In this paper, we give a new effective superpotential that makes clear Bethe/Gauge correspondence between 2d (and 3d) SO/Sp gauge theories and open XXX (and XXZ) spin chains with diagonal boundary conditions, and also works in the case of 2d (and 3d) BCN-type gauge theories which is not previously discussed in the literature. Especially, for exceptional Lie algebras F4, G2, we give the effective superpotential and vacuum equations. For E6,7,8, we only give theirs effective superpotential for convenience.
This note is an extension of [DZ23] where the supersymmetric vacua of three-dimensional N = 2 gauge theories with matter are shown to be in one-to-one correspondence with the eigenstate of 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 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.
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