  We introduce a class of new integrable lattice models labeled by a pair of
positive integers N and r. The integrable model is obtained from the Gauge/YBE
correspondence, which states the equivalence of the 4d N=1 S^1 \times S^3/Z_r
index of a large class of SU(N) quiver gauge theories with the partition
function of 2d classical integrable spin models. The integrability of the model
(star-star relation) is equivalent with the invariance of the index under the
Seiberg duality. Our solution to the Yang-Baxter equation is one of the most
general known in the literature, and reproduces a number of known integrable
models. Our analysis identifies the Yang-Baxter equation with a particular
duality (called the Yang-Baxter duality) between two 4d N=1 supersymmetric
quiver gauge theories. This suggests that the integrability goes beyond 4d lens
indices and can be extended to the full physical equivalence among the IR fixed
points.
