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

TL;DR

This paper explores the structure of VOAs derived from 3d $ abla=4$ quiver gauge theories, revealing their connection to Higgs branches and affine $ ext{W}$-algebras, and introduces a systematic construction method.

Contribution

It introduces a new perspective on H-twisted VOAs as uplifted Higgs branches, constructs explicit homomorphisms to affine $ ext{W}$-algebras, and develops an algorithm for free-field realizations of VOAs from quivers.

Findings

H-twisted VOAs can be seen as the chiralization of Higgs branches.

Constructed explicit homomorphisms to affine $ ext{W}$-algebras.

Developed an algorithm for free-field realizations of VOAs.

Abstract

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 can be regarded as the ''chiralization'' of the extended Higgs branch: many of the ingredients of the Higgs branch are naturally ''uplifted'' into the VOAs, while conversely the Higgs branch can be recovered as the associated variety of the VOA. We also discuss the connection of our VOA with affine $\mathcal{W}$-algebras. For example, we construct an explicit homomorphism from an affine $\mathcal{W}$-algebra $\mathcal{W}^{-n+1}(\mathfrak{gl}_n,f_{\mathrm{min}})$ into the H-twisted VOA for $T^{[2,1^{n-2}]}_{[1^n]}[\mathrm{SU}(n)]$ theories. Motivated by the relation with affine $\mathcal{W}$-algebras, we introduce a reduction procedure for the quiver diagram, and use this to give an algorithm to systematically construct novel free-field realizations for VOAs associated with general linear quivers.