  We study 't Hooft lines and the associated $\mathcal{L}$- operators in
topological 4D Chern-Simons theory with gauge symmetry given by the exceptional
groups E$_{6}$ and E$_{7}$. We give their oscillator realisations and propose
topological gauge quivers encoding the properties of these topological lines
where Darboux coordinates are interpreted in terms of topological fundamental
matter. Other related aspects are also described.
