Construction of mirror pairs Calabi-Yau orbifolds of the Berglund-Hubsch type

TL;DR

The paper presents a general algorithm to enumerate all Berglund–Hubsch–type Calabi–Yau orbifolds and their mirrors by constructing admissible deformations and maximal diagonal symmetry groups, and then pairing deformations with mirror deformations via the transpose matrix. It provides a concrete six-step procedure to determine $W_0$ deformations, invariant groups $G_{ ext{adm}}^{i}$ and $G_{ ext{adm}}^{T,i}$, and the mirror deformation sets $L_i$, enabling the computation of $h^{2,1}$ and the generation count through $N_{ ext{gen}}=|h^{2,1}(Y_i)-h^{2,1}(X_i)|$ with $h^{2,1}(X_i)=|R_i|$ and $h^{2,1}(Y_i)=|L_i|$. The authors demonstrate the method on a Fermat-type model with $A=\mathrm{diag}(2,4,6,18,36)$ and $\mathbf{k}=(18,9,6,2,1)$, obtaining 16 mirror pairs $(R_i,L_i)$ with explicit deformation sets, group orders, and generation/singlet counts, including a representative $Y_{15}$ with $|G^{T,15}_{\text{adm}}|=144$. This work provides a practical framework for identifying Calabi–Yau topologies that yield three generations and potentially observable dark-matter candidates in heterotic compactifications.

Abstract

In this paper we have developed general algorithm for finding all orbifolds of Berglund-Hubsch-type Calabi-Yau manifolds and their mirrors. An explicit construction is formulated for finding all admissible deformations and groups defining mirror pairs of orbifolds. Then using our algorithm for one of the Calabi-Yau manifolds, defined by a Fermat-type polynomial, we found all mirror pairs of orbifolds. For this model, for each pair of orbifolds, the number of generations and the number of singlets i.e. particles participating only in gravitational interactions (dark matter particles) were found.