The Spectral Problem for the q-Knizhnik-Zamolodchikov Equation

We analyze the spectral problem for the q-Knizhnik-Zamolodchikov equations for Uq(ŝl2)(0 < q ≤ 1) at level zero. The case of 2-point functions in the fundamental representation is studied in detail. The scattering states are found explicitly in terms of continuous q-Jacobi polynomials. The corresponding S-matrix is shown to coincide, up to a trivial factor, with the kink-antikink S-matrix in the spin2 XXZ antiferromagnet. In conformal field theory, the matrix elements of products of vertex operators between suitable vacuum states obey the Knizhnik-Zamolodchikov (KZ) equations [1]. These are first order differential equations akin to the Dirac and Bargmann-Wigner equations [2]. Recently Frenkel and Reshetikhin [3] have derived a q-analogue of the KZ equation for quantum affine algebras. This is a considerable generalization and these q-KZ equations are no longer differential, but rather difference equations. In this paper we study in detail the spectral problem for the q-KZ equation of Uq(ŝl2) in the fundamental representation at level zero. The scattering states are explicitly found in terms of continuous q-Jacobi polynomials. From the known asymptotics of these polynomials we derive an S-matrix which coincides, up to a trivial factor, with the S-matrix for kink-antikink scattering in the spin 1 2 XXZ antiferromagnet. Even in the “classical” limit q → 1, this spectral problem yields a non-trivial S-matrix, corresponding of course Work supported in part by the NSF: PHY-91-23780 Permanent address: Institute of Chemical Physics, Kosygina Str. 4, SU-117334, Moscow, Russia