The web of Calabi-Yau hypersurfaces in toric varieties

TL;DR

This paper demonstrates that Calabi-Yau hypersurfaces in toric varieties form a connected web, simplifying the proof of their moduli space connectedness using maximal polytopes and multiple weight systems.

Contribution

It identifies multiple weight systems in toric Calabi-Yau hypersurfaces and proves their connectedness within the web of CICYs, advancing understanding of their moduli space.

Findings

All Calabi-Yau manifolds with these properties are connected.

The approach simplifies the proof of connectedness via singular limits.

Almost completes the proof of connectedness for toric Calabi-Yau hypersurfaces.

Abstract

Recent results on duality between string theories and connectedness of their moduli spaces seem to go a long way toward establishing the uniqueness of an underlying theory. For the large class of Calabi-Yau 3-folds that can be embedded as hypersurfaces in toric varieties the proof of mathematical connectedness via singular limits is greatly simplified by using polytopes that are maximal with respect to certain single or multiple weight systems. We identify the multiple weight systems occurring in this approach. We show that all of the corresponding Calabi-Yau manifolds are connected among themselves and to the web of CICY's. This almost completes the proof of connectedness for toric Calabi-Yau hypersurfaces.