. Fiz. 31 (1980) 138] [INSPIRE].
  67. [67]
    M. Lüscher, Does the topological susceptibility in lattice σ-models scale according to the perturbative renormalization group?, Nucl. Phys. B 200 (1982) 61 [INSPIRE].CrossRefGoogle Scholar
  68. [68]
    M. Lüscher, Volume dependence of the energy spectrum in massive quantum field theories I. Stable particle states, Commun. Math. Phys. 104 (1986) 177 [INSPIRE].
  69. [69]
    M. Lüscher, Volume dependence of the energy spectrum in massive quantum field theories II. Scattering states, Commun. Math. Phys. 105 (1986) 153 [INSPIRE].
  70. [70]
    C. Ahn, J. Balog and F. Ravanini, NLIE for the sausage model, arXiv:1701.08933 [INSPIRE].
  71. [71]
    A.M. Polyakov, Interaction of Goldstone particles in two-dimensions. Applications to ferromagnets and massive Yang-Mills fields, Phys. Lett. B 59 (1975) 79 [INSPIRE].
  72. [72]
    S. Hikami and E. Brézin, Three loop calculations in the two-dimensional nonlinear σ-model, J. Phys. A 11 (1978) 1141 [INSPIRE].Google Scholar
  73. [73]
    P. Hasenfratz, M. Maggiore and F. Niedermayer, The exact mass gap of the O(3) and O(4) nonlinear σ-models in D = 2, Phys. Lett. B 245 (1990) 522 [INSPIRE].MathSciNetCrossRefGoogle Scholar
  74. [74]
    M. Lüscher, P. Weisz and U. Wolff, A numerical method to compute the running coupling in asymptotically free theories, Nucl. Phys. B 359 (1991) 221 [INSPIRE].CrossRefGoogle Scholar
  75. [75]
    D.-S. Shin, A determination of the mass gap in the O(N ) σ-model, Nucl. Phys. B 496 (1997) 408 [hep-lat/9611006] [INSPIRE].
  76. [76]
    J. Balog and A. Hegedus, The finite size spectrum of the 2-dimensional O(3) nonlinear σ-model, Nucl. Phys. B 829 (2010) 425 [arXiv:0907.1759] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  77. [77]
    G.V. Dunne and M. Ünsal, Resurgence and trans-series in quantum field theory: the CP N −1 model, JHEP 11 (2012) 170 [arXiv:1210.2423] [INSPIRE].
  78. [78]
    A. Dabholkar, Strings on a cone and black hole entropy, Nucl. Phys. B 439 (1995) 650 [hep-th/9408098] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  79. [79]
    V.A. Fateev and A.B. Zamolodchikov, Integrable perturbations of Z N parafermion models and O(3) σ-model, Phys. Lett. B 271 (1991) 91 [INSPIRE].CrossRefGoogle Scholar
  80. [80]
    V.A. Fateev, The σ-model (dual) representation for a two-parameter family of integrable quantum field theories, Nucl. Phys. B 473 (1996) 509 [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  81. [81]
    V.V. Bazhanov, G.A. Kotousov and S.L. Lukyanov, Winding vacuum energies in a deformed O(4) σ-model, Nucl. Phys. B 889 (2014) 817 [arXiv:1409.0449] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  82. [82]
    V.V. Bazhanov, S.L. Lukyanov and B.A. Runov, Vacuum energy of the Bukhvostov-Lipatov model, Nucl. Phys. B 911 (2016) 863 [arXiv:1607.04839] [INSPIRE].CrossRefzbMATHGoogle Scholar
  83. [83]
    V.V. Bazhanov, S.L. Lukyanov and B.A. Runov, Bukhvostov-Lipatov model and quantum-classical duality, arXiv:1711.09021 [INSPIRE].
  84. [84]
    J.M. Maillet, New integrable canonical structures in two-dimensional models, Nucl. Phys. B 269 (1986) 54 [INSPIRE].MathSciNetCrossRefGoogle Scholar
  85. [85]
    F.A. Smirnov, Quasiclassical study of form-factors in finite volume, hep-th/9802132 [INSPIRE].
  86. [86]
    S.L. Lukyanov, Finite temperature expectation values of local fields in the sinh-Gordon model, Nucl. Phys. B 612 (2001) 391 [hep-th/0005027] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  87. [87]
    A.B. Zamolodchikov, On the thermodynamic Bethe ansatz equation in sinh-Gordon model, J. Phys. A 39 (2006) 12863 [hep-th/0005181] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  88. [88]
    A.G. Bytsko and J. Teschner, Quantization of models with non-compact quantum group symmetry: modular XXZ magnet and lattice sinh-Gordon model, J. Phys. A 39 (2006) 12927 [hep-th/0602093] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  89. [89]
    J. Teschner, On the spectrum of the sinh-Gordon model in finite volume, Nucl. Phys. B 799 (2008) 403 [hep-th/0702214] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  90. [90]
    G. Borot, A. Guionnet and K.K. Kozlowski, Asymptotic expansion of a partition function related to the sinh-model, arXiv:1412.7721 [INSPIRE].
  91. [91]
    Y. Ikhlef, J.L. Jacobsen and H. Saleur, An integrable spin chain for the SL(2, R)/U(1) black hole σ-model, Phys. Rev. Lett. 108 (2012) 081601 [arXiv:1109.1119] [INSPIRE].CrossRefGoogle Scholar
  92. [92]
    H. Boos, M. Jimbo, T. Miwa, F. Smirnov and Y. Takeyama, Hidden Grassmann structure in the XXZ model, Commun. Math. Phys. 272 (2007) 263 [hep-th/0606280] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  93. [93]
    H. Boos, M. Jimbo, T. Miwa, F. Smirnov and Y. Takeyama, Hidden Grassmann structure in the XXZ model II: creation operators, Commun. Math. Phys. 286 (2009) 875 [arXiv:0801.1176] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  94. [94]
