Free-Field Construction of Heterotic String Compactified on Calabi-Yau Orbifolds via Correspondence with $\mathcal{N}{=}2$ SCFT Minimal Models

We establish a correspondence between the free-field construction and the minimal-model construction of the Calabi--Yau sector of the four-dimensional heterotic string compactified on Berglund--Hübsch type Calabi--Yau manifolds and their orbifolds.

For Fermat-type polynomials the Calabi--Yau vertex operators expressed in terms of free fields are shown to correspond to products of primary fields of $\mathcal{N}{=}2$ minimal models.

Using this correspondence we verify modular invariance of the free-field construction and extend it to Berglund--Hübsch Calabi--Yau orbifolds, deriving the conditions on complete vertex operators that parallel those of the minimal-model construction.