New approaches for studying conformal embeddings and collapsing levels for $W$--algebras

In this paper we prove a general result saying that under certain hypothesis an embedding of an affine vertex algebra into an affine $W$--algebra is conformal if and only if their central charges coincide. This result extends our previous result obtained in the case of minimal affine $W$-algebras. We also find a sufficient condition showing that certain conformal levels are collapsing. This new condition enables us to find some levels $k$ where $W_k(sl(N), x, f )$ collapses to its affine part when $f$ is of hook or rectangular type. Our methods can be applied to non-admissible levels. In particular, we prove Creutzig's conjecture on the conformal embedding in the hook type $W$-algebra $W_k(sl(n+m), x, f_{m,n})$ of its affine vertex subalgebra. Quite surprisingly, the problem of showing that certain conformal levels are not collapsing turns out to be very difficult. In the cases when $k$ is admissible and conformal, we prove that $W_k(sl(n+m), x, f_{m,n})$ is not collapsing. Then, by generalizing the results on semi-simplicity of conformal embeddings from our previous papers, we find many cases in which $W_k(sl(n+m), x, f_{m,n})$ is semi-simple as a module for its affine subalgebra at conformal level and we provide explicit decompositions.