Journal of High Energy Physics

, 2013:95 | Cite as

On form factors and Macdonald polynomials

  • Michael LashkevichEmail author
  • Yaroslav Pugai


We are developing the algebraic construction for form factors of local operators in the sinh-Gordon theory proposed in [1]. We show that the operators corresponding to the null vectors in this construction are given by the degenerate Macdonald polynomials with rectangular partitions and the parameters t = −q on the unit circle. We obtain an integral representation for the null vectors and discuss its simple applications.


Integrable Field Theories Quantum Groups Exact S-Matrix 


  1. [1]
    B. Feigin and M. Lashkevich, Form factors of descendant operators: free field construction and reflection relations, J. Phys. A 42 (2009) 304014 [arXiv:0812.4776] [INSPIRE].MathSciNetGoogle Scholar
  2. [2]
    A. Belavin, A.M. Polyakov and A. Zamolodchikov, Infinite conformal symmetry in two-dimensional quantum field theory, Nucl. Phys. B 241 (1984) 333 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  3. [3]
    A.B. Zamolodchikov and Al.B. Zamolodchikov, Structure constants and conformal bootstrap in Liouville field theory, Nucl. Phys. B 477 (1996) 577 [hep-th/9506136] [INSPIRE].
  4. [4]
    H. Awata, Y. Matsuo, S. Odake and J. Shiraishi, Collective field theory, Calogero-Sutherland model and generalized matrix models, Phys. Lett. B 347 (1995) 49 [hep-th/9411053] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  5. [5]
    Al. Zamolodchikov, Two point correlation function in scaling Lee-Yang model, Nucl. Phys. B 348 (1991) 619 [INSPIRE].
  6. [6]
    Al. Zamolodchikov, Higher equations of motion in Liouville field theory, Int. J. Mod. Phys. A 19S2 (2004) 510 [hep-th/0312279] [INSPIRE].
  7. [7]
    M. Lashkevich, Resonances in sinh- and sine-Gordon models and higher equations of motion in Liouville theory, J. Phys. A 45 (2012) 455403 [arXiv:1111.2547] [INSPIRE].MathSciNetADSGoogle Scholar
  8. [8]
    S.L. Lukyanov and A.B. Zamolodchikov, Exact expectation values of local fields in quantum sine-Gordon model, Nucl. Phys. B 493 (1997) 571 [hep-th/9611238] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  9. [9]
    M. Karowski and P. Weisz, Exact form-factors in (1 + 1)-dimensional field theoretic models with soliton behavior, Nucl. Phys. B 139 (1978) 455 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  10. [10]
    F.A. Smirnov, Form-factors in completely integrable models of quantum field theory, Adv. Ser. Math. Phys. 14 (1992) 1 [INSPIRE].CrossRefGoogle Scholar
  11. [11]
    A. Koubek and G. Mussardo, On the operator content of the sinh-Gordon model, Phys. Lett. B 311 (1993) 193 [hep-th/9306044] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  12. [12]
    S.L. Lukyanov, Form-factors of exponential fields in the sine-Gordon model, Mod. Phys. Lett. A 12 (1997) 2543 [hep-th/9703190] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  13. [13]
    J.L. Cardy and G. Mussardo, Form-factors of descendent operators in perturbed conformal field theories, Nucl. Phys. B 340 (1990) 387 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  14. [14]
    A. Koubek, A method to determine the operator content of perturbed conformal field theories, Phys. Lett. B 346 (1995) 275 [hep-th/9501028] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  15. [15]
    O. Babelon, D. Bernard and F. Smirnov, Quantization of solitons and the restricted sine-Gordon model, Commun. Math. Phys. 182 (1996) 319 [hep-th/9603010] [INSPIRE].MathSciNetADSCrossRefzbMATHGoogle Scholar
