Quantum sine(h)-Gordon model and classical integrable equations

  • S. L. Lukyanov
  • A. B. Zamolodchikov


We study a family of classical solutions of modified sinh-Gordon equation, \( {\partial_z}{\partial_{\bar{z}}}\eta - {{\text{e}}^{2\eta }} + p(z)p\left( {\bar{z}} \right)\,{{\text{e}}^{ - 2\eta }} = 0 \) with p(z) = z 2α s 2α . We show that certain connection coefficients for solutions of the associated linear problem coincide with the Q-function of the quantum sine-Gordon (α > 0) or sinh-Gordon (α < −1) models.


Field Theories in Lower Dimensions Integrable Equations in Physics Bethe Ansatz Integrable Field Theories 


  1. [1]
    T.T. Wu, B.M. McCoy, C.A. Tracy and E. Barouch, Spin spin correlation functions for the two-dimensional Ising model: exact theory in the scaling region, Phys. Rev. B 13 (1976) 316 [SPIRES].ADSGoogle Scholar
  2. [2]
    P. Fendley and H. Saleur, N = 2 supersymmetry, Painleve III and exact scaling functions in 2D polymers, Nucl. Phys. B 388 (1992) 609 [hep-th/9204094] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  3. [3]
    A.B. Zamolodchikov, Painleve III and 2D polymers, Nucl. Phys. B 432 (1994) 427 [hep-th/9409108] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  4. [4]
    S. Cecotti, P. Fendley, K.A. Intriligator and C. Vafa, A new supersymmetric index, Nucl. Phys. B 386 (1992) 405 [hep-th/9204102] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  5. [5]
    S. Cecotti and C. Vafa, Topological antitopological fusion, Nucl. Phys. B 367 (1991) 359 [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  6. [6]
    P.P. Kulish, Factorization of the classical and quantum s matrix and conservation laws, Theor. Math. Phys. 26 (1976) 132 [Teor. Mat. Fiz. 26 (1976) 198] [SPIRES].CrossRefGoogle Scholar
  7. [7]
    C.-N. Yang and C.P. Yang, Thermodynamics of a one-dimensional system of bosons with repulsive delta-function interaction, J. Math. Phys. 10 (1969) 1115 [SPIRES].zbMATHCrossRefADSGoogle Scholar
  8. [8]
    A.B. Zamolodchikov, On the thermodynamic Bethe ansatz equations for reflectionless ADE scattering theories, Phys. Lett. B 253 (1991) 391 [SPIRES].MathSciNetADSGoogle Scholar
  9. [9]
    C. Destri and H.J. de Vega, New thermodynamic Bethe ansatz equations without strings, Phys. Rev. Lett. 69 (1992) 2313 [SPIRES].zbMATHCrossRefMathSciNetADSGoogle Scholar
  10. [10]
    C. Destri and H.J. de Vega, Non-linear integral equation and excited-states scaling functions in the sine-Gordon model, Nucl. Phys. B 504 (1997) 621 [hep-th/9701107] [SPIRES].CrossRefADSGoogle Scholar
  11. [11]
    A. Klm̈per, M. Bathcelor and P.A. Pearce, Central charges of the 6- and 19-vertex models with twisted boundary conditions, J. Phys. A 24 (1991) 3111.ADSGoogle Scholar
  12. [12]
    D. Bernard, 2 and A. Leclair, Quantum group symmetries and nonlocal currents in 2D QFT, Commun. Math. Phys. 142 (1991) 99 [SPIRES].zbMATHCrossRefMathSciNetADSGoogle Scholar
  13. [13]
    A.I. Babenko, All constant mean curvature tori in R 3 , S 3 , H 3 in terms of theta-functions, Math. Ann. 290 (1991) 209.CrossRefMathSciNetGoogle Scholar
  14. [14]
    D. Gaiotto, G.W. Moore and A. Neitzke, Four-dimensional wall-crossing via three-dimensional field theory, arXiv:0807.4723 [SPIRES].
  15. [15]
    D. Gaiotto, G.W. Moore and A. Neitzke, Wall-crossing, Hitchin systems and the WKB approximation, arXiv:0907.3987 [SPIRES].
  16. [16]
    L.F. Alday and J. Maldacena, Null polygonal Wilson loops and minimal surfaces in anti-de-Sitter space, JHEP 11 (2009) 082 [arXiv:0904.0663] [SPIRES].CrossRefADSGoogle Scholar
  17. [17]
    L.F. Alday, D. Gaiotto and J. Maldacena, Thermodynamic Bubble ansatz, arXiv:0911.4708 [SPIRES].
  18. [18]
    L.F. Alday, J. Maldacena, A. Sever and P. Vieira, Y-system for scattering amplitudes, arXiv:1002.2459 [SPIRES].
  19. [19]
    L.D. Faddeev and L.A. Takhtajan, Hamiltonian methods in the theory of solitons, Springer Series in Soviet Mathematics, Springer, Berlin Germany (1987) [SPIRES].zbMATHGoogle Scholar
  20. [20]
    P. Dorey and R. Tateo, Anharmonic oscillators, the thermodynamic Bethe ansatz and nonlinear integral equations, J. Phys. A 32 (1999) L419 [hep-th/9812211] [SPIRES].MathSciNetADSGoogle Scholar
  21. [21]
    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] [SPIRES].zbMATHCrossRefMathSciNetADSGoogle Scholar
  22. [22]
    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] [SPIRES].zbMATHCrossRefMathSciNetADSGoogle Scholar
  23. [23]
    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] [SPIRES].zbMATHCrossRefMathSciNetADSGoogle Scholar
  24. [24]
    A. Voros, Spectral zeta functions, Adv. Stud. Pure Math. 21 (1992) 327.MathSciNetGoogle Scholar
