On the equivalence of Batyrev and Berglund-Hübsch-Krawitz Mirror symmetry constructions

Speaker: Alexander Belavin

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 XM in the weighted projective space, cut out by a quasi-homogeneous polynomial. The first, Berglund-Hübsch-Krawitz (BHK) construction, uses another
weighted projective space and the quotient of a new hypersurface XMT inside it by some dual group G^T. In the second, Batyrev's 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 simple evidence of the equivalence of these two constructions. We also discuss the phenomenon of coincidence of periods of the Berglund–Hübsch–Krawitz multiple mirrors.