The surprising connection between exactly solved lattice models and discrete holomorphicity

Over the past few years it has been discovered that an “observable” can be set up on the lattice which obeys the discrete Cauchy-Riemann equations. The ensuing condition of discrete holomorphicity leads to a system of linear equations which can be solved to yield the Boltzmann weights of the underlying lattice model. Surprisingly, these are the well known Boltzmann weights which satisfy the star-triangle or Yang-Baxter equations at criticality. This connection has been observed for a number of exactly solved models. I briefly review these developments and discuss how this connection can be made explicit in the context of the Z_N model. I also discuss how discrete holomorphicity has been used in recent breakthroughs in the rigorous proof of some key results in the theory of planar self-avoiding walks.