Classification of irreducible highest weight modules for the parafermion vertex algebras N_k(\mathfraksl₂) at arbitrary level

2026/08/03 by Drazen Adamovic, Qing Wang
Mathematics · #math.QA #math.RT #msc:17B69

paper · pdf

37 pages

arxiv created 2026/08/03 · arxiv updated 2026/08/04

Abstract

Let N^k(\mathfraksl₂) be the universal parafermion vertex algebra and N_k(\mathfraksl₂) its simple quotient. We classify all irreducible highest weight N^k(\mathfraksl₂)-modules for every k≠ 0. We give a presentation of Zhu's algebra A(N^k(\mathfraksl₂)) 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(\mathfraksl₂)) 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(\mathfraksl₂)-modules are parametrized by a two-parameter family L_k[x,y], (x,y)∈\mathbb C², constructed using a free-field realization. We also prove that each N^k(\mathfraksl₂)-module L_k[x,y] can be realized as an N^k(\mathfraksl₂)-submodule of an irreducible weight module M for the universal affine vertex algebra V^k(\mathfraksl₂). At non-integral admissible levels, we prove that L_k[x,y] is an N_k(\mathfraksl₂)-module if and only if the associated V^k(\mathfraksl₂)-module M is an L_k(\mathfraksl₂)-module. This gives the classification of irreducible highest weight N_k(\mathfraksl₂)-modules at all non-integral admissible levels. We also classify the irreducible highest weight modules for the simple parafermion algebra N_-2(\mathfraksl₂) at the critical level. At positive integral levels, the irreducible N_k(\mathfraksl₂)-modules were previously classified by Arakawa, Lam, and Yamada.

Citations

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