[Paper Review] Towards the Mirror Symmetry for Calabi-Yau Complete intersections in Gorenstein Toric Fano Varieties

Lev Borisov|ArXiv.org|Oct 2, 1993

Algebraic Geometry and Number TheoryMathematics1 references99 citations

TL;DR

This paper proposes a combinatorial duality for reflexive lattice polyhedra that generalizes Batyrev's polar duality to Calabi-Yau complete intersections in Gorenstein toric Fano varieties. By introducing nef-partitions of the vertices of a reflexive polytope and constructing dual polyhedra via piecewise linear functions, the authors establish a mirror symmetry correspondence between families of Calabi-Yau complete intersections in dual toric varieties, conjecturally realizing mirror symmetry for this broader class of Calabi-Yau manifolds.

ABSTRACT

We propose a combinatorical duality for lattice polyhedra which conjecturally gives rise to the pairs of mirror symmetric families of Calabi-Yau complete intersections in toric Fano varieties with Gorenstein singularities. Our construction is a generalization of the polar duality proposed by Batyrev for the case of hypersurfaces.

Motivation & Objective

• To extend Batyrev's polar duality for hypersurfaces to complete intersections in Gorenstein toric Fano varieties.
• To provide a systematic construction of mirror symmetric families of Calabi-Yau complete intersections beyond the hypersurface case.
• To define a combinatorial involution on nef-partitions of reflexive polyhedra that induces mirror symmetry.
• To establish a duality between reflexive polyhedra and their duals via piecewise linear functions and vertex partitions.
• To conjecture that this duality yields mirror pairs of Calabi-Yau complete intersections in dual toric Fano varieties.

Proposed method

• Define a reflexive polyhedron Δ in Mℝ containing 0 in its interior, with its dual Δ* in Nℝ as {y ∈ Nℝ | ⟨x,y⟩ ≥ -1 for all x ∈ Δ}.
• Introduce a nef-partition of the vertices of Δ into disjoint subsets E₁,…,Er such that there exist integral convex Σ[Δ]-piecewise linear functions φ₁,…,φᵣ with φᵢ(eⱼ)=1 if eⱼ ∈ Eᵢ and 0 otherwise.
• Construct r convex polyhedra Δᵢ = Conv({0} ∪ Eᵢ) and r dual polyhedra ∇ᵢ = {y ∈ Nℝ | ⟨x,y⟩ ≥ -φᵢ(x)} for i=1,…,r.
• Define the dual reflexive polyhedron ∇ = Conv(∇₁ ∪ ⋯ ∪ ∇ᵣ), showing that ∇ is reflexive and ∇* = Δ₁ + ⋯ + Δᵣ.
• Prove that the dual of the dual construction recovers the original, establishing an involution on the set of reflexive polyhedra with nef-partitions.
• Conjecture that this duality induces mirror symmetry for Calabi-Yau complete intersections in the toric Fano varieties PΔ* and P∇*.

Experimental results

Research questions

• RQ1 Can Batyrev's polar duality for Calabi-Yau hypersurfaces be generalized to complete intersections in Gorenstein toric Fano varieties?
• RQ2 What combinatorial structure on reflexive polyhedra gives rise to mirror symmetry for complete intersections?
• RQ3 How can nef-partitions of the vertices of a reflexive polytope be used to define dual families of Calabi-Yau complete intersections?
• RQ4 Is there a natural duality between the polyhedral data of two dual toric Fano varieties that induces mirror symmetry for their Calabi-Yau complete intersections?
• RQ5 Does the construction of dual polyhedra via piecewise linear functions yield a self-inverse involution on the set of reflexive polyhedra with nef-partitions?

Key findings

• The dual of the dual construction recovers the original reflexive polyhedron with its nef-partition, establishing an involution on the set of reflexive polyhedra with nef-partitions.
• The construction satisfies Δ* = ∇₁ + ⋯ + ∇ᵣ and ∇* = Δ₁ + ⋯ + Δᵣ, showing a symmetric duality between the two polyhedral families.
• The dual polyhedron ∇ is reflexive, ensuring that the mirror construction remains within the class of Gorenstein toric Fano varieties.
• The vertices of the dual polyhedra ∇ᵢ are precisely the vertices of ∇, and the dual partition E′₁,…,E′ᵣ forms a nef-partition of ∇, closing the duality loop.
• The duality induces a mirror symmetry correspondence between families of Calabi-Yau complete intersections in PΔ* and P∇*, as conjectured.
• The duality is realized via piecewise linear functions φᵢ and their duals ψᵢ, with ψᵢ(y) = -minₓ∈Δᵢ⟨x,y⟩, ensuring compatibility between the dual constructions.