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

TL;DR

The paper addresses extending explicit field constructions in orbifolds of products of $N=(2,2)$ minimal models to include ADE-type D and E invariants. It employs spectral flow twisting by the admissible group $G_{ ext{adm}}$ and conformal bootstrap to generate a complete orbifold field spectrum, subsequently defining a dual admissible group $G_{ ext{adm}}^{*}$ and a mirror spectral flow to realize a mirror isomorphism between the original and dual state spaces. The main contributions are the explicit construction of orbifold fields labeled by $G_{ ext{adm}}^{*}$, the generalized duality (a Berglund–Hubsch–Krawitz-like relation) for ADE composites, and concrete examples including $ extbf{A}_{2} extbf{E}_7^{3}$ and Gepner-type three-generation models, with a clear link to Calabi–Yau complete intersections. The results advance a conformal bootstrap perspective on mirror symmetry for ADE orbifolds and provide explicit state-level tools for Calabi–Yau sigma-models, with potential applications to Type II and Heterotic compactifications and correlation-function computations.

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 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.