Classification of irreducible highest weight modules for the parafermion vertex algebras $N_k({\mathfrak{sl}}_2)$ at arbitrary level

Abstract

Let $N^k({\mathfrak{sl}}_2)$ be the universal parafermion vertex algebra and $N_k({\mathfrak{sl}}_2)$ its simple quotient. We classify all irreducible highest weight $N^k({\mathfrak{sl}}_2)$-modules for every $k\ne 0$. We give a presentation of Zhu's algebra $A(N^k({\mathfrak{sl}}_2))$ as a quotient of a polynomial algebra in four variables by an ideal generated by three explicit polynomials. This presentation also shows that $A(N^k({\mathfrak{sl}}_2))$ is a free module of rank three over the polynomial subalgebra generated by the classes of the fields of weights two and three. The irreducible highest weight $N^k({\mathfrak{sl}}_2)$-modules are parametrized by a two-parameter family $L_k[x,y]$, $(x,y)\in {\mathbb{C}}^2$, constructed using a free-field realization. We also prove that each $N^k({\mathfrak{sl}}_2)$-module $L_k[x,y]$ can be realized as an $N^k({\mathfrak{sl}}_2)$-submodule of an irreducible weight module $M$ for the universal affine vertex algebra $V^k({\mathfrak{sl}}_2)$. At non-integral admissible levels, we prove that $L_k[x,y]$ is an $N_k({\mathfrak{sl}}_2)$-module if and only if the associated $V^k({\mathfrak{sl}}_2)$-module $M$ is an $L_k({\mathfrak{sl}}_2)$-module. This gives the classification of irreducible highest weight $N_k({\mathfrak{sl}}_2)$-modules at all non-integral admissible levels. We also classify the irreducible highest weight modules for the simple parafermion algebra $N_{-2}({\mathfrak{sl}}_2)$ at the critical level. At positive integral levels, the irreducible $N_k({\mathfrak{sl}}_2)$-modules were previously classified by Arakawa, Lam, and Yamada.

37 pages