Spectral curves and parameterizations of a discrete integrable 3-dimensional model

Abstract

We consider a discrete classical integrable model on the 3-dimensional cubic lattice. The solutions of this model can be used to parameterize the Boltzmann weights of the different 3-dimensional spin models. We have found the general solution of this model constructed in terms of the theta-functions defined on an arbitrary compact algebraic curve. The imposing of the periodic boundary conditions fixes the algebraic curve. We have shown that in this case the curve coincides with the spectral one of the auxiliary linear problem. In the case when the curve is a rational one, the soliton solutions have been constructed.