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We study lens partitions functions for the three-dimensional $ N=2$supersymmetric gauge theories on $S_b^3/Zr$. We consider an equality as a newhyperbolic hypergeometric solution to the star-star relation via the gauge/YBEcorrespondence. The correspondence allows the construction of integrablelattice spin models of statistical mechanics by the use of integral identities.Additionally, we obtain new hyperbolic hypergeometric integral identities ofgauge theories.

Lens Partition Functions and Integrability Properties