A new result for 3D Bethe/Gauge correspondence

In this paper, we have established the general relationship between 3D [Formula: see text] gauge theories and quantum spin chains [X.-M. Ding and T. Zhang, Bethe/Gauge correspondence for ABCDEFG-type 3d gauge theories, J. High Energy Phys. 04 (2023) 036, arXiv:2303.03102], known as the Bethe/Gauge correspondence. In this paper, we propose to derive the correspondence with alternative setting for a different boundary conditions, and clarify the distinctions between the two approaches. We argue that these differences originate from a form of Langlands duality between Lie algebras of types [Formula: see text] and [Formula: see text].