Communications in Mathematical Physics

, Volume 336, Issue 2, pp 953–986 | Cite as

Boundary Quantum Knizhnik–Zamolodchikov Equations and Bethe Vectors

  • Nicolai Reshetikhin
  • Jasper Stokman
  • Bart Vlaar


Solutions to boundary quantum Knizhnik–Zamolodchikov equations are constructed as bilateral sums involving “off-shell” Bethe vectors in case the reflection matrix is diagonal and only the 2-dimensional representation of \({{\mathcal{U}}_q(\widehat{{\mathfrak{sl}}}_2)}\) is involved. We also consider their rational and classical degenerations.


Meromorphic Function Vertex Operator Spin Chain Meromorphic Solution Monodromy Operator 
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  1. 1.
    Babujian, H.M., Flume, R.: Off-shell Bethe ansatz equation for Gaudin magnets and solutions of Knizhnik–Zamolodchikov equations. Modern Phys. Lett. A 9(22), 2029–2039 (1994)CrossRefADSzbMATHMathSciNetGoogle Scholar
  2. 2.
    Bailey, W.N.: Series of hypergeometric type which are infinite in both directions. Quart. J. Math. (Oxford) 7, 105–115 (1936)CrossRefADSGoogle Scholar
  3. 3.
    Bajnok, Z., Palla, L., Takács, G.: On the boundary form factor program. Nucl. Phys. B 750(3), 179–212 (2006)CrossRefADSzbMATHGoogle Scholar
  4. 4.
    Baxter, R.J.: Exactly Solved Models in Statistical Mechanics. Academic Press, Inc., London (1982)zbMATHGoogle Scholar
  5. 5.
    Buchstaber, V.M., Felder, G., Veselov, A.P.: Elliptic Dunkl operators, root systems, and functional equations. Duke Math. J. 76(3), 885–911 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Cardy, J.: Conformal invariance and surface critical behavior. Nucl. Phys. B 240, 514–532 (1984)CrossRefADSGoogle Scholar
  7. 7.
    Cherednik, I.: Quantum Knizhnik–Zamolodchikov equations and affine root systems. Comm. Math. Phys. 150, 109–136 (1992)CrossRefADSzbMATHMathSciNetGoogle Scholar
  8. 8.
    Cherednik, I.: Factorizing particles on a half line, and root systems. Theoret. Math. Phys. 61(1), 977–983 (1984)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Cherednik, I.: Monodromy representations for the generalized KZ equations and Hecke algebras. Publ. Res. Inst. Math. Sci. Kyoto Univ. 27, 711–726 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Cherednik, I.: A unification of Knizhnik–Zamolodchikov and Dunkl operators via affine Hecke algebras. Invent. Math. 106, 411–431 (1991)CrossRefADSzbMATHMathSciNetGoogle Scholar
  11. 11.
    Cherednik, I.: Double affine Hecke algebras, Knizhnik–Zamolodchikov equations, and Macdonald’s operators. Int. Math. Res. Not. 9 171–180 (1992)Google Scholar
  12. 12.
    Cherednik, I.: Induced representations of double affine Hecke algebras and applications Math. Res. Lett. 1, 319–337 (1994)zbMATHMathSciNetGoogle Scholar
  13. 13.
    Corrigan, E., Dorey, P.E., Rietdijk, R.H., Sasaki, R.: Affine Toda field theory on a half-line. Phys. Lett. B 333(1-2), 83–91 (1994)CrossRefADSMathSciNetGoogle Scholar
  14. 14.
    Delfino, G., Mussardo, G., Simonetti, P.: Scattering theory and correlation functions in statistical models with a line of defect. Nucl. Phys. B 432(3), 518–550 (1994)CrossRefADSMathSciNetGoogle Scholar
  15. 15.
    Di Francesco, Ph., Zinn-Justin, P.: Quantum Knizhnik–Zamolodchikov equation: reflecting boundary conditions and combinatorics. J. Stat. Mech. Theory Exp. 12, P12009 (2007)Google Scholar
