Communications in Mathematical Physics

, Volume 160, Issue 1, pp 33–48 | Cite as

Aq-deformation of Wakimoto modules, primary fields and screening operators

  • Atsushi Matsuo


Theq-vertex operators of Frenkel and Reshetikhin are studied by means of aq-deformation of the Wakimoto module for the quantum affine algebraU q \((\widehat{\mathfrak{s}\mathfrak{l}}_2 )\) at an arbitrary levelk≠0, −2. A Fock-module version of theq-deformed primary field of spinj is introduced, as well as the screening operators which (anti-)commute with the action ofU q \((\widehat{\mathfrak{s}\mathfrak{l}}_2 )\) up to a total difference of a field. A proof of the intertwining property is given for theq-vertex operators corresponding to the primary fields of spinj∉1/2Z≦0. A sample calculation of the correlation function is also given.


Neural Network Statistical Physic Correlation Function Complex System Nonlinear Dynamics 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [ABE] Abada, A., Bougourzi, A.H., El Gradechi, M.A.: Deformation of the Wakimoto construction. Preprint (1992)Google Scholar
  2. [ATY] Awata, H., Tsuchiya, A., Yamada, Y.: Integral formulas for the WZNW correlation functions. Nucl. Phys.B365, 680 (1991)CrossRefGoogle Scholar
  3. [BF] Bernard, D., Felder, G.: Fock representations and BRST cohomology inSL(2) current algebra. Commun. Math. Phys.127, 145 (1990)CrossRefGoogle Scholar
  4. [Bi] Bilal, A.: Bosonization ofZ N parafermions andsu(2)N Kac-Moody algebra. Phys. Lett.B226, 272 (1989)CrossRefGoogle Scholar
  5. [Bo] Bougourzi, A.H.: Uniqueness of the bosonization of theU q(su(2) k) quantum current algebra. Preprint 1993Google Scholar
  6. [DJMM] Date, E., Jimbo, M., Matsuo, A., Miwa, T.: Hypergeometric type integrals and thesl(2, C) Knizhnik-Zamolodchikov equations. Intern. J. Mod. Phys.B4, 1049 (1990)CrossRefGoogle Scholar
  7. [DFJMN] Davies, B., Foda, O., Jimbo, M., Miwa, T., Nakayashiki, A.: Diagonalization of the XXZ Hamiltonian by vertex operators. Preprint, RIMS-873, 1992Google Scholar
  8. [D1] Drinfeld, V.G.: Hopf algebras and the quantum Yang-Baxter equation. Sov. Math. Dokl.32, 254 (1985)Google Scholar
  9. [D2] Drinfeld, V.G.: A new realization of Yangians and quantum affine algebras. Sov. Math. Dokl.36, 212 (1988)Google Scholar
  10. [FF] Feigin, B., Frenkel, E.: Semi-infinite Weil complex and the Virasoro algebra, Commun. Math. Phys.137, 617 (1991)Google Scholar
  11. [FLMSSx] Frau, M., Lerda, A., McCarthy, J.G., Sciuto, S., Sidenius, J.: Free field representation for\(\widehat{SU(2)}_k \) WZNW models on Riemann surface. Phys. Lett. B245, 453 (1990)CrossRefGoogle Scholar
  12. [FJ] Frenkel, I.B., Jing, N.H.: Vertex representations of quantum affine algebras. Proc. Nat'l Acad. Sci. USA85, 9373 (1988)Google Scholar
  13. [FR] Frenkel, I.B., Reshetikhin, N.Yu.: Quantum affine algebras and holonomic difference equations. Commun. Math. Phys.146, 1 (1992)Google Scholar
  14. [FMS] Friedan, D., Martinec, E., Shenker, S.: Conformal invariance, supersymmetry and string theory. Nucl. Phys.B271, 93 (1986)CrossRefGoogle Scholar
