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 W_Y is determined from the properties of the polynomial W_X specifying the Calabi-Yau manifold X. The polynomial W_Y is determined in an explicit form. The group of symmetry of the polynomial W_Y is found from its form and it is tested whether it coincides with that predicted by the Berglund—Hubsch—Krawitz construction.