Quiver ${\text{W}}_{{\boldsymbol{{\epsilon}}}_{\mathbf{1}},{\boldsymbol{{\epsilon}}}_{\mathbf{2}}}$ algebras of 4D $\boldsymbol{\mathcal{N}}\mathbf{=}\mathbf{2}$ gauge theories

We construct an e-deformation of W algebras, corresponding to the additive version of quiver W q,t algebras which feature prominently in the 5d version of the BPS/CFT correspondence and refined topological strings on toric Calabi-Yau's. This new type of algebras fill in the missing intermediate level between q-deformed and ordinary W algebras. We show that e-deformed W algebras are spectral duals of conventional W algebras, in particular the e-deformed conformal blocks manifestly reproduce instanton partition functions of 4d N = 2 quiver gauge theories in the full Ω-background and give dual integral representations of ordinary W conformal blocks.