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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.

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

Citations:

Zbl 0788.46077
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References:

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