  15. [IIJMNT] Idzumi, M. Iohara, K., Jimbo, M., Miwa, T., Nakashima, T., Tokihiro, T.: Quantum affine symmetry in vertex models. Preprint 1992Google Scholar
  16. [I] Ito, K.:N=2 super Coulomb gas formalism. Nucl. Phys.B 332, 566 (1990)CrossRefGoogle Scholar
  17. [JNS] Jayaraman, T., Narain, K.S., Sarmadi, M.H.:SU(2)k WZW andZ k parafermion models on the torus. Nucl. PhysB 343, 418 (1990)CrossRefGoogle Scholar
  18. [J1] Jimbo, M.: Aq-difference analogue ofU(g) and the Yang-Baxter equation. Lett. Math. Phys.10, 63 (1985)CrossRefGoogle Scholar
  19. [J2] Jimbo, M.: Lecture at University of TokyoGoogle Scholar
  20. [JMMN] Jimbo, M., Miki, K., Miwa, T., Nakayashiki, A.: Correlation functions of the XXZ model for Δ<−1. Preprint, RIMS-877, 1992Google Scholar
  21. [KQS] Kato, A., Quano, Y., Shiraishi, J.: Free boson representation ofq-vertex operators and their correlation functions. Preprint, UT-618, 1992Google Scholar
  22. [Ki] Kimura, K.: On free boson representation of quantum affine algebraU q \((\widehat{\mathfrak{s}\mathfrak{l}}_2 )\). Preprint 1992Google Scholar
  23. [KZ] Knizhnik, V.G., Zamolodchikov, A.B.: Current algebra and Wess-Zumino models in two dimensions. Nucl. Phys.B247, 83 (1984)CrossRefGoogle Scholar
  24. [Ku] Kuroki, G.: Fock space representations of affine Lie algebras and integral representations in the Wess-Zumino-Witten model. Commun. Math. Phys.142, 511 (1991)CrossRefGoogle Scholar
  25. [L] Lusztig, G.: Quantum deformation of certain simple modules over enveloping algebras. Adv. Math.70, 237 (1988)CrossRefGoogle Scholar
  26. [M1] Matsuo, A.: Jackson integrals of Jordan-Pochhammer type and quantum Knizhnik-Zamolodchikov equations. Commun. Math. Phys.151, 263 (1993)CrossRefGoogle Scholar
  27. [M2] Matsuo, A.: Quantum algebra structure of certain Jackson integrals. Preprint, 1992Google Scholar
  28. [M3] Matsuo, A.: Free field representation of the quantum affine algebraU q \((\widehat{\mathfrak{s}\mathfrak{l}}_2 )\). Preprint, 1992Google Scholar
  29. [N] Nemeschansky, D.: Feigin-Fuchs representation of\(\widehat{su}(2)_k \) Kac-Moody algebra. Phys. Lett.B 224, 121 (1989)CrossRefGoogle Scholar
  30. [R] Reshetikhin, N.Yu.: Jackson type integrals, Bethe vectors, and solutions to a difference analog of the Knizhnik-Zamolodchikov system. Lett. Math. Phys.26, 153 (1992), Preprint 1992CrossRefGoogle Scholar
  31. [S] Shiraishi, J.: Free boson representation ofU q \((\widehat{\mathfrak{s}\mathfrak{l}}_2 )\). Preprint, UT-617, 1992Google Scholar
  32. [TK] Tsuchiya, A., Kanie, K.: Vertex operators in conformal field theory onP 1 and monodromy representations of braid groups. Adv. Stud. Pure. Math.16, 297 (1988)Google Scholar
  33. [W] Wakimoto, M.: Fock representation of the affine Lie algebraA 1(1). Commun. Math. Phys.104, 605 (1986)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 1994

Authors and Affiliations

  • Atsushi Matsuo
    • 1
  1. 1.Department of MathematicsNagoya UniversityNagoyaJapan

Personalised recommendations