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We study Vertex Operator Algebras (VOAs) obtained from the H-twist of 3d$\mathcal{N}=4$ linear quiver gauge theories. We find that H-twisted VOAs canbe regarded as the ''chiralization'' of the extended Higgs branch: many of theingredients of the Higgs branch are naturally ''uplifted'' into the VOAs, whileconversely the Higgs branch can be recovered as the associated variety of theVOA. We also discuss the connection of our VOA with affine$\mathcal{W}$-algebras. For example, we construct an explicit homomorphism froman affine $\mathcal{W}$-algebra$\mathcal{W}^{-n+1}(\mathfrak{gl}_n,f_{\mathrm{min}})$ into the H-twisted VOAfor $T^{[2,1^{n-2}]}_{[1^n]}[\mathrm{SU}(n)]$ theories. Motivated by therelation with affine $\mathcal{W}$-algebras, we introduce a reduction procedurefor the quiver diagram, and use this to give an algorithm to systematicallyconstruct novel free-field realizations for VOAs associated with general linearquivers.

Affine $\mathcal{W}$-algebras and Miura maps from 3d $\mathcal N=4$ non-Abelian quiver gauge theories