On a discrete version of the P¹ sigma model and surfaces immersed in ³

We present a discretization of the P¹ sigma model. We show that the discrete P¹ sigma model is described by a nonlinear partial second-order difference equation with rational nonlinearity. To derive discrete surfaces immersed in three-dimensional Euclidean space a 'complex' lattice is introduced. The so-obtained surfaces are characterized in terms of the quadrilateral cross-ratio of four surface points. In this way we prove that all surfaces associated with the discrete P¹ sigma model are of constant mean curvature. An explicit example of such discrete surfaces is constructed.