  25. [25]
    A. Voros, Exact resolution method for general 1D polynomial Schrödinger equation, J. Phys. A 32 (1999) 5993 [math-ph/9902016] [SPIRES].MathSciNetADSGoogle Scholar
  26. [26]
    V.V. Bazhanov, S.L. Lukyanov and A.B. Zamolodchikov, Spectral determinants for Schrödinger equation and Q-operators of conformal field theory, J. Stat. Phys. 102 (2001) 567 [hep-th/9812247] [SPIRES].zbMATHCrossRefMathSciNetGoogle Scholar
  27. [27]
    A.B. Zamolodchikov, On the thermodynamic Bethe ansatz equation in sinh-Gordon model, J. Phys. A 39 (2006) 12863 [hep-th/0005181] [SPIRES].MathSciNetADSGoogle Scholar
  28. [28]
    S.L. Lukyanov, Finite temperature expectation values of local fields in the sinh-Gordon model, Nucl. Phys. B 612 (2001) 391 [hep-th/0005027] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  29. [29]
    J. Teschner, On the spectrum of the sinh-Gordon model in finite volume, Nucl. Phys. B 799 (2008) 403 [hep-th/0702214] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  30. [30]
    P. Dorey and R. Tateo, On the relation between Stokes multipliers and the T-Q systems of conformal field theory, Nucl. Phys. B 563 (1999) 573 [Erratum ibid. B 603 (2001) 581] [hep-th/9906219] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  31. [31]
    V.V. Bazhanov, S.L. Lukyanov and A.B. Zamolodchikov, Higher-level eigenvalues of Q-operators and Schrödinger equation, Adv. Theor. Math. Phys. 7 (2004) 711 [hep-th/0307108] [SPIRES].MathSciNetGoogle Scholar
  32. [32]
    R.J. Baxter, Generalized ferroelectric model on a square lattice, Stud. Appl. Math. 50 (1971) 51 [SPIRES].MathSciNetGoogle Scholar
  33. [33]
    R.J. Baxter, Exactly solved models in statistical mechanics, Academic Press Inc., London U.K. (1982).zbMATHGoogle Scholar
  34. [34]
    E.K. Sklyanin, L. Takhtajan and L.D. Faddeev, The quantum inverse problem method. I, Theor. Math. Phys. 40 (1979) 688 [Teor. Mat. Fiz. 40 (1979) 194] [SPIRES].CrossRefGoogle Scholar
  35. [35]
    N.Y. Reshetikhin, A method of functional equations in the theory of exactly solvable quantum systems, Lett. Math. Phys. 7 (1983) 205 [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  36. [36]
    A.B. Zamolodchikov, Mass scale in the sine-Gordon model and its reductions, Int. J. Mod. Phys. A 10 (1995) 1125 [SPIRES].ADSGoogle Scholar
  37. [37]
    E.K. Sklyanin, The quantum Toda chain, Lect. Notes Phys. 226 (1985) 196 [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  38. [38]
    E.K. Sklyanin, Quantum inverse scattering method. Selected topics, in Quantum groups and quantum integrable systems, World Scientific, U.S.A. (1992), pg. 63 [hep-th/9211111] [SPIRES].Google Scholar
  39. [39]
    F.A. Smirnov, Quasi-classical study of form factors in finite volume, hep-th/9802132 [SPIRES].
  40. [40]
    I.M. Gel’fand and L.A. Dikii, Asymptotic behaviour of the resolvent of Sturm-Liouville equations and the algebra of the Korteweg-De Vries equations, Russ. Math. Surveys 30 (1975) 77 [Usp. Mat. Nauk 30 (1975) 67] [SPIRES].zbMATHCrossRefMathSciNetADSGoogle Scholar
  41. [41]
    V.V. Bazhanov, S.L. Lukyanov and A.B. Zamolodchikov, Quantum field theories in finite volume: excited state energies, Nucl. Phys. B 489 (1997) 487 [hep-th/9607099] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  42. [42]
    M. Takahashi and M. Suzuki, One-dimensional anisotropic Heisenberg model at finite temperatures, Prog. Theor. Phys. 48 (1972) 2187 [SPIRES].CrossRefADSGoogle Scholar
  43. [43]
    B.M. McCoy, C.A. Tracy and T.T. Wu, Painleve functions of the third kind, J. Math. Phys. 18 (1977) 1058 [SPIRES].zbMATHCrossRefMathSciNetADSGoogle Scholar
  44. [44]
    A.B. Zamolodchikiv, unpublished notes (2001).Google Scholar
  45. [45]
    V.A. Fateev and S.L. Lukyanov, Boundary RG flow associated with the AKNS soliton hierarchy, J. Phys. A 39 (2006) 12889 [hep-th/0510271] [SPIRES].MathSciNetADSGoogle Scholar
  46. [46]
    G. Feverati, F. Ravanini and G. Tak’acs, Nonlinear integral equation and finite volume spectrum of sine-Gordon theory, Nucl. Phys. B 540 (1999) 543 [hep-th/9805117] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  47. [47]
    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] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  48. [48]
    M. Jimbo, T. Miwa and F. Smirnov, On one-point functions of descendants in sine-Gordon model, arXiv:0912.0934 [SPIRES].

Copyright information

© SISSA, Trieste, Italy 2010

Authors and Affiliations

  1. 1.NHETC, Department of Physics and AstronomyRutgers UniversityPiscatawayU.S.A.
  2. 2.L.D. Landau Institute for Theoretical PhysicsChernogolovka, Moscow regionRussia

Personalised recommendations