Dual Polyhedra and Mirror Symmetry for Calabi-Yau Hypersurfaces in Toric Varieties

Abstract

We consider families $\mathcal{F}(\Delta )$ consisting of complex $(n-1)$-dimensional projective algebraic compactifications of $\Delta$-regular affine hypersurfaces $Z_f$ defined by Laurent polynomials $f$ with a fixed $n$-dimensional Newton polyhedron $\Delta$ in $n$-dimensional algebraic torus $\mathbf{T}=({\mathbf{C}}^{\ast }{)}^n$. If the family $\mathcal{F}(\Delta )$ defined by a Newton polyhedron $\Delta$ consists of $(n-1)$-dimensional Calabi-Yau varieties, then the dual, or polar, polyhedron ${\Delta }^{\ast }$ in the dual space defines another family $\mathcal{F}({\Delta }^{\ast })$ 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 $\mathcal{F}(\Delta )$ and $\mathcal{F}({\Delta }^{\ast })$.

43 pages, Latex

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• A Generalized Construction of Mirror Manifolds
• Quantum Cohomology Rings of Toric Manifolds
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• On Stokes Matrices of Calabi-Yau hypersurfaces
• Tautological systems under the conifold transition on $G(2,4)$
• Towards the Doran-Harder-Thompson conjecture via the Gross-Siebert program
• Fano horospherical variety
• On complete intersections with trivial canonical class
• Toward the classification of higher-dimensional toric Fano varieties
• Equivalence of Mirror Families Constructed from Toric Degenerations of Flag Varieties
• On toric Calabi-Yau hypersurfaces fibered by weighted K3 hypersurfaces
• Rational Ehrhart Theory
• On the moments of moments of random matrices and Ehrhart polynomials
• Root bundles: Applications to F-theory Standard Models
• Vafa-Witten Theory: Invariants, Floer Homologies, Higgs Bundles, a Geometric Langlands Correspondence, and Categorification (String Math 2022 Proceedings)
• Normal Forms and Tyurin Degenerations of K3 Surfaces Polarised by a Rank 18 Lattice
• Stringy Hodge numbers and Virasoro algebra
• First coniveau notch of the Dwork family and its mirror
• Classification of Gorenstein Toric Del Pezzo Varieties in arbitrary dimension
• K3 surfaces from configurations of six lines in ${\mathbb{P}}^2$ and mirror symmetry II --- ${\lambda }_{K3}$-functions ---
• All Abelian Quotient C.I.-Singularities Admit Projective Crepant Resolutions in All Dimensions
• Mirror symmetry for tropical hypersurfaces and patchworking
• Superpotentials of D-branes in Calabi-Yau Manifolds with Several Moduli by Mirror Symmetry and Blown-up
• On smooth Calabi-Yau threefolds of Picard number two
• Coincidences between Calabi-Yau manifolds of Berglund-Hubsch type and Batyrev polytopes
• On mirrors of elliptically fibered K3 surfaces
• (0,2) String Compactifications: The Higgs Mechanism
• Polytope duality for families of $K3$ surfaces associated to singularities $Q_{16}$ and $S_{16}$
• Lattice duality for coupling pairs admitting polytope duality with trivial toric contribution
• Complete intersection Calabi--Yau threefolds in Hibi toric varieties and their smoothing
• Families of Calabi-Yau hypersurfaces in $\mathbb{Q}$-Fano toric varieties