Calabi–Yau manifold

E129502

Kähler manifold Ricci-flat manifold Riemannian manifold algebraic variety complex manifold geometric object projective variety

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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All labels observed (5)

Label	Occurrences
Calabi–Yau compactifications	1
Calabi–Yau manifold canonical	1
Calabi–Yau manifolds	1
K3 surface	1
mirror Calabi–Yau manifold	1

Statements (63)

Predicate	Object
instanceOf	Kähler manifold ⓘ Ricci-flat manifold ⓘ Riemannian manifold ⓘ algebraic variety ⓘ complex manifold ⓘ geometric object ⓘ projective variety ⓘ
appearsIn	heterotic string compactifications ⓘ superstring compactification from 10D to 4D ⓘ type II string compactifications ⓘ
centralConceptIn	Strominger–Yau–Zaslow conjecture ⓘ mirror symmetry conjecture ⓘ string phenomenology ⓘ
developedBy	Eugenio Calabi ⓘ Shing-Tung Yau ⓘ
dimension	complex dimension n ≥ 1 ⓘ
example	Calabi–Yau manifold self-linksurface differs ⓘ surface form: K3 surface complete intersection Calabi–Yau threefold ⓘ complex torus with trivial canonical bundle and appropriate holonomy ⓘ quintic threefold in ℙ^4 ⓘ
fieldOfStudy	algebraic geometry ⓘ complex geometry ⓘ differential geometry ⓘ string theory ⓘ
hasInvariant	Euler characteristic ⓘ Hodge numbers ⓘ Kähler cone ⓘ Kähler moduli ⓘ Picard number ⓘ complex structure moduli ⓘ fundamental group ⓘ
hasProperty	Kähler form is closed ⓘ admits a Ricci-flat Kähler metric ⓘ admits a nowhere-vanishing holomorphic volume form ⓘ c1 = 0 in H^2(M,ℝ) ⓘ c1(M)=0 in H^2(M,ℤ) for algebraic Calabi–Yau ⓘ holonomy contained in SU(n) ⓘ vanishing first Chern class ⓘ
hasStructure	Calabi–Yau metric ⓘ covariantly constant spinor ⓘ holomorphic tangent bundle ⓘ trivial canonical bundle ⓘ
implies	Ricci curvature tensor vanishes ⓘ first Betti number b1 = 0 for simply connected case ⓘ preservation of some supersymmetry in compactifications ⓘ
mirrorTo	Calabi–Yau manifold self-linksurface differs ⓘ surface form: mirror Calabi–Yau manifold
namedAfter	Eugenio Calabi ⓘ Shing-Tung Yau ⓘ
relatedConcept	G2 manifold ⓘ Kähler–Einstein metric ⓘ canonical bundle ⓘ holonomy group SU(n) ⓘ special holonomy ⓘ
typicalDimension	complex dimension 3 in string theory ⓘ
usedIn	complex algebraic geometry ⓘ mathematical physics ⓘ mirror symmetry ⓘ string compactification ⓘ string theory ⓘ superstring theory ⓘ supersymmetric field theories ⓘ topological string theory ⓘ
YauTheorem	existence of Ricci-flat Kähler metric given c1=0 ⓘ

Referenced by (5)

Full triples — surface form annotated when it differs from this entity's canonical label.

Kähler manifold → hasExample → Calabi–Yau manifold ⓘ

Calabi–Yau manifold → example → Calabi–Yau manifold self-linksurface differs ⓘ

this entity surface form: K3 surface

Calabi–Yau manifold → mirrorTo → Calabi–Yau manifold self-linksurface differs ⓘ

this entity surface form: mirror Calabi–Yau manifold

Seiberg–Witten theory → relatedTo → Calabi–Yau manifold ⓘ

this entity surface form: Calabi–Yau compactifications

Shing-Tung Yau → knownFor → Calabi–Yau manifold ⓘ

this entity surface form: Calabi–Yau manifolds