Mirror Symmetry for Calabi-Yau Hypersurfaces in Weighted P_4 and Extensions of Landau Ginzburg Theory

Philip Candelas; Sheldon Katz; Xenia de la Ossa

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Mirror Symmetry for Calabi-Yau Hypersurfaces in Weighted P_4 and Extensions of Landau Ginzburg Theory

Philip Candelas, Xenia de la Ossa, Sheldon Katz

High Energy Physics - Theory · Submitted 13 Dec 1994

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The authors' main claim By means of Batyrev's construction we have checked that each of the 7555 manifolds does indeed have a mirror.

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arXiv IDhep-th/9412117 (version 1)

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CitationPhilip Candelas, Sheldon Katz, Xenia de la Ossa. "Mirror Symmetry for Calabi-Yau Hypersurfaces in Weighted P_4 and Extensions of Landau Ginzburg Theory." arXiv:hep-th/9412117 (1994). https://pith.science/paper/JZUSZOBA

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What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.50 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.93 · compares with prior results 0.88 · mentions code or data availability 0.02.

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Recently two groups have listed all sets of weights (k_1,...,k_5) such that the weighted projective space P_4^{(k_1,...,k_5)} admits a transverse Calabi-Yau hypersurface. It was noticed that the corresponding Calabi-Yau manifolds do not form a mirror symmetric set since some 850 of the 7555 manifolds have Hodge numbers (b_{11},b_{21}) whose mirrors do not occur in the list. By means of Batyrev's construction we have checked that each of the 7555 manifolds does indeed have a mirror. The `missing mirrors' are constructed as hypersurfaces in toric varieties. We show that many of these manifolds may be interpreted as non-transverse hypersurfaces in weighted P_4's, ie, hypersurfaces for which dp vanishes at a point other than the origin. This falls outside the usual range of Landau--Ginzburg theory. Nevertheless Batyrev's procedure provides a way of making sense of these theories.

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What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Recently two groups have listed all sets of weights (k_1,...,k_5) such that the weighted projective space P_4^{(k_1,...,k_5)} admits a transverse Calabi-Yau hypersurface.	background 1.00	0.04	0.03	asserted 1.00
s1	It was noticed that the corresponding Calabi-Yau manifolds do not form a mirror symmetric set since some 850 of the 7555 manifolds have Hodge numbers (b_{11},b_{21}) whose mirrors do not occur in the list.	gap 0.60	0.15	0.11	asserted 1.00
s2	By means of Batyrev's construction we have checked that each of the 7555 manifolds does indeed have a mirror.	result 0.75	0.91	0.24	asserted 1.00
s3	The `missing mirrors' are constructed as hypersurfaces in toric varieties.	result 0.64	0.72	0.17	asserted 1.00
s4	We show that many of these manifolds may be interpreted as non-transverse hypersurfaces in weighted P_4's, ie, hypersurfaces for which dp vanishes at a point other than the origin.	interpretation 0.59	0.88	0.27	hedged possibility 0.81
s5	This falls outside the usual range of Landau--Ginzburg theory.	interpretation 0.46	0.11	0.05	asserted 1.00
s6	Nevertheless Batyrev's procedure provides a way of making sense of these theories.	interpretation 0.79	0.40	0.31	asserted 0.83

What kind of paper the classifier reports this is: theoretical result 0.95 · evidence from proof or derivation 0.73.

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Original abstract

Recently two groups have listed all sets of weights (k_1,...,k_5) such that the weighted projective space P_4^{(k_1,...,k_5)} admits a transverse Calabi-Yau hypersurface. It was noticed that the corresponding Calabi-Yau manifolds do not form a mirror symmetric set since some 850 of the 7555 manifolds have Hodge numbers (b_{11},b_{21}) whose mirrors do not occur in the list. By means of Batyrev's construction we have checked that each of the 7555 manifolds does indeed have a mirror. The `missing mirrors' are constructed as hypersurfaces in toric varieties. We show that many of these manifolds may be interpreted as non-transverse hypersurfaces in weighted P_4's, ie, hypersurfaces for which dp vanishes at a point other than the origin. This falls outside the usual range of Landau--Ginzburg theory. Nevertheless Batyrev's procedure provides a way of making sense of these theories.

