Mirror Symmetry for Calabi-Yau Hypersurfaces in Weighted P_4 and Extensions of Landau Ginzburg Theory

TL;DR

This paper investigates mirror symmetry for Calabi-Yau hypersurfaces in weighted projective spaces, extending Landau-Ginzburg theory to non-transverse cases using Batyrev's construction and toric varieties.

Contribution

It demonstrates that many non-transverse hypersurfaces can be interpreted within mirror symmetry, expanding Landau-Ginzburg theory beyond its traditional scope.

Findings

Confirmed mirror pairs for all 7555 Calabi-Yau manifolds.

Constructed missing mirrors as hypersurfaces in toric varieties.

Extended Landau-Ginzburg framework to non-transverse hypersurfaces.

Abstract

Recently two groups have listed all sets of weights (k_1,...,k_5) such that the weighted projective space P_4^{(k_1,...,k_5)} 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 (b_{11},b_{21}) whose mirrors do not occur in the list. By means of Batyrev's 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 P_4's, ie, hypersurfaces for which dp vanishes at a point other than the origin. This falls outside the usual range of Landau--Ginzburg theory. Nevertheless Batyrev's procedure provides a way of making sense of these theories.