't Hooft lines of ADE-type and Topological Quivers

We investigate 4D Chern-Simons theory with ADE gauge symmetries in the presence of interacting Wilson and 't Hooft line defects. We analyse the intrinsic properties of these lines' coupling and explicate the building of oscillator-type Lax matrices verifying the RLL integrability equation. We propose gauge quiver diagrams Q$_{G}^{\mu }$ encoding the topological data carried by the Lax operators and give several examples where Darboux coordinates are interpreted in terms of topological bi-fundamental matter. We exploit this graphical description $\left( i\right)$ to give new results regarding solutions in representations beyond the fundamentals of $sl_{N}$, $$ and $e_{6,7}$, and $\left( ii\right)$ to classify the Lax operators for simply laced symmetries in a unified E$_{7}$ CS theory. For quick access, a summary list of the leading topological quivers Q$$ is given in the conclusion section [Figures 29.(a-e), 30.(a-d) and 31.(a-d)].