Discrete holomorphicity and quantized affine algebras

We consider non-local currents in the context of quantized affine algebras, following the construction introduced by Bernard and Felder. In the case of U_q(A₁⁽¹⁾) and U_q(A₂⁽²⁾), these currents can be identified with configurations in the six-vertex and Izergin--Korepin nineteen-vertex models. Mapping these to their corresponding Temperley--Lieb loop models, we directly identify non-local currents with discretely holomorphic loop observables. In particular, we show that the bulk discrete holomorphicity relation and its recently derived boundary analogue are equivalent to conservation laws for
non-local currents.