Cosets from equivariant 𝒲-algebras

Abstract

The equivariant W \mathcal W -algebra of a simple Lie algebra g \mathfrak {g} is a BRST reduction of the algebra of chiral differential operators on the Lie group of g \mathfrak {g} . We construct a family of vertex algebras A [ g , κ , n ] A[\mathfrak {g}, \kappa , n] as subalgebras of the equivariant W \mathcal W -algebra of g \mathfrak {g} tensored with the integrable affine vertex algebra L n ( g ˇ ) L_n(\check {\mathfrak {g}}) of the Langlands dual Lie algebra g ˇ \check {\mathfrak {g}} at level n ∈ Z > 0 n\in \mathbb {Z}_{>0} . They are conformal extensions of the tensor product of an affine vertex algebra and the principal W \mathcal {W} -algebra whose levels satisfy a specific relation. When g \mathfrak {g} is of type A D E ADE , we identify A [ g , κ , 1 ] A[\mathfrak {g}, \kappa , 1] with the affine vertex algebra V κ − 1 ( g ) ⊗ L 1 ( g ) V^{\kappa -1}(\mathfrak {g}) \otimes L_1(\mathfrak {g}) , giving a new and efficient proof of the coset realization of the principal W \mathcal W -algebras of type A D E ADE .