Affine weighted star trees with central parameter k are classified by reducing the positive-semidefinite null-vector condition to the Egyptian-fraction equation sum 1/(r_i+1) = m-k for each fixed (m,k).
Dual Polyhedra and Mirror Symmetry for Calabi-Yau Hypersurfaces in Toric Varieties
7 Pith papers cite this work. Polarity classification is still indexing.
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abstract
We consider families ${\cal F}(\Delta)$ 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 ${\bf T} =({\bf C}^*)^n$. If the family ${\cal F}(\Delta)$ defined by a Newton polyhedron $\Delta$ consists of $(n-1)$-dimensional Calabi-Yau varieties, then the dual, or polar, polyhedron $\Delta^*$ in the dual space defines another family ${\cal F}(\Delta^*)$ 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 ${\cal F}(\Delta)$ and ${\cal F}(\Delta^*)$.
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UNVERDICTED 7representative citing papers
Quasi-finite Feynman integrals produce sparse Fano and reflexive polytopes that encode degenerate Calabi-Yau varieties and link to del Pezzo surfaces, K3 surfaces, and Calabi-Yau threefolds.
QCD axions constrain F-theory base threefolds to have rigid or flux-rigidified divisors, yielding typical axion masses around 10^{-9} eV and decay constants near 10^{15} GeV in allowed regions.
Transpolar pairs involving VEX multitopes yield smooth toric spaces whose Chern classes satisfy Todd-Hirzebruch identities and belong to deformation families of generalized complete intersections.
Deformations in algebraic superstring models indicate a non-algebraic generalization that aligns with mirror duality requirements.
A roadmap paper describing potential applications of numerical Ricci-flat Calabi-Yau metrics to heterotic string phenomenology and mathematical questions in special geometry.
Lecture notes providing a technical introduction to naturalness problems and the string theory landscape for graduate students.
citing papers explorer
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Star-Shaped Integral Cartan-Type Matrices and an Egyptian-Fraction Classification of Affine Weighted Trees
Affine weighted star trees with central parameter k are classified by reducing the positive-semidefinite null-vector condition to the Egyptian-fraction equation sum 1/(r_i+1) = m-k for each fixed (m,k).
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Fano and Reflexive Polytopes from Feynman Integrals
Quasi-finite Feynman integrals produce sparse Fano and reflexive polytopes that encode degenerate Calabi-Yau varieties and link to del Pezzo surfaces, K3 surfaces, and Calabi-Yau threefolds.
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Constraining F-theory Model Building with QCD Axions
QCD axions constrain F-theory base threefolds to have rigid or flux-rigidified divisors, yielding typical axion masses around 10^{-9} eV and decay constants near 10^{15} GeV in allowed regions.
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Chern Characteristics and Todd-Hirzebruch Identities for Transpolar Pairs of Toric Spaces
Transpolar pairs involving VEX multitopes yield smooth toric spaces whose Chern classes satisfy Todd-Hirzebruch identities and belong to deformation families of generalized complete intersections.
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Beyond Algebraic Superstring Compactification: Part II
Deformations in algebraic superstring models indicate a non-algebraic generalization that aligns with mirror duality requirements.
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What to do with a Ricci-flat Calabi--Yau metric?
A roadmap paper describing potential applications of numerical Ricci-flat Calabi-Yau metrics to heterotic string phenomenology and mathematical questions in special geometry.
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Lectures on Naturalness, String Landscape and Multiverse
Lecture notes providing a technical introduction to naturalness problems and the string theory landscape for graduate students.