Bethe/Gauge Correspondence for ABCD quiver Gauge Theories and Spin Chains

his note is an extension of [DZ23] where the supersymmetric vacua of three-dimensional $\mathcal{N}=2$ gauge theories with matter are shown to be in one-to-one correspondence with the eigenstate of $\text{XXZ}$ integrable spin chain Hamiltonians with open boundary conditions. We consider the $A_{2}$ quiver gauge theory, which is the simplest non-trivial quiver gauge theory, and $sl_{3}$ open $\text{XXZ}$ spin chain with diagonal boundary condition. We demonstrate the correspondence between the vacuum equations of different gauge groups and Bethe ansatz equations with different boundary parameters. We furthermore push forward the program to the general linear quiver gauge theory.