ADE SUBALGEBRAS OF THE TRIPLET VERTEX ALGEBRA 𝒲(p): D-SERIES

Abstract

We are continuing our study of ADE-orbifold subalgebras of the triplet vertex algebra 𝒲(p) initiated in D. Adamović, X. Lin and A. Milas, ADE subalgebras of the triplet vertex algebra 𝒲(p): Am-series, Commun. Contemp. Math.15 (2013), Article ID: 1350028, 1–30. This part deals with the dihedral series. First, subject to a certain constant term identity, we classify all irreducible modules for the vertex algebra [Formula: see text], the ℤ2-orbifold of the singlet vertex algebra [Formula: see text]. Then, we classify irreducible modules and determine Zhu's and C2-algebra for the vertex algebra 𝒲(p)D2. A general method for construction of twisted 𝒲(p)-modules is also introduced. We also discuss classification of twisted [Formula: see text]-modules including the twisted Zhu's algebra [Formula: see text], which is of independent interest. The category of admissible Ψ-twisted [Formula: see text]-modules is expected to be semisimple. We also prove C2-cofiniteness of 𝒲(p)Dm for all m, and give a conjectural list of irreducible 𝒲(p)Dm-modules. Finally, we compute characters of the relevant irreducible modules and describe their modular closure.