Recently, there has been observed an interesting correspondence between supersymmetric quiver gauge theories with four supercharges and integrable lattice models of statistical mechanics such that the two-dimensional spin lattice is the quiver diagram, the partition function of the lattice model is the partition function of the gauge theory and the Yang-Baxter equation expresses the identity of partition functions for dual pairs. This correspondence is a powerful tool which enables us to generate new integrable models. The aim of the present paper is to give a short account on a progress in integrable lattice models which has been made due to the relationship with supersymmetric gauge theories.MSC classes: 81T60, 16T25, 14K25, 33D60, 33E20, 33D90, 39A13

We present a multi-spin solution to the Yang-Baxter equation. The solution corresponds to the integrable lattice spin model of statistical mechanics with positive Boltzmann weights and parameterized in terms of the basic hypergeometric functions. We obtain this solution from a non-trivial basic hypergeometric sum-integral identity which originates from the equality of supersymmetric indices for certain three-dimensional N = 2 Seiberg dual theories.

We show that the equality of 2d N =(2,2) supersymmetric indices in Seibergtype duality leads to a new integrable Ising-type model. The emergence of the new model is the result of correspondence between the supersymmetric SU (2) quiver gauge theories and the Yang-Baxter equation. Using this correspondence, we solve the star-triangle relation and obtain the two-dimensional exactly solvable spin model. The model corresponding to our solution possesses continuous spin variables on the circle and the Boltzmann weights are demonstrated in terms of the Jacobi theta function. Using the solution of the star-triangle relation, we also construct interaction-round-a-face and vertex models.