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 compactified on the torus of $E\left(8\right)\times SO\left(10\right)$ 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, starting from the mirror pair of Batyrev lattices corresponding to a given CY manifold, we construct vertex operators of the complete physical theory as cohomology of Borisov differentials that correspond to points of reflexive Batyrev polyhedra. In particular, we show how the number of 27, $\overline{27}$ and Singlet representations of $E\left(6\right)$ is determined by the data of reflexive Batyrev polytope that determines this CY-manifold.