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**On solutions of the KZ and qKZ equations at level zero.**
*(English)*
Zbl 0941.35089

The quantum qKZ equations appear naturally in the representation theory of quantum affine algebras as equations for matrix elements of products of vertex operators and later as equations for matrix elements of product of vertex operators. Their specializations are the equations for correlation functions in solvable lattice models and the equations for form factors in two-dimensional massive integrable models of quantum field theory.

In the paper under review the authors consider the rational qKZ equation associated with the Lie algebra \(\text{sl}_2\) at level zero which is the case of Smirnov’s equations for form factors.

There are several approaches to integral formulae for solutions of the qKZ equations at level zero. The first one was given by F. A. Smirnov [Form factors in completely integrable models of quantum field theories. World Scientific, Singapore (1992; Zbl 0788.46077)]. This construction can be seen as a deformation of the hyperelliptic integrals. They are expressed via certain polynomials and they are enumerated by periodic functions, which are arbitrary polynomials of exponentials of bounded degree. But this approach is closely related to the equations at level zero and fails to work in the case of the qKZ equation at nonzero level.

The second approach is not restricted only to the qKZ equations at level zero but cannot be directly applied to producing solutions in general. This approach for producing the integral formulae for solutions to the qKZ equations belonging to M. Jimbo and T. Miwa, S. Lukyanov, S. Khoroshkin, D. Lebedev and S. Pakuliak, is based on calculation a trace of a product of vertex operators over an infinite-dimensional representation of the quantum affine algebra \(U_q(\widehat {\text{sl}_2})\) or corresponding centrally extended Yangian double, using the bosonization technique.

A general approach to integral representations for solutions of the qKZ equations is based on the recent papers of V. Tarasov and A. Varchenko (1991–1997). It allows to describe effectively the total space of solutions of the qKZ equation at generic position, as well as to compute its transition matrices between asymptotic solutions. But, unfortunately, this approach cannot be applied immediately to the case of the qKZ equation at level zero. The authors make some modifications and state theorem (6.3) that describes all solutions of the qKZ equation at level zero. But they have no proof of this conjecture yet.

The general aim of the paper is to compare these three types of integral formulae. It is shown that Smirnov’s formula can be obtained from a general author’s formula (6.2) by a certain specialization of a periodic function due to a certain trick available only at level zero. The structure of the paper is the following.

In Section 2 the authors recall the definitions of the KZ and qKZ equations at level zero. In Section 3 they describe the necessary spaces of rational functions. Solutions of the KZ equation at level zero are considered in Section 4. In Section 5 the authors describe the space of “deformed cycles” (periodic functions) and the pairing between the spaces of rational and periodic functions given by the hypergeometric integral. In Section 6 the solutions of the qKZ equations at level zero are given. The traces of products of vertex operators are considered in Section 7.

In the paper under review the authors consider the rational qKZ equation associated with the Lie algebra \(\text{sl}_2\) at level zero which is the case of Smirnov’s equations for form factors.

There are several approaches to integral formulae for solutions of the qKZ equations at level zero. The first one was given by F. A. Smirnov [Form factors in completely integrable models of quantum field theories. World Scientific, Singapore (1992; Zbl 0788.46077)]. This construction can be seen as a deformation of the hyperelliptic integrals. They are expressed via certain polynomials and they are enumerated by periodic functions, which are arbitrary polynomials of exponentials of bounded degree. But this approach is closely related to the equations at level zero and fails to work in the case of the qKZ equation at nonzero level.

The second approach is not restricted only to the qKZ equations at level zero but cannot be directly applied to producing solutions in general. This approach for producing the integral formulae for solutions to the qKZ equations belonging to M. Jimbo and T. Miwa, S. Lukyanov, S. Khoroshkin, D. Lebedev and S. Pakuliak, is based on calculation a trace of a product of vertex operators over an infinite-dimensional representation of the quantum affine algebra \(U_q(\widehat {\text{sl}_2})\) or corresponding centrally extended Yangian double, using the bosonization technique.

A general approach to integral representations for solutions of the qKZ equations is based on the recent papers of V. Tarasov and A. Varchenko (1991–1997). It allows to describe effectively the total space of solutions of the qKZ equation at generic position, as well as to compute its transition matrices between asymptotic solutions. But, unfortunately, this approach cannot be applied immediately to the case of the qKZ equation at level zero. The authors make some modifications and state theorem (6.3) that describes all solutions of the qKZ equation at level zero. But they have no proof of this conjecture yet.

The general aim of the paper is to compare these three types of integral formulae. It is shown that Smirnov’s formula can be obtained from a general author’s formula (6.2) by a certain specialization of a periodic function due to a certain trick available only at level zero. The structure of the paper is the following.

