Calabi–Yau metric
E551968
A Calabi–Yau metric is a special Ricci-flat Kähler metric with SU(n) holonomy that endows Calabi–Yau manifolds with their characteristic geometric and physical properties.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Calabi–Yau metric canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T5837288 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Calabi–Yau metric Context triple: [Calabi–Yau manifold, hasStructure, Calabi–Yau metric]
-
A.
Calabi–Yau manifold
A Calabi–Yau manifold is a special type of complex manifold with vanishing first Chern class that plays a central role in string theory compactifications and complex algebraic geometry.
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B.
Kähler manifold
A Kähler manifold is a complex manifold equipped with a Hermitian metric whose associated symplectic form is closed, making it simultaneously a complex, Riemannian, and symplectic manifold in a compatible way.
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C.
Kobayashi metric
The Kobayashi metric is an intrinsic pseudometric in complex analysis that measures hyperbolic distance on complex manifolds and generalizes the Poincaré metric to higher dimensions.
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D.
Bergman metric
The Bergman metric is a canonical Kähler metric on complex domains derived from the Bergman kernel, widely used in several complex variables and complex differential geometry.
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E.
Kähler–Ricci flow
Kähler–Ricci flow is a geometric evolution equation that deforms Kähler metrics on complex manifolds according to their Ricci curvature, playing a central role in complex differential geometry and the study of canonical metrics.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Calabi–Yau metric Target entity description: A Calabi–Yau metric is a special Ricci-flat Kähler metric with SU(n) holonomy that endows Calabi–Yau manifolds with their characteristic geometric and physical properties.
-
A.
Calabi–Yau manifold
A Calabi–Yau manifold is a special type of complex manifold with vanishing first Chern class that plays a central role in string theory compactifications and complex algebraic geometry.
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B.
Kähler manifold
A Kähler manifold is a complex manifold equipped with a Hermitian metric whose associated symplectic form is closed, making it simultaneously a complex, Riemannian, and symplectic manifold in a compatible way.
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C.
Kobayashi metric
The Kobayashi metric is an intrinsic pseudometric in complex analysis that measures hyperbolic distance on complex manifolds and generalizes the Poincaré metric to higher dimensions.
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D.
Bergman metric
The Bergman metric is a canonical Kähler metric on complex domains derived from the Bergman kernel, widely used in several complex variables and complex differential geometry.
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E.
Kähler–Ricci flow
Kähler–Ricci flow is a geometric evolution equation that deforms Kähler metrics on complex manifolds according to their Ricci curvature, playing a central role in complex differential geometry and the study of canonical metrics.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
Hermitian metric
ⓘ
Kähler metric ⓘ Ricci-flat metric ⓘ Riemannian metric ⓘ |
| appearsIn |
mirror symmetry
ⓘ
topological string theory ⓘ |
| compatibleWith |
complex structure
ⓘ
symplectic structure ⓘ |
| constructedBy | solving a complex Monge–Ampère equation in a fixed Kähler class ⓘ |
| definedFor | complex n-dimensional manifolds ⓘ |
| definedOn | Calabi–Yau manifold ⓘ |
| dimensionOfHolonomyGroup | n^2-1 ⓘ |
| ensures |
N=1 supersymmetry in four-dimensional effective theories from heterotic strings
ⓘ
N=1 supersymmetry in four-dimensional effective theories from type II strings ⓘ preservation of some supersymmetry in compactification ⓘ |
| existsIf | manifold is compact Kähler with vanishing first Chern class ⓘ |
| generalizes | flat metric on complex tori with trivial holonomy subgroup of SU(n) ⓘ |
| guaranteedBy |
Yau's proof of the Calabi conjecture
ⓘ
linked to:
Calabi conjecture
|
| hasAssociatedObject |
Kähler form
ⓘ
holomorphic volume form ⓘ |
| hasConsequence | vanishing of the beta function in certain sigma models ⓘ |
| hasHolonomy |
SU(n)
ⓘ
linked to:
special unitary group SU(n)
|
| hasProperty |
Kähler form is closed
ⓘ
Levi-Civita connection has holonomy contained in SU(n) ⓘ Ricci curvature equal to zero ⓘ admits a covariantly constant spinor ⓘ admits a parallel holomorphic volume form ⓘ holonomy is exactly SU(n) for generic Calabi–Yau manifolds ⓘ is determined by its Kähler potential locally ⓘ is real-analytic in harmonic coordinates ⓘ volume form is parallel with respect to Levi-Civita connection ⓘ |
| implies |
first Chern class of the manifold is zero
ⓘ
manifold is Calabi–Yau ⓘ |
| relatedTo |
Calabi conjecture
ⓘ
G2 holonomy via dimensional reduction ⓘ Spin(7) holonomy via dimensional reduction ⓘ special holonomy ⓘ |
| satisfies |
Einstein field equations with zero cosmological constant in Euclidean signature
ⓘ
Monge–Ampère type equation in local coordinates ⓘ |
| uniqueUpTo |
Kähler class
ⓘ
linked to:
Kähler cone
overall scaling in a fixed Kähler class ⓘ |
| usedIn |
nonlinear sigma models in quantum field theory
ⓘ
string theory compactification ⓘ superstring theory ⓘ supersymmetric compactifications ⓘ |
| usedToDefine |
Kähler moduli space
ⓘ
linked to:
Kähler cone
complex structure moduli space ⓘ moduli space of Ricci-flat Kähler metrics ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Calabi–Yau metric Description of subject: A Calabi–Yau metric is a special Ricci-flat Kähler metric with SU(n) holonomy that endows Calabi–Yau manifolds with their characteristic geometric and physical properties.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.