On the equivalence of Batyrev and BHK Mirror symmetry constructions

TL;DR

This paper investigates the relationship between two prominent mirror symmetry constructions for Calabi-Yau orbifolds, providing evidence that the Berglund-H"ubsch-Krawitz and Batyrev methods are essentially equivalent.

Contribution

The paper offers a straightforward demonstration of the equivalence between the BHK and Batyrev mirror symmetry constructions for Calabi-Yau orbifolds.

Findings

Evidence supporting the equivalence of BHK and Batyrev constructions

Clarification of the relationship between hypersurfaces in weighted projective spaces and toric varieties

Insight into the duality of group actions in mirror symmetry

Abstract

We consider the connection between two constructions of the mirror partner for the Calabi-Yau orbifold. This orbifold is defined as a quotient by some suitable subgroup $G$ of the phase symmetries of the hypersurface $ X_M $ in the weighted projective space, cut out by a quasi-homogeneous polynomial $W_M$. The first, Berglund-H\"ubsch-Krawitz (BHK) construction, uses another weighted projective space and the quotient of a new hypersurface $X_{M^T}$ inside it by some dual group $G^T$. In the second, Batyrev construction, the mirror partner is constructed as a hypersurface in the toric variety defined by the reflexive polytope dual to the polytope associated with the original Calabi-Yau orbifold. We give a simple evidence of the equivalence of these two constructions.