[Paper Review] Constructing new Calabi-Yau 3-folds and their mirrors via conifold transitions

Victor V. Batyrev, Maximilian Kreuzer|ArXiv.org|Feb 22, 2008

Advanced Algebra and Geometry22 references4 citations

TL;DR

This paper constructs 68 new topologically distinct Calabi–Yau 3-folds with h¹¹ = 1 via conifold transitions from toric hypersurfaces with conifold singularities, using smoothings of 210 reflexive 4-polytopes. It proposes a mirror construction via flat deformations and computes 28 new Picard–Fuchs operators for 1-parameter families, generalizing mirror symmetry beyond toric geometry and enabling quantum cohomology computations.

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.

Motivation & Objective

• To construct new Calabi–Yau 3-folds with small Picard numbers using conifold transitions from toric hypersurfaces.
• To generalize mirror symmetry beyond toric geometry by proposing a mirror construction via smoothing of singular Calabi–Yau hypersurfaces.
• To compute Picard–Fuchs operators for 1-parameter families of mirror Calabi–Yau 3-folds, enabling quantum cohomology and instanton number computations.
• To classify and enumerate topologically distinct Calabi–Yau 3-folds arising from smoothable toric hypersurfaces with conifold singularities.
• To explore the relationship between diffeomorphism types and Picard–Fuchs operators, testing whether rational variable changes preserve instanton numbers.

Proposed method

• Identify reflexive 4-polytopes with only triangles and minimal-volume parallelograms as 2-faces, yielding 198,849 such polytopes.
• Use Namikawa’s criterion to select 30,241 reflexive 4-polytopes whose associated Calabi–Yau hypersurfaces are smoothable via flat deformation.
• Construct Calabi–Yau 3-folds as smoothings of generic hypersurfaces in Gorenstein toric Fano 4-folds defined by these polytopes.
• Apply toric geometry and MPPC (maximal partial projective crepant) desingularization to compute Hodge numbers and topological invariants.
• Derive Picard–Fuchs operators for 1-parameter families by direct evaluation of the principal period and mirror map.
• Verify that different Picard–Fuchs operators for the same diffeomorphism type are related by rational changes of variables, preserving instanton numbers.

Experimental results

Research questions

• RQ1 Which reflexive 4-polytopes yield Calabi–Yau hypersurfaces that are smoothable via flat deformation, and how many such smooth Calabi–Yau 3-folds arise?
• RQ2 Can a mirror construction be systematically defined for non-toric Calabi–Yau 3-folds using conifold transitions and toric degenerations?
• RQ3 How are Picard–Fuchs operators related across different representatives of the same diffeomorphism type, and do they yield identical instanton numbers?
• RQ4 What is the topological diversity of Calabi–Yau 3-folds with h¹¹ = 1 constructed via conifold transitions from toric hypersurfaces?
• RQ5 Can the proposed mirror construction be extended to complete intersections and higher-genus topological string amplitudes?

Key findings

• The authors identify 210 reflexive 4-polytopes that define 68 topologically distinct Calabi–Yau 3-folds with h¹¹ = 1.
• Among these, 28 diffeomorphism types yield new 1-parameter families for which Picard–Fuchs operators are explicitly computed.
• For the diffeomorphism type X⁴⁵₁₄₄,₁₂₀, two different Picard–Fuchs operators are related by the transformation z → z/(1 + 4z), and both yield the same instanton numbers: n(0) = {3744, 50112, 1656320, ...} up to 8th order.
• For the diffeomorphism type X⁵¹₂₀₀,₁₄₀, two operators are related by z → z/(1 − 4z), and both produce identical instanton numbers: n(0) = {2600, 25600, 530000, ...} up to 8th order.
• The results confirm that rational changes of variables preserve instanton numbers, supporting the consistency of the proposed mirror construction.
• The study reveals that diffeomorphism types with small h¹² may have up to five distinct Picard–Fuchs operators, suggesting a rich structure in the moduli space of mirror families.