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 [3]. We also find a sufficient condition showing that certain conformal levels are collapsing. This new condition enables us to find some levels k where Wk(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 [18] on the conformal embedding in the hook type W-algebra Wk(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 Wk(sl(n+m), x, f m,n) is not collapsing. Then, by generalizing the results on semi-simplicity of conformal embeddings from [2], [5], we find many cases in which Wk(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.