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 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 [18] 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 [2], [5], 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.