MIRROR SYMMETRY FOR CALABI-YAU HYPERSURFACES IN WEIGHTED P-4 AND EXTENSIONS OF LANDAU-GINZBURG THEORY

Recently two groups have listed all sets of weights k=(k(1),...,k(5)) such that the weighted projective space P-4(k) admits a transverse Cal abi-Yau hypersurface. It was noticed that the corresponding Calabi-Yau manifolds do not form a mirror symmetric set since some 850 of the 75 55 manifolds have Hedge numbers (b(11), b(21)) whose mirrors do not oc cur in the list. By means of Batyrev's construction we have checked th at 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 hype rsurfaces in weighted P-4's, i.e. hypersurfaces for which dp vanishes at a point other than the origin. This falls outside the usual range o f Landau-Ginzburg theory. Nevertheless Batyrev's procedure provides a way of making sense of these theories.