Duality via convolution of W-algebras

Creutzig, Thomas; Linshaw, Andrew R.; Nakatsuka, S.; Sato, Ryo

doi:10.48550/arxiv.2203.01843

Cited by 1 publication

(2 citation statements)

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“…The same methods can also be applied for hook type W -algebras of types B, C, D and for hook type W -superalgebras (see [22,Table 1] for the list of such vertex algebras). We hope to study these cases in our forthcoming papers.…”

Section: Conformal Vs Collapsing Levelsmentioning

confidence: 99%

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New Approaches for Studying Conformal Embeddings and Collapsing Levels for W–Algebras

Adamović

Frajria

Papi

2023

International Mathematics Research Notices

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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.

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Section: Conformal Vs Collapsing Levelsmentioning

confidence: 99%

“…Hook type W -algebras and rectangular W -algebras. Hook type W -algebras recently appeared in [37] as coset vertex algebras, and also in the context of dualities and trialities of various vertex algebras [19,20,22]. In these papers the authors mainly studied the generic level case.…”

Section: Introductionmentioning

confidence: 99%