Bethe/Gauge Correspondence for ABCDEFG-type 3d Gauge Theories

In this paper, we give a new effective superpotential that makes clear Bethe/Gauge correspondence between 2d (and 3d) $\text{SO/Sp}$ gauge theories and open $\text{XXX}$ (and $\text{XXZ}$) spin chains with diagonal boundary conditions, and also works in the case of 2d (and 3d) $\text{BC}_{N}$-type gauge theories which is not previously discussed in the literature. Especially, for exceptional Lie algebras $\text{F}_{4}$, $\text{G}_{2}$, we give the effective superpotential and vacuum equations. For $\text{E}_{6,7,8}$, we only give theirs effective superpotential for convenience.