New solvable lattice models in three dimensions

Abstract

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-dimen-sional lattice so far solved. The second is the chiral Potts model on the square lattice and its generalization associated with theU _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 threedimensional simple cubic lattice consisting ofn square-lattice layers with anN- valued (N⩾2) spin at each site. Further, in theN=2 case this three-dimen-sional model reduces (after a modification of the boundary conditions) to the Zamolodchikov model we mentioned above.