The tetrahedron equation is a three-dimensional generalization of the Yang-Baxter equation. Its solutions define integrable three-dimensional lattice models of statistical mechanics and quantum field theory. Their integrability is not related to the size of the lattice, therefore the same solution of the tetrahedron equation defines different integrable models for different finite periodic cubic lattices. Obviously, any such three-dimensional model can be viewed as a two-dimensional integrable model on a square lattice, where the additional third dimension is treated as an internal degree of freedom. Therefore every solution of the tetrahedron equation provides an infinite sequence of integrable 2d models differing by the size of this "hidden third dimension". In this paper we construct a new solution of the tetrahedron equation, which provides in this way the two-dimensional solvable models related to finitedimensional highest weight representations for all quantum affine algebra U q ( sl(n)), where the rank n coincides with the size of the hidden dimension. These models are related with an anisotropic deformation of the sl(n)-invariant Heisenberg magnets. They were extensively studied for a long time, but the hidden 3d structure was hitherto unknown. Our results lead to a remarkable exact "rank-size" duality relation for the nested Bethe Ansatz solution for these models. Note also, that the above solution of the tetrahedron equation arises in the quantization of the "resonant three-wave scattering" model, which is a well-known integrable classical system in 2 + 1 dimensions.

The Faddeev-Volkov solution of the star-triangle relation is connected with the modular double of the quantum group U q (sl 2 ). It defines an Ising-type lattice model with positive Boltzmann weights where the spin variables take continuous values on the real line. The free energy of the model is exactly calculated in the thermodynamic limit. The model describes quantum fluctuations of circle patterns and the associated discrete conformal transformations connected with the Thurston's discrete analogue of the Riemann mappings theorem. In particular, in the quasi-classical limit the model precisely describe the geometry of integrable circle patterns with prescribed intersection angles.

We describe a scheme of constructing classical integrable models in 2 + 1-dimensional discrete space-time, based on the functional tetrahedron equation-equation that makes manifest the symmetries of a model in local form. We construct a very general "block-matrix model" together with its algebro-geometric solutions, study its various particular cases, and also present a remarkably simple scheme of quantization for one of those cases.

The Faddeev-Volkov model is an Ising-type lattice model with positive Boltzmann weights where the spin variables take continuous values on the real line. It serves as a lattice analog of the sinh-Gordon and Liouville models and intimately connected with the modular double of the quantum group Uq(sl2). The free energy of the model is exactly calculated in the thermodynamic limit. In the quasi-classical limit c → +∞ the model describes quantum fluctuations of discrete conformal transformations connected with the Thurston's discrete analogue of the Riemann mappings theorem. In the strongly-coupled limit c → 1 the model turns into a discrete version of the D = 2 Zamolodchikov's "fishing-net" model.

In this paper we formulate an integrable model on the simple cubic lattice. The N -valued spin variables of the model belong to edges of the lattice. The Boltzmann weights of the model obey the vertex type Tetrahedron Equation. In the thermodynamic limit our model is equivalent to the Bazhanov -Baxter Model. In the case when N = 2 we reproduce the Korepanov's and Hietarinta's solutions of the Tetrahedron equation as some special cases.