Cosets from equivariant W-algebras

TL;DR

This paper constructs a new family of vertex algebras from equivariant W-algebras and affine vertex algebras, revealing their structure as conformal extensions related to Lie algebra duality.

Contribution

It introduces a novel family of vertex algebras as subalgebras of equivariant W-algebras combined with affine vertex algebras, expanding understanding of their structure and relations.

Findings

Constructed vertex algebras $A[rak{g}, ka, n]$ as subalgebras of equivariant W-algebras tensor affine vertex algebras.

Showed these algebras are conformal extensions of tensor products of affine and principal W-algebras.

Established specific level relations for the conformal extensions.

Abstract

The equivariant $\mathcal{W}$-algebra of a simple Lie algebra $\mathfrak{g}$ is a BRST reduction of the algebra of chiral differential operators on the Lie group of $\mathfrak{g}$. We construct a family of vertex algebras $A[\mathfrak{g}, \kappa, n]$ as subalgebras of the equivariant $\mathcal{W}$-algebra of $\mathfrak{g}$ tensored with the integrable affine vertex algebra $L_n(\check{\mathfrak{g}})$ of the Langlands dual Lie algebra $\check{\mathfrak{g}}$ at level $n\in \mathbb{Z}_{>0}$. They are conformal extensions of the tensor product of an affine vertex algebra and the principal $\mathcal{W}$-algebra whose levels satisfy a specific relation.