Quiver W${}_{ϵ1,ϵ2}$ algebras of 4d N = 2 gauge theories

We construct an $ϵ$-deformation of W algebras, corresponding to the additive version of quiver W${}_{q,t^{-1}}$ 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 $ϵ$-deformed W algebras are spectral duals of conventional W algebras, in particular the $ϵ$-deformed conformal blocks manifestly reproduce instanton partition functions of 4d N = 2 quiver gauge theories in the full $\Omega$-background and give dual integral representations of ordinary W conformal blocks.