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

TLDR: In this paper, a realisation of the universal/simple Bershadsky-Polyakov vertex algebra as subalgebras of the tensor product of the Zamolodchikov vertex algebra and an isotropic lattice vertex algebra is presented.

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 $$\mathfrak {osp} (1 \vert 2)$$ given in Adamovic (Commun Math Phys 366:1025–1067, 2019). Relaxed highest-weight modules are likewise constructed, conditions for their irreducibility are established, and their characters are explicitly computed, generalising the character formulae of Kawasetsu and Ridout (Commun Math Phys 368:627–663, 2019).