Calabi-Yau manifolds up to dimension 4 have found application in physics in heterotic string/ F-theory duality, N = 1 string vacua that admit both a heterotic description and a dual F-theory description. We are interested in a heterotic string on a Calabi-Yau orbifold. For a singular Calabi-Yau orbifolds of Fermat type, we consider a variant of the Borcea-Voisin construction of Calabi-Yau manifolds from an elliptic curve and some Calabi-Yau orbifolds.
For this we define Roan pairs of elements of a group and its dual in the Berglund-Hübsch-Krawits setting of the Landau-Ginzburg/Landau-Ginzburg mirror symmetry, and efficiently compute Hodge numbers of Calabi-Yau manifolds. For orbifolds of Fermat type, our formulas for Hodge numbers resemble the Batyrev-Borisov for orbifolds of toric hypersurfaces, but are different and give different sets of monomials for deformations.
The talk is based on joint work with Alexander Belavin and Maxim Malyutin. | 
