W-algebras as conformal extensions of affine VOAs

We provide a criterion for a vertex operator superalgebra homomorphism from an affine vertex algebra to another vertex superalgebra to be conformal, and an additional criterion that guarantees that this homomorphism is surjective.

This situation is applied to W-algebras and W-superalgebras and we list all cases where our criterion applies.

This gives many new examples of W-algebras that collapse to affine vertex algebras or are conformal extensions.

In particular, we provide many examples of simple W-algebras at non-admissible levels that collapse to admissible level affine vertex algebras.