Conformal invariance in two-dimensional discrete field theory

A discretized massless wave equation in two dimensions, on an appropriately chosen square lattice, exactly reproduces the solutions of the corresponding continuous equations. We show that the reason for this exact solution property is the discrete analogue of conformal invariance present in the model, and find more general field theories on a two-dimensional lattice that exactly solve their continuous limit equations. These theories describe in general nonlinearly coupled bosonic and fermionic fields and are similar to the Wess-Zumino-Witten model.