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

TL;DR

This paper constructs a large class of new Calabi-Yau 3-folds with small Picard numbers, proposes mirror constructions via conifold transitions, and computes new Picard-Fuchs operators for their mirror families.

Contribution

It introduces a novel method for constructing Calabi-Yau 3-folds and their mirrors using conifold transitions and toric hypersurfaces, expanding known examples significantly.

Findings

Identified 68 topologically distinct Calabi-Yau 3-folds with h11=1.

Found 30241 reflexive 4-polytopes with smoothable Calabi-Yau hypersurfaces.

Computed new Picard-Fuchs operators for mirror families.

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.