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Cited by 5 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

GLSM's for gerbes (and other toric stacks)
hep-th 2005-02 accept novelty 7.0 of 10

Gauged linear sigma models with nonminimal U(1) charges describe toric stacks, and their infrared physics depends only on the stack, not on the chosen presentation.
Ricci-Flat Mirror Hypersurfaces in Spaces of General Type
hep-th 2025-01 conditional novelty 5.0 of 10

Laurent-polynomial anticanonical sections can smooth Calabi-Yau hypersurfaces in general-type toric/torus manifolds and yield transposition mirrors with computable GLSM data.
Chern Characteristics and Todd-Hirzebruch Identities for Transpolar Pairs of Toric Spaces
hep-th 2024-03 unverdicted novelty 5.0 of 10

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.
Beyond Algebraic Solutions to Stringy Spacetime
hep-th 2026-05 unverdicted novelty 3.0 of 10

Generalizations beyond algebraic geometry in string theory remain aligned with mirror symmetry, support quantitative analysis, and point to deeper symplectic geometry connections.
The Calabi-Yau Landscape: from Geometry, to Physics, to Machine-Learning
hep-th 2018-12 conditional novelty 2.0 of 10

The draft is a pedagogical survey connecting Calabi-Yau geometry, string-theory model building, computational algebraic geometry, and machine-learning applications, with code examples and dataset pointers for beginners.

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12 arXiv papers cite this in 23 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 16 sentences without a confident label

Builds on it 2 papers

Throughout this paper, we focus on this distinctly algebraic quality, and defer its relation to the subtler complex-analytic property of smoothness (related to Cauchy-Riemann conditions and similar) for later.] Calabi-Yau hypersurfaces, and (b) automatically realize natural pairs of mirror Calabi-Yau $n$-folds generalizing earlier results [ref] , associated to “” pairs of polytopes.

A Generalized Construction of Calabi-Yau Models and Mirror Symmetry · Incentives, results and summary · 0.90
It generalizes to the involution of VEX s, $XX ^-4muX$ [ref] , including the rigorous results on 2-dimensional generalized legal loops [ref] , and encoding the mirror map between entire deformation families of Calabi–Yau (anticanonical) hypersurfaces, $X[c_1]MM[c_1]$, vastly generalizing the transposition mirror construction [ref] :

Chern Characteristics and Todd-Hirzebruch Identities for Transpolar Pairs of Toric Spaces · The Mirror-Symmetric Framework · 0.97

Compares with it 1 paper

Given our choice of vertices and monomials [ref], this chain of equalities is in complete agreement with the detailed analysis in Section 3 of Ref. [ref] .

A Generalized Construction of Calabi-Yau Models and Mirror Symmetry · Mirror models · 0.91

Cites it as background 4 papers

The hypersurface equation defining the potential Calabi-Yau in the weighted projective space then becomes a generic hypersurface in the toric variety's anticanonical divisor class [ref] ; with alternative interpretations as non-transverse hypersurfaces in weighted projective spaces [ref] .

Calabi-Yau Four/Five/Six-folds as $\mathbb{P}^n_\textbf{w}$ Hypersurfaces: Machine Learning, Approximation, and Generation · Background · 0.94
Such Landau-Ginzburg orbifolds have been studied in the context of mirror symmetry and LG/CY-correspondence in [ref] .

D-brane central charge and Landau-Ginzburg orbifolds · Two-parameter family 2 · 0.98
These, when plotted in terms of their Hodge numbers $h^1,1 - h^2,1$ against $h^1,1 + h^2,1$, gave rise to the famous funnel shape with left-right symmetry signifying mirror symmetry [ref] .

Calabi-Yau Volumes and Reflexive Polytopes · Introduction · 0.80

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