Bershadsky-Polyakov vertex algebras at positive integer levels and duality

Abstract

We study the simple Bershadsky-Polyakov algebra ${\mathcal{W}}_k={\mathcal{W}}_k(s{l}_3,f_{\theta })$ 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 $L_{k^{\prime }}(osp(1|2))$ for $k^{\prime }=-5/4$. Using the free-field realization of $L_{k^{\prime }}(osp(1|2))$ 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 ${\mathcal{W}}_k$.

24 pages