In Section 2 the authors recall the definitions of the KZ and qKZ equations at level zero. In Section 3 they describe the necessary spaces of rational functions. Solutions of the KZ equation at level zero are considered in Section 4. In Section 5 the authors describe the space of “deformed cycles” (periodic functions) and the pairing between the spaces of rational and periodic functions given by the hypergeometric integral. In Section 6 the solutions of the qKZ equations at level zero are given. The traces of products of vertex operators are considered in Section 7.

Reviewer: Valentina Golubeva (Moskva)

### MSC:

35Q53 | KdV equations (Korteweg-de Vries equations) |

32G34 | Moduli and deformations for ordinary differential equations (e.g., Knizhnik-Zamolodchikov equation) |

17B81 | Applications of Lie (super)algebras to physics, etc. |

39A70 | Difference operators |

81T10 | Model quantum field theories |

81R50 | Quantum groups and related algebraic methods applied to problems in quantum theory |

17B69 | Vertex operators; vertex operator algebras and related structures |

### Keywords:

quantum Knizhnik-Zamolodchikov difference equations at level zero; rational solutions of the Yang-Baxter equations; integral formulae for solutions of the qKZ equations### Citations:

Zbl 0788.46077
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\textit{A. Nakayashiki} et al., Ann. Inst. Henri Poincaré, Phys. Théor. 71, No. 4, 459--496 (1999; Zbl 0941.35089)

### References:

[1] | A. Chervov , Traces of creating-annihilating operators and Fredholm’s formulas , Preprint ( 1997 ) 1-20, q-alg/9703017. |

[2] | B. Davies , O. Foda , M. Jimbo , T. Miwa and A. Nakayashiki , Diagonalization of the XXZ Hamiltonian by vertex operators , Comm. Math. Phys. 151 ( 1993 ) 89 - 153 . Article | MR 1201657 | Zbl 0769.17020 · Zbl 0769.17020 · doi:10.1007/BF02096750 |

[3] | E. Date , M. Jimbo , A. Matsuo and T. Miwa , Hypergeometric type integrals and the sl(2, C) Knizhnik-Zamolodchikov equations , Int. J. Mod. Phys. B 4 ( 1990 ) 1049 - 1057 . MR 1064760 | Zbl 0722.33007 · Zbl 0722.33007 · doi:10.1142/S0217979290000528 |

[4] | I. Frenkel and N. Reshetikhin , Quantum affine algebras and holonomic difference equations , Comm. Math. Phys. 146 ( 1992 ) 1 - 60 . Article | MR 1163666 | Zbl 0760.17006 · Zbl 0760.17006 · doi:10.1007/BF02099206 |

[5] | K. Iohara and M. Kohno , A central extension of Yangian double and its vertex representations , Lett. Math. Phys. 37 ( 1996 ) 319 - 328 . MR 1392590 | Zbl 0867.17013 · Zbl 0867.17013 |

[6] | M. Jimbo , T. Kojima , T. Miwa and Y.-H. Quano , Smirnov’s integrals and the quantum Knizhnik-Zamolodchikov equation of level 0 , J. Phys. A 27 ( 1994 ) 3267 - 3283 . MR 1282167 | Zbl 0842.17030 · Zbl 0842.17030 · doi:10.1088/0305-4470/27/9/036 |

[7] | M. Jimbo and T. Miwa , Algebraic analisys of solvable lattice models, Conference Board of the Mathem . Sciences , Regional Conference Series in Mathematics 85 ( 1995 ). MR 1308712 | Zbl 0828.17018 · Zbl 0828.17018 |

[8] | M. Jimbo and T. Miwa , Quantum KZ equation with |q| = 1 and correlation functions of the XXZ model in the gapless regime , J. Phys. A 29 ( 1996 ) 2923 - 2958 . MR 1398600 | Zbl 0896.35114 · Zbl 0896.35114 · doi:10.1088/0305-4470/29/12/005 |

[9] | S. Khoroshkin , Central extension of the Yangian double , in: Collection SMF, Colloque ”Septièmes Rencontres du Contact Franco-Belge en Algèbre”, June 1995 , Reins; Preprint ( 1996 ) 1-12, q-alg/9602031. MR 1384643 | Zbl 0887.17007 · Zbl 0887.17007 |

[10] | S. Khoroshkin , D. Lebedev and S. Pakuliak , Traces of intertwining operators for the Yangian double , Lett. Math. Phys. 41 ( 1997 ) 31 - 47 . MR 1455030 | Zbl 0973.17022 · Zbl 0973.17022 · doi:10.1023/A:1007372113466 |

