Explicit construction of states in orbifolds of products of $N=2$ Superconformal ADE Minimal models

Abstract: We generalize the explicit construction of fields in orbifolds of products of $N=(2,2)$ minimal models, developed by A.Belavin, V.Belavin and S.Parkhomenko to include minimal models with D and E-type modular invariants. It is shown that spectral flow twisting by the elements of admissible group $G_{\text{adm}}$, which is used in the construction of the orbifold, is consistent with the nondiagonal pairing of D and E-type minimal models. We obtain the complete set of fields of the orbifold from the mutual locality and other requirements of the conformal bootstrap. The collection of mutually local primary fields is labeled by the elements of mirror group $G^{*}_{\text{adm}}$. The permutation of $G_{\text{adm}}$ and $G^*_{\text{adm}}$ is given by the mirror spectral flow construction of the fields and transforms the original orbifold into a mirror one. This transformation is by construction an isomorphism of $N=(2,2)$ models. We illustrate our construction for the orbifolds of $\textbf{A}_{2}\textbf{E}_7^{3}$ model.