Wilson-’t Hooft lines as transfer matrices

We establish a correspondence between class of Wilson-'t Hooft lines in four-dimensional $\mathcal{N} = 2$ supersymmetric gauge theories described by circular quivers and transfer matrices constructed from dynamical L-operators for trigonometric quantum integrable systems. compute the vacuum expectation values twisted product space $S^1 \times_\epsilon \mathbb{R}^2 \times \mathbb{R}$ localization show that they are equal to Wigner transforms matrices. A variant AGT implies an identification with Verlinde operators Toda theory, which we also verify. explain how these field theory setups related Chern-Simons via embedding into string dualities.