Chiral SU(2)k currents as local operators in vertex models and spin chains

The six-vertex model and its spin-S descendants obtained from the fusion procedure are well-known lattice discretizations of the SU(2)k WZW models, with k = 2S. It is shown that, in these models, it is possible to exhibit a local observable on the lattice that behaves as the chiral current Ja(z) in the continuum limit. The observable is built out of generators of the su(2) Lie algebra acting on a small (finite) number of lattice sites. The construction works also for the multi-critical quantum spin chains related to the vertex models, and is verified numerically for S = 1/2 and S = 1 using Bethe ansatz and form factors techniques. (paper)