NEW SOLVABLE LATTICE MODELS IN 3 DIMENSIONS

In this paper we establish a remarkable connection between two seemingly unrelated topics in the area of solvable lattice models. The first is the Zamolodchikov model, which is the only nontrivial model on a three-dimensional lattice so far solved. The second is the chiral Potts model on the square lattice and its generalization associated with the U(q)(Sl(n)) algebra, which is of current interest due to its connections with high-genus algebraic curves and with representations of quantum groups at roots of unity. We show that this last "sl(n)-generalized chiral Potts model" can be interpreted as a model on a three-dimensional simple cubic lattice consisting of n square-lattice layers with an N-valued (N greater-than-or-equal-to 2) spin at each site. Further, in the N = 2 case this three-dimensional model reduces (after a modification of the boundary conditions) to the Zamolodchikov model we mentioned above.