Free field construction of Heterotic string compactified on Calabi-Yau manifolds of Berglund-Hubsch type in the Batyrev-Borisov combinatorial approach

Heterotic string models in $4$-dimensions are the hybrid theories of a left-moving $N=1$ fermionic string whose additional $6$-dimensions are compactified on a $N=2$ SCFT theory with the central charge $9$, and a right-moving bosonic string, whose additional dimensions are also compactified on $N=2$ SCFT theory with the central charge $9$, and the remaining $13$ dimensions of which form the torus of $E(8)\times SO(10)$ Lie algebra. The important class of exactly solvable Heterotic string models considered earlier by D. Gepner corresponds to the products of $N=2$ minimal models with the total central charge $c=9$. These models are known to describe Heterotic string models compactified on Calabi-Yau manifolds, which belong a special subclass of general CY manifolds of Berglund-Hubsch type. We generalize this construction to all cases of compactifications on Calabi-Yau manifolds of general Berglund-Hubsch type, using Batyrev-Borisov combinatorial approach. In particular, we show how the number of $27$, $\overline{27}$ and Singlet representations of $E(6)$ is determined by the data of reflexive Batyrev polytope that determines this CY-manifold.