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

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 dual 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 maps the space of states of the original $G_{\text{adm}}$ orbifold onto the space of states of $G^*_{\text{adm}}$ orbifold. We show that this transformation is by construction a mirror isomorphism of spaces of states. Thus, mirror isomorphism of states is built into the construction. We illustrate our approach for the orbifolds of $\textbf{A}_{2}\textbf{E}_7^{3}$ model.