Mirror Pairs of Quintic Orbifolds

Abstract

Two constructions of mirror pairs of Calabi-Yau manifolds are compared by example of quintic orbifolds $$\mathcal{Q}$$ . The first, Berglund—Hubsch—Krawitz, construction is as follows. If X is the factor of the hypersurface $$\mathcal{Q}$$ by a certain subgroup H′ of the maximum allowed group SL, the mirror manifold Y is defined as the factor by the dual subgroup H,′T. In the second, Batyrev, construction, the toric manifold T containing the mirror Y as a hypersurface specified by zeros of the polynomial WY is determined from the properties of the polynomial WX specifying the Calabi-Yau manifold X. The polynomial WY is determined in an explicit form. The group of symmetry of the polynomial WY is found from its form and it is tested whether it coincides with that predicted by the Berglund—Hubsch—Krawitz construction.