Modularity of Bershadsky-Polyakov minimal models

TL;DR

This paper investigates the modular properties and fusion rules of Bershadsky-Polyakov minimal models, revealing parallels with affine $rak{sl}_2$ models but involving Zamolodchikov's $ ext{W}_3$ minimal models.

Contribution

It combines previous representation theory results with inverse quantum hamiltonian reduction to analyze modularity and fusion in Bershadsky-Polyakov models.

Findings

Derived modular transformation properties of characters.

Established Grothendieck fusion rules for these models.

Connected the structure to $ ext{W}_3$ 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 Adamovi\'{c}'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.