A formal Bethe/gauge correspondence for classical gauge groups

We study a formal algebraic Bethe/Gauge matching between Nekrasov-Shatashvili (NS)-limit vacuum equations extracted from four-dimensional \\mathcal N=1 𝒩 = 1 localization on D^2× T^2 D 2 × T 2 and the homogeneous two-term Bethe equations of a special open XYZ spin-chain sector. The main new calculation is an explicit four-dimensional elliptic construction for the C_N C N gauge group. We also present the A_N A N , B_N B N , and D_N D N cases in the same normalization, thereby giving a uniform comparison across the classical series. The Bethe equations used in the matching are not the most general finite-size off-diagonal equations with an inhomogeneous term. We further observe that, after the NS scaling, the formal vacuum equations no longer depend explicitly on the chiral multiplet R R -charge. The NS scaling should not be interpreted as a physical cancellation or bypassing of four-dimensional anomalies, but as a convenient way to derive the Bethe/Gauge correspondence. Outside the anomaly-free locus, the equations considered here are used only as formal semiclassical algebraic expressions to be compared with the Bethe equations. This formal matching is compatible with the standard dimensional hierarchy: $ N=(2,~2) $ ↔ ↔ \\text{XXX spin chain,} XXX spin chain, \\text{three-dimensional}\\ \\mathcal N=2\\ \\text{theory} three-dimensional 𝒩 = 2 theory ↔ ↔ $ $ \\text{four-dimensional}\\ \\mathcal N=1\\ \\text{theory} four-dimensional 𝒩 = 1 theory ↔ ↔ \\text{XYZ spin chain.} XYZ spin chain.