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Journal of Statistical Physics

, Volume 102, Issue 3–4, pp 883–921 | Cite as

Free Field Construction for the ABF Models in Regime II

  • Michio Jimbo
  • Hitoshi Konno
  • Satoru Odake
  • Yaroslav Pugai
  • Jun'ichi Shiraishi
Article

Abstract

The Wakimoto construction for the quantum affine algebra U\(_q\)(\((\widehat{\mathfrak{s}\mathfrak{l}_2 })\)) admits a reduction to the q-deformed parafermion algebras. We interpret the latter theory as a free field realization of the Andrews–Baxter–Forrester models in regime II. We give multi-particle form factors of some local operators on the lattice and compute their scaling limit, where the models are described by a massive field theory with \(\mathbb{Z}\)\(_k\) symmetric minimal scattering matrices.

integrable lattice model ABF model in regime II free field construction vertex operator approach q-deformed parafermion form factor \(\mathbb{Z}\)\(_k\) symmetric scattering deformed W algebra 

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REFERENCES

  1. 1.
    Vl. S. Dotsenko and V. A. Fateev, Conformal algebra and multipoint correlation functions in 2D statistical models, Nucl. Phys. B 240(FS12):312–348 (1984).Google Scholar
  2. 2.
    M. Jimbo and T. Miwa, Algebraic Analysis of Solvable Lattice Models, CBMS Regional Conference Series in Mathematics, Vol. 85 (AMS, 1994).Google Scholar
  3. 3.
    S. Lukyanov and Y. Pugai, Multi-point local height probabilities in the integrable RSOS model, Nucl. Phys. B 473(FS):631–658 (1996).Google Scholar
  4. 4.
    G. E. Andrews, R. J. Baxter, and P. J. forrester, Eight-vertex SOS model and generalized Rogers-Ramanujan-type identities, J. Stat. Phys. 35:193–266 (1984).Google Scholar
  5. 5.
    M. Jimbo, T. Miwa, and M. Okado, Solvable lattice models whose states are dominant integral weights of A n−1(1), Lett. Math. Phys. 14:123–131 (1987).Google Scholar
  6. 6.
    Y. Asai, M. Jimbo, T. Miwa, and Y. Pugai, Bosonization of vertex operators for the A n−1(1) face model, J. Phys. A 29:6595–6616 (1996).Google Scholar
  7. 7.
    B. Feigin, M. Jimbo, T. Miwa, A. Odesskii, and Y. Pugai, Algebra of screening operators for the deformed W n algebra, Commun. Math. Phys. 191:501–541 (1998).Google Scholar
  8. 8.
    A. B. Zamolodchikov, and V. A. Fateev, Nonlocal (parafermion) currents in two-dimensional conformal quantum field theory and self-dual critical points in ℤN-symmetric statistical models, Sov. Phys. JETP 62(2):215–225 (1985).Google Scholar
  9. 9.
    A. B. Zamolodchikov, Integrals of motion in scaling 3–state Potts model field theory, Int. J. Mod. Phys. A 3:743–750 (1988).Google Scholar
  10. 10.
    A. M. Tsvelik, The exact solution of 2D Z N invariant statistical models, Nucl. Phys. B 305:675–684 (1988).Google Scholar
  11. 11.
    V. V. Bazhanov and N. Yu. Reshetikhin, Scattering amplitudes in off-critical models and RSOS integrable models, Prog. Theor. Phys. Supplement 102:301–318 (1990).Google Scholar
  12. 12.
    V. A. Fateev, Integrable deformations in ℤN-symmetrical models of the conformal quantum field theory, Int. J. Mod. Phys. A 6:2109–2132 (1991).Google Scholar
  13. 13.
    T. R. Klassen and E. Melzer, Purely elastic scattering theories and their ultraviolet limits, Nucl. Phys. B 338:485–528 (1990).Google Scholar
  14. 14.
    A. Matsuo, A q-deformation of Wakimoto modules, primary fields and screening operators, Comm. Math. Phys. 161:33–48 (1994).Google Scholar
  15. 15.