  16. 16.
    Dougall, J.: On Vandermonde’s theorem and some more general expansions. Proc. Edin. Math. Soc. 25, 114–132 (1906)CrossRefGoogle Scholar
  17. 17.
    Etingof, P.I., Frenkel, I.B., Kirillov, Jr. A.A.: Lectures on representation theory and Knizhnik–Zamolodchikov equations, Math. Surveys and Monographs, 58. Am. Math. Soc., Providence, RI (1998)Google Scholar
  18. 18.
    Faddeev, L.D., Sklyanin, E.K., Takhtajan, L.A.: Quantum inverse problem method. I. Theoret. Mat. Fiz. 40(2), 194–220 (1979)MathSciNetGoogle Scholar
  19. 19.
    Feigin, B., Frenkel, E.: Semi-infinite Weil complex and the Virasoro algebra Comm. Math. Phys. 137(3), 617–639 (1991)CrossRefADSzbMATHMathSciNetGoogle Scholar
  20. 20.
    Filali, G., Kitanine, N.: Spin chains with non-diagonal boundaries and trigonometric SOS model with reflecting end. SIGMA 7, Paper 012, 22 pages (2011)Google Scholar
  21. 21.
    Frenkel, E., E., Ben-Zvi, D.: Vertex algebras and algebraic curves 2nd edn, Mathematical Surveys and Monographs, 88, Am. Math. Soc., Providence, RI (2004)Google Scholar
  22. 22.
    Frenkel, I., Reshetikhin, N.: Quantum affine algebras and holonomic difference equations. Comm. Math. Phys. 146, 1–60 (1992)CrossRefADSzbMATHMathSciNetGoogle Scholar
  23. 23.
    Gasper, G., Rahman, M.: Basic Hypergeometric Series 2nd edn, Encycl. Math. Appl. 96, Cambridge University Press (2004)Google Scholar
  24. 24.
    Ghoshal, S., Zamolodchikov, A.: Boundary S-matrix and boundary state in two-dimensional integrable quantum field theory. Int. J. Mod. Phys. A 9(21), 3841–3885 (1994)CrossRefADSzbMATHMathSciNetGoogle Scholar
  25. 25.
    Jimbo, M., Kedem, R., Kojima, T., Konno, H., Miwa, T.: XXZ chain with a boundary. Nucl. Phys. B 441(3), 437–470 (1995)CrossRefADSzbMATHMathSciNetGoogle Scholar
  26. 26.
    Jimbo, M., Kedem, R., Konno, H., Miwa, T., Weston, R.: Difference equations in spin chains with a boundary. Nucl. Phys. B 448(3), 429–456 (1995)CrossRefADSzbMATHMathSciNetGoogle Scholar
  27. 27.
    Kasatani, M.: Boundary quantum Knizhnik–Zamolodchikov equation. In: New trends in quantum integrable systems, 157–171, World sci. Publ., Hackensack, NJ (2011)Google Scholar
  28. 28.
    Kasatani, M., Takeyama, Y.: The quantum Knizhnik–Zamolodchikov equation and non-symmetric Macdonald polynomials. Funkcial. Ekvac. 50, 491–509 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  29. 29.
    Knizhnik, V.G., Zamolodchikov, A.B.: Current algebra and Wess-Zumino Model in two dimensions. Nucl. Phys. B 247(1), 83–103 (1984)CrossRefADSzbMATHMathSciNetGoogle Scholar
  30. 30.
    Kojima, T.: Free field approach to diagonalization of boundary transfer matrix: recent advances. J. Phys. Conf. Ser. 284, 012041 (2011)CrossRefADSGoogle Scholar
  31. 31.
    Mezincescu, L., Nepomechie, R.I.: Integrable open spin chains with non-symmetric R-matrices. J. Phys. A Math. Gen. 24, L17–L23 (1991)CrossRefADSzbMATHMathSciNetGoogle Scholar
  32. 32.
    Mezincescu, L., Nepomechie, R.I.: Fusion procedure for open chains. J. Phys. A Math. Gen. 25, 2533–2543 (1992)CrossRefADSMathSciNetGoogle Scholar
  33. 33.
    Pasquier, V.: Quantum incompressibility and Razumov Stroganov type conjectures. Ann. Henri Poincaré 7, 397–421 (2006)CrossRefADSzbMATHMathSciNetGoogle Scholar
  34. 34.
