Mirror symmetry for Calabi-Yau hypersurfaces in weighted P4 and extensions of Landau-Ginzburg theory Original Research Article

Recently two groups have listed all sets of weights k = (kl,…, k5) such that the weighted projective space P4k admits a transverse Calabi-Yau hypersurface. It was noticed that the corresponding Calabi-Yau manifolds do not form a mirror symmetric set since some 850 of the 7555 manifolds have Hodge numbers (b11, b21) whose mirrors do not occur in the list. By means of Batyrevʹs construction we have checked that each of the 7555 manifolds does indeed have a mirror. The ‘missing mirrors’ are constructed as hypersurfaces in toric varieties. We show that many of these manifolds may be interpreted as non-transverse hypersurfaces in weighted P4ʹs, i.e. hypersurfaces for which dp vanishes at a point other than the origin. This falls outside the usual range of Landau-Ginzburg theory. Nevertheless Batyrevʹs procedure provides a way of making sense of these theories.