Classifying relaxed highest-weight modules for admissible-level Bershadsky-Polyakov algebras

Abstract

The Bershadsky-Polyakov algebras are the minimal quantum hamiltonian reductions of the affine vertex algebras associated to ${\mathfrak{sl}}_3$ and their simple quotients have a long history of applications in conformal field theory and string theory. Their representation theories are therefore quite interesting. Here, we classify the simple relaxed highest-weight modules, with finite-dimensional weight spaces, for all admissible but nonintegral levels, significantly generalising the known highest-weight classifications [arxiv:1005.0185, arxiv:1910.13781]. In particular, we prove that the simple Bershadsky-Polyakov algebras with admissible nonintegral $\mathsf{k}$ are always rational in category $\mathcal{O}$, whilst they always admit nonsemisimple relaxed highest-weight modules unless $\mathsf{k}+\frac{3}{2}\in {\mathbb{Z}}_{\ge 0}$.

arXiv comment35 pages, 7 figures, 1 table

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Cited by in corpus (7)

• Parity and Spin CFT with boundaries and defects
• Modularity of Bershadsky-Polyakov minimal models
• Subregular W-algebras of type A
• Rigid tensor structure on big module categories for some $W$-(super)algebras in type $A$
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• Simple modules for Affine vertex algebras in the minimal nilpotent orbit