Weight module classifications for Bershadsky--Polyakov algebras

Abstract

A. The Bershadsky-Polyakov algebras are the subregular quantum hamiltonian reductions of the affine vertex operator algebras associated with 𝔰𝔩 3 . In [5], we realised these algebras in terms of the regular reduction, Zamolodchikov's W 3 -algebra, and an isotropic lattice vertex operator algebra. We also proved that a natural construction of relaxed highestweight Bershadsky-Polyakov modules has the property that the result is generically irreducible. Here, we prove that this construction, when combined with spectral flow twists, gives a complete set of irreducible weight modules whose weight spaces are finite-dimensional. This gives a simple independent proof of the main classification theorem of [30] for nondegenerate admissible levels and extends this classification to a category of weight modules. We also deduce the classification for the nonadmissible level k = − 7 3 , which is new.

Subregular W-algebras of type A

“…This inverse reduction was used in [12] to determine modular transformations and conjecture Grothendiek fusion rules for Bershadsky-Polyakov minimal models in terms of such data for W 3 minimal models. The same inverse reduction also facilitates a classification of irreducible weight modules (with finite-dimensional weight spaces) for W k ( 3 , ) and, for certain k, its simple quotient [13].…”

“…It was later shown by Adamović that this inverse reduction can be deployed to understand some of the representation theory of V k ( 2 ) and, at certain levels, L k ( 2 ) [1]. An analogous embedding for 3 was studied in [2], and an inverse to a partial reduction for 3 was studied in [11][12][13]. These inverse reductions have proven exceptionally useful in constructing interesting modules, as well as computing data important for conformal field theoretic applications.…”