Bershadsky-Polyakov vertex algebras at positive integer levels and duality

TL;DR

This paper classifies irreducible modules of Bershadsky-Polyakov algebras at positive integer levels, confirms a conjecture, and reveals a duality with certain affine vertex superalgebras, providing new free-field realizations.

Contribution

It classifies modules of Bershadsky-Polyakov algebras at positive levels and establishes a duality with affine vertex superalgebras, including explicit free-field realizations.

Findings

Confirmed the conjecture on irreducible modules classification.

Discovered a Kazama-Suzuki-type duality at level k=1.

Provided a free-field realization of the algebra and modules.

Abstract

We study the simple Bershadsky-Polyakov algebra $\mathcal W_k = \mathcal{W}_k(sl_3,f_{\theta})$ at positive integer levels and classify their irreducible modules. In this way we confirm the conjecture from arXiv:1910.13781. Next, we study the case $k=1$. We discover that this vertex algebra has a Kazama-Suzuki-type dual isomorphic to the simple afine vertex superalgebra $L_{k'} (osp(1 \vert 2))$ for $k'=-5/4$. Using the free-field realization of $L_{k'} (osp(1 \vert 2))$ from arXiv:1711.11342, we get a free-field realization of $\mathcal W_k$ and their highest weight modules. In a sequel, we plan to study fusion rules for $\mathcal W_k$.