Constructing new Calabi-Yau 3-folds and their mirrors via conifold transitions

TL;DR

The paper constructs new Calabi–Yau 3-folds $X$ and their mirrors $X^*$ via conifold transitions from toric hypersurfaces in 4-dimensional Gorenstein toric Fano varieties $\mathbb{P}_{\Delta}$, associated to reflexive 4-polytopes $\Delta$. It extends toric-degeneration mirror constructions from Grassmannians and Flag manifolds to this toric setting and analyzes conifold transitions on both $X$ and $X^*$. It shows there exist $198849$ reflexive 4-polytopes with 2-faces restricted to unimodular triangles or parallelograms of minimal volume; Namikawa's smoothing criterion identifies $30241$ polytopes whose hypersurfaces are smoothable, yielding $210$ polytopes defining $68$ Calabi–Yau 3-folds with $h_{11}=1$. The authors also explain the mirror construction and compute new Picard–Fuchs operators for the resulting 1-parameter families, enabling study of quantum cohomology and non-toric mirrors.

Abstract

We construct a surprisingly large class of new Calabi-Yau 3-folds $X$ with small Picard numbers and propose a construction of their mirrors $X^*$ using smoothings of toric hypersurfaces with conifold singularities. These new examples are related to the previously known ones via conifold transitions. Our results generalize the mirror construction for Calabi-Yau complete intersections in Grassmannians and flag manifolds via toric degenerations. There exist exactly 198849 reflexive 4-polytopes whose 2-faces are only triangles or parallelograms of minimal volume. Every such polytope gives rise to a family of Calabi-Yau hypersurfaces with at worst conifold singularities. Using a criterion of Namikawa we found 30241 reflexive 4-polytopes such that the corresponding Calabi-Yau hypersurfaces are smoothable by a flat deformation. In particular, we found 210 reflexive 4-polytopes defining 68 topologically different Calabi--Yau 3-folds with $h_{11}=1$. We explain the mirror construction and compute several new Picard--Fuchs operators for the respective 1-parameter families of mirror Calabi-Yau 3-folds.