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Smirnov’s integrals and the quantum Knizhnik-Zamolodchikov equation of level 0. (English) Zbl 0842.17030

From the introduction: In this paper, following [F. A. Smirnov, Form factors in completely integrable models of quantum field theory (Singapore: World Scientific) (1992; Zbl 0788.46077)]the authors give an integral formula for solutions to the quantum Knizhnik-Zamolodchikov (KZ) equation [I. B. Frenkel and N. Yu. Reshetikhin, Quantum affine algebras and holonomic difference equations, Commun. Math. Phys. 146, 1-60 (1992; Zbl 0760.17006)]for the quantum affine algebra \(U_q (\widehat {\mathfrak {sl}}_2)\) when the spin is \(1/2\), the level is 0 and \(|q|<1\).
In [loc. cit.], Smirnov gave an integral formula for the form factors of the sine-Gordon model. His method involved solving a system of difference equations for a vector-valued function in \(N\) variables \((\beta_1, \dots, \beta_N)\), which takes values in the \(N\)-fold tensor product of the spin \(1/2\) representation \(\mathbb{C}^2\) of \(U_q (\widehat {\mathfrak {sl}}_2)\). The total space \(\mathbb{C}^2 \otimes \cdots \otimes \mathbb{C}^2\) splits into the subspaces of fixed total spins \((l- n)/2\) where \(l+n =N\) and \(0\leq n\leq N\). Further, an integral formula was given for the case \(n=l= N/2\) (\(N\) even) and \(|q|=1\). In a private communication to the present authors, Smirnov showed the modified formula for the case \(|q|<1\), which is given at the end of the present paper. The authors’ main contribution is to generalize Smirnov’s formula to the case of an arbitrary total spin.

MSC:

17B37 Quantum groups (quantized enveloping algebras) and related deformations
39A10 Additive difference equations
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
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