  16. [16]
    O. Babelon, D. Bernard and F. Smirnov, Null vectors in integrable field theory, Commun. Math. Phys. 186 (1997) 601 [hep-th/9606068] [INSPIRE].MathSciNetADSCrossRefzbMATHGoogle Scholar
  17. [17]
    M. Jimbo, T. Miwa and Y. Takeyama, Counting minimal form-factors of the restricted sine-Gordon model, Moscow Math. J. 4 (2004) 787 [math-ph/0303059] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  18. [18]
    M. Jimbo, T. Miwa, E. Mukhin and Y. Takeyama, Form-factors and action of \( \mathrm{U}\left( {-{1^{{{1 \left/ {2} \right.}}}}} \right)\left( {\widetilde{\mathrm{sl}}(2)} \right) \) on infinite cycles, Commun. Math. Phys. 245 (2004) 551 [math/0305323] [INSPIRE].MathSciNetADSCrossRefzbMATHGoogle Scholar
  19. [19]
    G. Delfino and G. Niccoli, Form-factors of descendant operators in the massive Lee-Yang model, J. Stat. Mech. 0504 (2005) P04004 [hep-th/0501173] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  20. [20]
    G. Delfino and G. Niccoli, Isomorphism of critical and off-critical operator spaces in two-dimensional quantum field theory, Nucl. Phys. B 799 (2008) 364 [arXiv:0712.2165] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  21. [21]
    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].MathSciNetADSCrossRefzbMATHGoogle Scholar
  22. [22]
    M. Jimbo, T. Miwa and F. Smirnov, Fermionic structure in the sine-Gordon model: form factors and null-vectors, Nucl. Phys. B 852 (2011) 390 [arXiv:1105.6209] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  23. [23]
    A.A. Belavin, V.A. Belavin, A.V. Litvinov, Y.P. Pugai and Al.B. Zamolodchikov, On correlation functions in the perturbed minimal models M (2, 2n + 1), Nucl. Phys. B 676 (2004) 587 [hep-th/0309137] [INSPIRE].
  24. [24]
    V.A. Fateev, V.V. Postnikov and Y.P. Pugai, On scaling fields in Z N Ising models, JETP Lett. 83 (2006) 172 [hep-th/0601073] [INSPIRE].ADSCrossRefGoogle Scholar
  25. [25]
    V.A. Fateev and Y.P. Pugai, Correlation functions of disorder fields and parafermionic currents in Z N Ising models, J. Phys. A 42 (2009) 304013 [arXiv:0909.3347] [INSPIRE].MathSciNetGoogle Scholar
  26. [26]
    A. Fring, G. Mussardo and P. Simonetti, Form-factors for integrable Lagrangian field theories, the sinh-Gordon theory, Nucl. Phys. B 393 (1993) 413 [hep-th/9211053] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  27. [27]
    J. Shiraishi, H. Kubo, H. Awata and S. Odake, A quantum deformation of the Virasoro algebra and the Macdonald symmetric functions, Lett. Math. Phys. 38 (1996) 33 [q-alg/9507034] [INSPIRE].MathSciNetADSCrossRefzbMATHGoogle Scholar
  28. [28]
    S. Lukyanov and Y. Pugai, Bosonization of ZF algebras: direction toward deformed Virasoro algebra, J. Exp. Theor. Phys. 82 (1996) 1021 [hep-th/9412128] [INSPIRE].ADSGoogle Scholar
  29. [29]
    G. Delfino and G. Niccoli, The composite operator \( T\overline{T} \) in sinh-Gordon and a series of massive minimal models, JHEP 05 (2006) 035 [hep-th/0602223] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  30. [30]
    I.G. Macdonald, Symmetric functions and hall polynomials, 2nd edition, Oxford University Press, Oxford U.K. (1995).zbMATHGoogle Scholar

Copyright information

© SISSA, Trieste, Italy 2013

Authors and Affiliations

  1. 1.Landau Institute for Theoretical PhysicsChernogolovkaRussia
  2. 2.Moscow Institute of Physics and TechnologyDolgoprudny of Moscow RegionRussia

Personalised recommendations