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.


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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