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 Fehily et al. (Comm Math Phys 385:859–904, 2021), 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.