Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: New approaches for studying conformal embeddings and collapsing levels
  for $W$--algebras   In this paper we prove a general result saying that under mild 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 show collapsing of $W_k(sl(N), x, f )$ for $f$ 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.

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The categories of this paper are: cmp-lg
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The categories of this paper are: chem-ph
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The categories of this paper are: stat
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The categories of this paper are: patt-sol
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The categories of this paper are: atom-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: solv-int
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The categories of this paper are: gr-qc
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The categories of this paper are: q-alg
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The categories of this paper are: cond-mat
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The categories of this paper are: dg-ga
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The categories of this paper are: chao-dyn
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The categories of this paper are: acc-phys
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The categories of this paper are: hep-ph
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The categories of this paper are: quant-ph
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The categories of this paper are: nucl-th