  We investigate the correspondence between two dimensional topological gauge
theories and quantum integrable systems discovered by Moore, Nekrasov,
Shatashvili. This correspondence means that the hidden quantum integrable
structure exists in the topological gauge theories. We showed the
correspondence between the G/G gauged WZW model and the phase model in JHEP 11
(2012) 146 (arXiv:1209.3800). In this paper, we study a one-parameter
deformation for this correspondence and show that the G/G gauged WZW model
coupled to additional matters corresponds to the q-boson model. Furthermore, we
investigate this correspondence from a viewpoint of the commutative Frobenius
algebra, the axiom of the two dimensional topological quantum field theory.
