Discrete holomorphicity and integrability in loop models with open boundaries

Understand

We consider boundary conditions compatible with discrete holomorphicity for the dilute O(n) and C_2^(1) loop models.

• In each model, for a general set of boundary plaquettes, multiple types of loops can appear.
• A generalisation 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.