[11] | T. Kojima and Y.-H. Quano , Quantum Knizhnik-Zamolodchikov equation for Uq(sln) and integral formula , J. Phys. A 27 ( 1994 ) 6807 - 6826 . MR 1306832 | Zbl 0844.17011 · Zbl 0844.17011 · doi:10.1088/0305-4470/27/20/018 |

[12] | S. Lukyanov , Free field representation for massive integrable models , Comm. Math. Phys. 167 ( 1995 ) 183 - 226 . Article | MR 1316504 | Zbl 0818.46079 · Zbl 0818.46079 · doi:10.1007/BF02099357 |

[13] | K. Mimachi , A solution to quantum Knizhnik-Zamolodchikov equations and its application to eigenvalue problems of the Macdonald type , Duke Math. J. 85 ( 1996 ) 635 - 658 . Article | MR 1422360 | Zbl 0889.17009 · Zbl 0889.17009 · doi:10.1215/S0012-7094-96-08524-5 |

[14] | E. Mukhin and A. Varchenko , On algebraic equations satisfied by hypergeometric solutions of the qKZ equation , Preprint ( 1997 ) 1-20, q-alg/9710040; Quantization of the space of conformal blocks Preprint ( 1997 ) 1-9, q-alg/9710039; The quantized Knizhnik-Zamolodchikov equation in tensor products of irreducible sl(2)-modules Preprint ( 1997 ) 1-32, q-alg/9709026. · Zbl 1157.33329 |

[15] | A. Nakayashiki , Integral and theta formulae for solutions of slN Knizhnik-Zamolodchikov equation at level zero , Publ. Res. Inst. Math. Sci. 34 ( 1998 ) 439 - 486 . MR 1658129 | Zbl 0959.32028 · Zbl 0959.32028 · doi:10.2977/prims/1195144514 |

[16] | A. Nakayashiki , Some integral formulas for the solutions of the sl2 dKZ equation with level -4 , Int. J. Mod. Phys. A 9 ( 1994 ) 5673 - 5687 . MR 1306785 | Zbl 0985.82504 · Zbl 0985.82504 · doi:10.1142/S0217751X94002326 |

[17] | F.A. Smirnov , Form Factors in Completely Integrable Field Theories , World Scientific , Singapore , 1992 . MR 1253319 · Zbl 0788.46077 |

[18] | F. Smirnov , Counting the local fields in SG theory , Nucl. Phys. B 453 ( 1995 ) 807 - 824 . MR 1358784 | Zbl 1003.81545 · Zbl 1003.81545 · doi:10.1016/0550-3213(95)00423-P |

[19] | F. Smirnov , Dynamical symmetries of massive integrable models. I , Adv. Ser. Math. Phys. 16 ( 1992 ) 813 - 837 . MR 1187577 · Zbl 0925.17027 · doi:10.1142/S0217751X92004063 |

[20] | F. Smirnov , Lectures on integrable massive models of quantum field theory , in: M.-L. Ge and B.-H. Zhao, eds., Nankai Lectures on Mathem. Physics , World Scentific , Singapore , 1990 , pp. 1 - 68 . MR 1145242 |

[21] | V. Schechtman and A. Varchenko , Arrangements of hyperplanes and Lie algebra homology , Invent. Math. 106 ( 1991 ) 139 - 194 . MR 1123378 | Zbl 0754.17024 · Zbl 0754.17024 · doi:10.1007/BF01243909 |

[22] | V. Tarasov and A. Varchenko , Jackson integral representations of solutions of the quantized Knizhnik-Zamolodchikov equation , St. Petersburg Math. J. 6 ( 1995 ) 275 - 313 . MR 1290820 | Zbl 0824.33012 · Zbl 0824.33012 |

[23] | V. Tarasov and A. Varchenko , Geometry of q-hypergeometric functions as a bridge between Yangians and quantum affine algebras , Invent. Math. 128 ( 1997 ) 501 - 588 . MR 1452432 | Zbl 0877.33013 · Zbl 0877.33013 · doi:10.1007/s002220050151 |

[24] | V. Tarasov and A. Varchenko , Geometry of q-hypergeometric functions, quantum affine algebras and elliptic quantum groups , Preprint ( 1997 ) 1-69; to appear in Asterisque , q-alg/9703044. arXiv | MR 1646561 · Zbl 0938.17012 |

[25] | A. Varchenko , Quantized Knizhnik-Zamolodchikov equations, quantum Yang-Baxter equation, and difference equations for q-hypergeometric functions , Comm. Math. Phys. 162 ( 1994 ) 499 - 528 . Article | MR 1277474 | Zbl 0807.17014 · Zbl 0807.17014 · doi:10.1007/BF02101745 |

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