Spin Chain Integrability as a Supersymmetric Gauge Duality

We establish a novel correspondence between 4D $\mathcal{N}=1$ supersymmetric gauge theories on $D^2\times T^2$ and open XYZ spin chains with generalized boundary conditions, extending beyond previous 3D Bethe/gauge duality frameworks. Our primary contribution is the rigorous construction of the first exact duality between 4D BCD-type gauge theories and general-boundary XYZ spin chains governed by elliptic $R$-matrices. This framework provides a universal mechanism for resolving supersymmetric gauge theory/spin-chain duality across dimensional hierarchies: 2D $\mathcal{N}=(2,2)$ $⟷$ XXX spin chain, 3D $\mathcal{N}=2$ $⟷$ XXZ spin chain, 4D $\mathcal{N}=1$ $⟷$ XYZ spin chain, mediated through the $\mathrm{\Omega }$-deformation parameter $\epsilon$.