[Paper Review] Dual Polyhedra and Mirror Symmetry for Calabi-Yau Hypersurfaces in Toric Varieties

Victor V. Batyrev|ArXiv.org|Oct 5, 1993

Algebraic Geometry and Number Theory23 references936 citations

TL;DR

This paper introduces a duality between families of Calabi-Yau hypersurfaces in toric varieties using reflexive polyhedra and their polar duals. By defining a mirror map via the combinatorial involution Δ ↦ Δ*, the authors establish a correspondence that matches Hodge numbers h^{1,1} and h^{2,1} between dual families, providing a systematic construction of mirror pairs and extending known results from weighted projective spaces to general toric varieties.

ABSTRACT

We consider families ${\cal F}(Δ)$ consisting of complex $(n-1)$-dimensional projective algebraic compactifications of $Δ$-regular affine hypersurfaces $Z_f$ defined by Laurent polynomials $f$ with a fixed $n$-dimensional Newton polyhedron $Δ$ in $n$-dimensional algebraic torus ${\bf T} =({\bf C}^*)^n$. If the family ${\cal F}(Δ)$ defined by a Newton polyhedron $Δ$ consists of $(n-1)$-dimensional Calabi-Yau varieties, then the dual, or polar, polyhedron $Δ^*$ in the dual space defines another family ${\cal F}(Δ^*)$ of Calabi-Yau varieties, so that we obtain the remarkable duality between two {\em different families} of Calabi-Yau varieties. It is shown that the properties of this duality coincide with the properties of {\em Mirror Symmetry} discovered by physicists for Calabi-Yau $3$-folds. Our method allows to construct many new examples of Calabi-Yau $3$-folds and new candidats for their mirrors which were previously unknown for physicists. We conjecture that there exists an isomorphism between two conformal field theories corresponding to Calabi-Yau varieties from two families ${\cal F}(Δ)$ and ${\cal F}(Δ^*)$.

Motivation & Objective

• To establish a combinatorial duality between families of Calabi-Yau hypersurfaces in toric varieties using reflexive polyhedra.
• To generalize mirror symmetry constructions from weighted projective spaces to general toric varieties.
• To provide a systematic method for constructing new mirror pairs of Calabi-Yau 3-folds and their candidates.
• To prove that the Hodge diamond symmetry h^{1,1}(V) = h^{2,1}(V*) is realized via the dual polyhedron construction.
• To conjecture an isomorphism between conformal field theories associated with dual Calabi-Yau families.

Proposed method

• Define families F(Δ) of Calabi-Yau hypersurfaces in toric varieties P_Δ as compactifications of affine hypersurfaces defined by Laurent polynomials with fixed Newton polyhedron Δ.
• Introduce Δ-regularity to ensure singularities arise only from the ambient toric variety, enabling simultaneous resolution of all members in F(Δ).
• Characterize reflexive polyhedra Δ ⊂ M_Q as those for which the dual polyhedron Δ* ⊂ N_Q is also reflexive, inducing a duality involution Δ ↦ Δ*.
• Construct the mirror map MIR: F(Δ) → F(Δ*) via the polar duality of reflexive polyhedra.
• Use lattice dualities and quotient constructions to compute fundamental groups of Calabi-Yau hypersurfaces from reflexive simplices.
• Apply the method to weighted projective spaces and show that Fermat-type hypersurfaces and their quotients arise naturally from reflexive simplices.

Experimental results

Research questions

• RQ1 How can mirror symmetry for Calabi-Yau 3-folds be generalized beyond weighted projective spaces to arbitrary toric varieties?
• RQ2 What combinatorial condition on a Newton polyhedron ensures that its associated hypersurface is Calabi-Yau?
• RQ3 Does the duality between a reflexive polyhedron Δ and its polar Δ* induce a mirror symmetry correspondence in terms of Hodge numbers?
• RQ4 What is the fundamental group of a Calabi-Yau hypersurface arising from a reflexive simplex, and how is it related to the weights of the ambient weighted projective space?
• RQ5 Can the construction of mirror pairs via polar duality be extended to complete intersections and other toric compactifications?

Key findings

• The duality Δ ↦ Δ* between reflexive polyhedra induces a mirror map MIR: F(Δ) → F(Δ*) that exchanges Hodge numbers h^{1,1} and h^{2,1}, matching the physical mirror symmetry conjecture.
• All known mirror pairs of Calabi-Yau 3-folds from hypersurfaces in weighted projective spaces are special cases of this construction.
• The family F(Δ) of Calabi-Yau hypersurfaces in P_Δ consists of deformations of Fermat-type hypersurfaces when Δ is a reflexive simplex.
• The fundamental group π₁(Δ) of a reflexive simplex Δ with weights w is isomorphic to the kernel of a surjective homomorphism from (μ_{d₀} × ⋯ × μ_{dₙ})/μ_d to μ_d, where d_i = b_{ii} + 1.
• The order of the fundamental group π₁(Δ) is given by d₀d₁⋯dₙ / d², where d = lcm{d₀, ..., dₙ}.
• For n = 4, the construction recovers Roan’s result on mirror symmetry for Calabi-Yau 3-folds in weighted projective spaces P(w₀, ..., w₄), showing that F(Δ) consists of quotients of Fermat-type hypersurfaces by π₁(Δ, M).