Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 4 (2008), 049, 13 pages      arXiv:0802.1776      https://doi.org/10.3842/SIGMA.2008.049

Free Field Approach to Solutions of the Quantum Knizhnik-Zamolodchikov Equations

Kazunori Kuroki a and Atsushi Nakayashiki b
a) Department of Mathematics, Kyushu University, Hakozaki 6-10-1, Fukuoka 812-8581, Japan
b) Department of Mathematics, Kyushu University, Ropponmatsu 4-2-1, Fukuoka 810-8560, Japan

Received February 18, 2008, in final form May 27, 2008; Published online June 03, 2008

Abstract
Solutions of the qKZ equation associated with the quantum affine algebra Uq(^sl2) and its two dimensional evaluation representation are studied. The integral formulae derived from the free field realization of intertwining operators of q-Wakimoto modules are shown to coincide with those of Tarasov and Varchenko.

Key words: free field; vertex operator; qKZ equation; q-Wakimoto module.

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References

  1. Abada A., Bougourzi A.H., El Gradechi M.A., Deformation of the Wakimoto construction, Modern Phys. Lett. A 8 (1993), 715-724, hep-th/9209009.
  2. Awata H., Odake S., Shiraishi J., Free boson realization of Uq(^slN), Comm. Math. Phys. 162 (1994), 61-83, hep-th/9305146.
  3. Awata H., Tsuchiya A., Yamada Y., Integral formulas for the WZNW correlation functions, Nuclear Phys. B 365 (1991), 680-696.
  4. Bougourzi A.H., Weston R.A., Matrix elements of Uq(su(2)k) vertex operators via bosonization, Internat. J. Modern Phys. A 9 (1994), 4431-4447, hep-th/9305127.
  5. Felder G., BRST approach to minimal models, Nuclear Phys. B 317 (1989), 215-236.
  6. Frenkel I.B., Jing N.H., Vertex representations of quantum affine algebras, Proc. Nat. Acad. Sci. U.S.A. 85 (1988), 9373-9377.
  7. Frenkel I.B., Reshetikhin N.Yu., Quantum affine algebras and holonomic difference equations, Comm. Math. Phys. 146 (1992), 1-60.
  8. Idzumi M., Tokihiro T., Iohara K., Jimbo M., Miwa T., Nakashima T., Quantum affine symmetry in vertex models, Internat. J. Modern Phys. A 8 (1993), 1479-1511, hep-th/9208066.
  9. Jimbo M., Miwa T., Algebraic analysis of solvable lattice models, CBMS Regional Conference Series in Mathematics, Vol. 85, American Math. Soc., Providence, RI, 1995.
  10. Kato A., Quano Y.-H., Shiraishi J., Free boson representation of q-vertex operators and their correlation functions, Comm. Math. Phys. 157 (1993), 119-137, hep-th/9209015.
  11. Konno H., BRST cohomology in quantum affine algebra Uq(^sl2), Modern Phys. Lett. A 9 (1994), 1253-1265, hep-th/9310108.
  12. Konno H., Free-field representation of the quantum affine algebra Uq(^sl2) and form factors in the higher-spin XXZ model, Nuclear Phys. B 432 (1994), 457-486, hep-th/9407122.
  13. Konno H., An elliptic algebra Uq,p(^sl2) and the fusion RSOS model, Comm. Math. Phys. 195 (1998), 373-403, q-alg/9709013.
  14. Matsuo A., Quantum algebra structure of certain Jackson integrals, Comm. Math. Phys. 157 (1993), 479-498.
  15. Matsuo A., A q-deformation of Wakimoto modules, primary fields and screening operators, Comm. Math. Phys. 160 (1994), 33-48, hep-th/9212040.
  16. Shiraishi J., Free boson representation of quantum affine algebra, Phys. Lett. A 171 (1992), 243-248.
  17. Shiraishi J., Free boson realization of quantum affine algebras, PhD thesis, University of Tokyo, 1995.
  18. Schechtman V.V., Varchenko A.N., Integral representations of N-point conformal correlators in the WZW model, Max-Planck-Institut fur Mathematik, Preprint MPI/89-51, 1989.
  19. Schechtman V.V., Varchenko A.N., Hypergeometric solutions of Knizhnik-Zamolodchikov equations, Lett. Math. Phys. 20 (1990), 279-283.
  20. Tarasov V., Hypergeometric solutions of the qKZ equation at level zero, Czechoslovak J. Phys. 50 (2000), 193-200.
  21. Varchenko A.N., Tarasov V.O., Jackson integral representations of solutions of the quantized Knizhnik-Zamolodchikov equation, St. Petersburg Math. J. 6 (1995), 275-313, hep-th/9311040.
  22. Tarasov V., Varchenko A., Geometry of q-hypergeometric functions, quantum affine algebras and elliptic quantum groups, Astérisque 246 (1997), 1-135.
  23. Varchenko A., Quantized Knizhnik-Zamolodchikov equations, quantum Yang-Baxter equation, and difference equations for q-hypergeometric functions, Comm. Math. Phys. 162 (1994), 499-528.

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