${\mathfrak{g}\mathfrak{l}}_N$ Higgsed networks

We generalize the framework of Higgsed networks of intertwiners to the quantum toroidal algebra associated to Lie algebra ${\mathfrak{g}\mathfrak{l}}_N$. Using our formalism we obtain a systems of screening operators corresponding to W-algebras associated to toric strip geometries and reproduce partition functions of 3d theories on orbifolded backgrounds.