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We study integrable models in the context of the recently discoveredGauge/YBE correspondence, where the Yang-Baxter equation is promoted to aduality between two supersymmetric gauge theories. We study flavored ellipticgenus of 2d $\mathcal{N}=(2,2)$ quiver gauge theories, which theories aredefined from statistical lattices regarded as quiver diagrams. Our R-matricesare written in terms of theta functions, and simplifies considerably when thegauge groups at the quiver nodes are Abelian. We also discuss the modularityproperties of the R-matrix, reduction of 2d index to 1d Witten index, andstring theory realizations of our theories.

Integrability from 2d N=(2,2) Dualities