Modularity of Bershadsky-Polyakov minimal models

Abstract

The Bershadsky-Polyakov algebras are the original examples of nonregular W-algebras, obtained from the affine vertex operator algebras associated with ${\mathfrak{sl}}_3$ by quantum hamiltonian reduction. In [arXiv:2007.03917], we explored the representation theories of the simple quotients of these algebras when the level $\mathsf{k}$ is nondegenerate-admissible. Here, we combine these explorations with Adamović's inverse quantum hamiltonian reduction functors to study the modular properties of Bershadsky-Polyakov characters and deduce the associated Grothendieck fusion rules. The results are not dissimilar to those already known for the affine vertex operator algebras associated with ${\mathfrak{sl}}_2$, except that the role of the Virasoro minimal models in the latter is here played by the minimal models of Zamolodchikov's ${\mathsf{W}}_3$ algebras.

arXiv comment37 pages, 1 figure

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Cited by in corpus (8)

• Admissible-level ${\mathfrak{sl}}_3$ minimal models
• Subregular W-algebras of type A
• Rigid tensor structure on big module categories for some $W$-(super)algebras in type $A$
• On the structure of W-algebras in type A
• Orthosymplectic Feigin-Semikhatov duality
• Fusion rules and rigidity for weight modules over the simple admissible affine $\mathfrak{sl}(2)$ and $\mathcal{N}=2$ superconformal vertex operator superalgebras
• Associated varieties of simple affine VOAs $L_k(s{l}_3)$ and $W$-algebras $W_k(s{l}_3,f)$
• Connecting affine $\mathcal{W}$-algebras: A case study on ${\mathfrak{sl}}_4$