ADE subalgebras of the triplet vertex algebra W(p): D_m-series

Abstract: We are continuing our study of ADE-orbifold subalgebras of the triplet vertex algebra W(p). This part deals with the dihedral series. First, subject to a certain constant term identity, we classify all irreducible modules for the vertex algebra $\bar{M(1)} ^+$, the $\Z_2$--orbifold of the singlet vertex algebra $\bar{M(1)}$. Then we classify irreducible modules and determine Zhu's and $C_2$--algebra for the vertex algebra $\triplet ^{D_2}$. A general method for construction of twisted $\triplet$--modules is also introduced. We also discuss classification of twisted $\bar{M(1)}$--modules including the twisted Zhu's algebra $A_{\Psi} (\bar{M(1)})$, which is of independent interest. The category of admissible $\Psi$-twisted $\bar{M(1)}$-modules is expected to be semisimple. We also prove $C_2$-cofiniteness of $\triplet^{D_m}$ for all $m$, and give a conjectural list of irreducible $\triplet^{D_m}$-modules. Finally, we compute characters of the relevant irreducible modules and describe their modular closure.