    H. Konno, An elliptic algebra \(U_{p,q} \left( {\widehat{\mathfrak{s}}{\mathfrak{l}_2 }} \right)\) and the fusion RSOS model, Comm. Math. Phys. 195:373–403 (1998).Google Scholar
  16. 16.
    H. Awata, H. Kubo, S. Odake, and J. Shiraishi, Quantum \({\mathcal{W}}_N \) algebras and Macdonald polynomials, Comm. Math. Phys. 179:401–416 (1996).Google Scholar
  17. 17.
    B. L. Feigin and E. V. Frenkel, Quantum \({\mathcal{W}}\)-algebras and elliptic algebras, Comm. Math. Phys. 178:653–678 (1996).Google Scholar
  18. 18.
    O. Foda, M. Jimbo, T. Miwa, K. Miki, and A. Nakayashiki, Vertex operators in solvable lattice models, J. Math. Phys. 35:13–46 (1994).Google Scholar
  19. 19.
    V. G. Kac and D. H. Peterson, Infinite-dimensional Lie algebras, theta functions and modular forms, Adv. in Math. 53:125–264 (1984).Google Scholar
  20. 20.
    D. Bernard and G. Felder, Fock representations and BRST cohomology in SL(2) current algebra, Comm. Math. Phys. 127:145–168 (1990).Google Scholar
  21. 21.
    M. Jimbo, M. Lashkevich, T. Miwa, and Y. Pugai, Lukyanov's screening operators for the deformed Virasoro algebra, Phys. Lett. A 229:285–292 (1997).Google Scholar
  22. 22.
    R. Köberle and J. A. Swieca, Factorizable Z(N) models, Phys. Lett. B 86:209–210 (1979).Google Scholar
  23. 23.
    S. Lukyanov, Free field representation for massive integrable models, Comm. Math. Phys. 167:183–226 (1995).Google Scholar
  24. 24.
    V. Brazhnikov and S. Lukyanov, Angular quantization and form factors in massive integrable models, Nucl. Phys. B 512:616–636 (1998).Google Scholar
  25. 25.
    T. Oota, Functional equations of form factors for diagonal scattering theories, Nucl. Phys. B 466:361–382 (1996).Google Scholar
  26. 26.
    S. Lukyanov, Form-factors of exponential fields in the affine \(A_{N - 1}^{\left( 1 \right)} \) toda models, Phys. Lett. B 408:192–200 (1997).Google Scholar
  27. 27.
    T. Jayaraman, K. S. Narain, and M. H. Sarmadi, SU(2)k WZW and ℤk parafermion models on the torus, Nucl. Phys. B 343:418–449 (1990).Google Scholar
  28. 28.
    M. Karowski and P. Weisz, Exact form factors in (1 + 1)-dimensional field theoretic models with soliton behavior, Nucl. Phys. B 139:455–476 (1978).Google Scholar
  29. 29.
    S. Lukyanov, Form-factors of exponential fields in the sine-Gordon model, Mod. Phys. Lett. A 12:2543–2550 (1997).Google Scholar
  30. 30.
    M. Lashkevich, Scaling limit of the six vertex model in the framework of free field representation, JHEP 9710 003 (1997).Google Scholar
  31. 31.
    Al. B. Zamolodchikov, Two-point correlation function in scaling Lee-Yang model, Nucl. Phys. B 348:619–641 (1991).Google Scholar
  32. 32.
    A. N. Kirillov and F. A. Smirnov, ITF preprint, ITF-88–73R, Kiev (1988).Google Scholar
  33. 33.
    F. A. Smirnov, Quantum groups and generalized statistics in integrable models, Comm. Math. Phys. 132:415–439 (1990).Google Scholar

Copyright information

© Plenum Publishing Corporation 2001

Authors and Affiliations

  • Michio Jimbo
    • 1
  • Hitoshi Konno
    • 2
  • Satoru Odake
    • 3
  • Yaroslav Pugai
    • 4
  • Jun'ichi Shiraishi
    • 5
  1. 1.Division of Mathematics, Graduate School of ScienceKyoto UniversityKyotoJapan
  2. 2.Department of Mathematics, Faculty of Integrated Arts and SciencesHiroshima UniversityHigashi-HiroshimaJapan
  3. 3.Department of Physics, Faculty of ScienceShinshu UniversityMatsumotoJapan
  4. 4.Research Institute for Mathematical SciencesKyoto UniversityKyotoJapan
  5. 5.Graduate School of Mathematical SciencesThe University of TokyoTokyoJapan

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