Discrete holomorphicity and integrability in loop models with open boundaries

Abstract

We consider boundary conditions compatible with discrete holomorphicity for the dilute O(n) and C2(1) loop models. In each model, for a general set of boundary plaquettes, multiple types of loop can appear. A generalization of Smirnov's parafermionic observable is therefore required in order to maintain the discrete holomorphicity property in the bulk. We show that there exist natural boundary conditions for this observable which are consistent with integrability, that is to say, that by imposing certain boundary conditions, we obtain a set of linear equations whose solutions also satisfy the corresponding reflection equation. In both loop models, several new sets of integrable weights are f..