Quiver $\text{W}_{ε_1,ε_2}$ algebras of 4d $\mathcal{N}=2$ gauge theories

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