Cosets from equivariant $\mathcal {W}$-algebras

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.

When $\mathfrak {g}$ is of type $ADE$, we identify $A[\mathfrak {g}, \kappa , 1]$ with the affine vertex algebra $V^{\kappa -1}(\mathfrak {g}) \otimes L_1(\mathfrak {g})$, giving a new and efficient proof of the coset realization of the principal $\mathcal W$-algebras of type $ADE$.