    M. Jimbo, T. Miwa and F. Smirnov, Hidden Grassmann structure in the XXZ model III: introducing Matsubara direction, J. Phys. A 42 (2009) 304018 [arXiv:0811.0439] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  95. [95]
    H. Boos, M. Jimbo, T. Miwa and F. Smirnov, Hidden Grassmann structure in the XXZ model IV: CFT limit, Commun. Math. Phys. 299 (2010) 825 [arXiv:0911.3731] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  96. [96]
    M. Jimbo, T. Miwa and F. Smirnov, Hidden Grassmann structure in the XXZ model V: sine-Gordon model, Lett. Math. Phys. 96 (2011) 325 [arXiv:1007.0556] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  97. [97]
    P.B. Wiegmann, On the theory of non-Abelian Goldstone bosons in two-dimensions: exact solution of the O(3) nonlinear σ model, Phys. Lett. B 141 (1984) 217 [INSPIRE].CrossRefGoogle Scholar
  98. [98]
    J. Balog and A. Hegedus, TBA equations for excited states in the O(3) and O(4) nonlinear σ-model, J. Phys. A 37 (2004) 1881 [hep-th/0309009] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  99. [99]
    N. Gromov, V. Kazakov and P. Vieira, Finite volume spectrum of 2D field theories from Hirota dynamics, JHEP 12 (2009) 060 [arXiv:0812.5091] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  100. [100]
    A.B. Zamolodchikov and A.B. Zamolodchikov, Massless factorized scattering and σ-models with topological terms, Nucl. Phys. B 379 (1992) 602 [INSPIRE].MathSciNetCrossRefGoogle Scholar
  101. [101]
    B. Feigin, M. Jimbo, T. Miwa and E. Mukhin, Quantum toroidal \( \mathfrak{g}{\mathfrak{l}}_1 \) and Bethe ansatz, J. Phys. A 48 (2015) 244001 [arXiv:1502.07194] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  102. [102]
    M.N. Alfimov and A.V. Litvinov, On spectrum of ILW hierarchy in conformal field theory II: coset CFT’s, JHEP 02 (2015) 150 [arXiv:1411.3313] [INSPIRE].CrossRefGoogle Scholar
  103. [103]
    B. Feigin, M. Jimbo and E. Mukhin, Integrals of motion from quantum toroidal algebras, J. Phys. A 50 (2017) 464001 [arXiv:1705.07984].MathSciNetzbMATHGoogle Scholar
  104. [104]
    C. Klimčík, Integrability of the bi-Yang-Baxter σ-model, Lett. Math. Phys. 104 (2014) 1095 [arXiv:1402.2105] [INSPIRE].
  105. [105]
    B. Hoare, R. Roiban and A.A. Tseytlin, On deformations of AdS n × S n supercosets, JHEP 06 (2014) 002 [arXiv:1403.5517] [INSPIRE].CrossRefGoogle Scholar
  106. [106]
    F. Delduc, M. Magro and B. Vicedo, Integrable double deformation of the principal chiral model, Nucl. Phys. B 891 (2015) 312 [arXiv:1410.8066] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  107. [107]
    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].MathSciNetCrossRefzbMATHGoogle Scholar
  108. [108]
    C. Klimčík, Poisson-Lie T-duals of the bi-Yang-Baxter models, Phys. Lett. B 760 (2016) 345 [arXiv:1606.03016] [INSPIRE].
  109. [109]
    A. Litvinov and L. Spodyneiko, On W algebras commuting with a set of screenings, JHEP 11 (2016) 138 [arXiv:1609.06271] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  110. [110]
    V. Fateev, A. Litvinov and L. Spodyneiko, private communications, to be published.Google Scholar
  111. [111]
    S.L. Lukyanov and A.B. Zamolodchikov, Integrable circular brane model and Coulomb charging at large conduction, J. Stat. Mech. 0405 (2004) P05003 [hep-th/0306188] [INSPIRE].zbMATHGoogle Scholar
  112. [112]
    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].Google Scholar
  113. [113]
    N. Beisert et al., Review of AdS/CFT integrability: an overview, Lett. Math. Phys. 99 (2012) 3 [arXiv:1012.3982] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  114. [114]
    K.B. Efetov, Supersymmetry in disorder and chaos, Cambridge University Press, New York U.S.A., (1997).zbMATHGoogle Scholar
  115. [115]
    G. Albertini, Bethe-ansatz type equations for the Fateev-Zamolodchikov spin model, J. Phys. A 25 (1992) 1799.MathSciNetGoogle Scholar
  116. [116]
    S. Ray, Bethe ansatz study for ground state of Fateev-Zamolodchikov model, J. Math. Phys. 38 (1997) 1524.MathSciNetCrossRefzbMATHGoogle Scholar
  117. [117]
    A.B. Zamolodchikov and V.A. Fateev, Model factorized S matrix and an integrable Heisenberg chain with spin 1 (in Russian), Sov. J. Nucl. Phys. 32 (1980) 298 [Yad. Fiz. 32 (1980) 581] [INSPIRE].

Copyright information

© The Author(s) 2018

Authors and Affiliations

  • Vladimir V. Bazhanov
    • 1
  • Gleb A. Kotousov
    • 1
    • 2
  • Sergei L. Lukyanov
    • 2
    • 3
  1. 1.Department of Theoretical Physics, Research School of Physics and EngineeringAustralian National UniversityCanberraAustralia
  2. 2.NHETC, Department of Physics and AstronomyRutgers UniversityPiscatawayU.S.A.
  3. 3.L.D. Landau Institute for Theoretical PhysicsChernogolovkaRussia

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