A realisation of the Bershadsky--Polyakov algebras and their relaxed modules

Abstract

We present a realisation of the universal/simple Bershadsky--Polyakov vertex algebras as subalgebras of the tensor product of the universal/simple Zamolodchikov vertex algebras and an isotropic lattice vertex algebra. This generalises the realisation of the universal/simple affine vertex algebras associated to ${\mathfrak{sl}}_2$ and $\mathfrak{osp}(1|2)$ given in arXiv:1711.11342. Relaxed highest-weight modules are likewise constructed, conditions for their irreducibility are established, and their characters are explicitly computed, generalising the character formulae of arXiv:1803.01989.

23 pages; v2 24 pages --- added a few more details and remarks, fixed a few typos; to appear in Lett. Math. Phys