    Reshetikhin, N.: Jackson-type integrals, Bethe vectors, and solutions to a difference analog of the Knizhnik–Zamolodchikov system. Lett. Math. Phys. 26, 153–165 (1992)CrossRefADSzbMATHMathSciNetGoogle Scholar
  35. 35.
    Reshetikhin, N.: Integrable models of quantum one-dimensional magnets with \({{\rm O}(n)}\) and \({{\rm Sp}(2k)}\) symmetry. Theoret. Math. Phys. 63(3), 555–569 (1985)CrossRefMathSciNetGoogle Scholar
  36. 36.
    Reshetikhin, N.: The algebraic Bethe ansatz for \({{\rm SO}(N)}\) -invariant transfer matrices (Russian) Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 169 (1988), Voprosy Kvant. Teor. Polya i Statist. Fiz. 8, 122–140, 189; transl. in J. Soviet Math. 54(3), 940–951 (1991)Google Scholar
  37. 37.
    Reshetikhin, N., Stokman, J., Vlaar, B.: Boundary quantum Knizhnik–Zamolodchikov equations and fusion. Ann. Henri Poincaré (to appear). arXiv:1404.5492
  38. 38.
    Reshetikhin, N., Varchenko, A.: Quasiclassical asymptotics of solutions to the KZ equations. In: Geometry, topology, physics, 293–322, Conf. Proc. Lecture Notes Geom. Topology, IV, Int. Press, Cambridge, MA (1995)Google Scholar
  39. 39.
    Ruijsenaars, S.N.M.: First order analytic difference equations and integrable quantum systems. J. Math. Phys. 38(2), 1069–1146 (1997)CrossRefADSzbMATHMathSciNetGoogle Scholar
  40. 40.
    Schechtman, V.V., Varchenko, A.N.: Arrangements of hyperplanes and Lie algebra homology. Invent. Math. 106(1), 139–194 (1991)CrossRefADSzbMATHMathSciNetGoogle Scholar
  41. 41.
    Sklyanin, E.K.: Boundary conditions for integrable quantum systems. J. Phys. A Math. Gen. 21, 2375–2389 (1988)CrossRefADSzbMATHMathSciNetGoogle Scholar
  42. 42.
    Smirnov, F.: A general formula for soliton form factors in the quantum sine-Gordon model. J. Phys. A 19(10), L575–L578 (1986)CrossRefADSzbMATHGoogle Scholar
  43. 43.
    Stokman, J.V.: Quantum affine Knizhnik–Zamolodchikov equations and quantum spherical functions, I. Int. Math. Res. Not. IMRN 5, 1023–1090 (2011)Google Scholar
  44. 44.
    Stokman, J., Vlaar, B.: Koornwinder polynomials and the XXZ spin chain (Dedicated to the 80th birthday of Dick Askey). Journal of Approximation Theory (2014), doi: 10.1016/j.jat.2014.03.003, arXiv:1310.5545
  45. 45.
    Takhtadzhan, L.A., Faddeev, L.D.: The quantum method of the inverse problem and the Heisenberg XYZ model. Russian Math. Surveys 34(5), 11–68 (1979)CrossRefADSGoogle Scholar
  46. 46.
    Varchenko, A.N., Tarasov, V.O.: Jackson integral representations for solutions to the quantized Knizhnik–Zamolodchikov equation. St. Petersburg Math. J. 6(2), 275–313 (1995)MathSciNetGoogle Scholar
  47. 47.
    Tsuchiya, A., Kanie, Y.: Vertex operators in conformal field theory on \({\mathbf{P}^1}\) and monodromy representations of braid group. Adv. Stud. Pure Math. 16, 297–372 (1988)MathSciNetGoogle Scholar
  48. 48.
    Weston, R.: Correlation functions and the boundary qKZ equation in a fractured XXZ chain. J. Stat. Mech. Theory Exp. 12, P12002 (2012)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Nicolai Reshetikhin
    • 1
    • 2
    • 3
  • Jasper Stokman
    • 2
    • 4
  • Bart Vlaar
    • 2
  1. 1.Department of MathematicsUniversity of CaliforniaBerkeleyUSA
  2. 2.KdV Institute for MathematicsUniversity of AmsterdamAmsterdamThe Netherlands
  3. 3.ITMO UniversitySaint PetersburgRussia
  4. 4.IMAPP, Radboud UniversityNijmegenThe Netherlands

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