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The six-vertex model and its spin-$S$ descendants obtained from the fusionprocedure 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 alocal observable on the lattice that behaves as the chiral current $J^a(z)$ inthe continuum limit. The observable is built out of generators of the su$(2)$Lie algebra acting on a small (finite) number of lattice sites. Theconstruction works also for the multi-critical quantum spin chains related tothe vertex models, and is verified numerically for $S=1/2$ and $S=1$ usingBethe Ansatz and form factors techniques.

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