Calabi–Yau metric

E551968

Hermitian metric Kähler metric Ricci-flat metric Riemannian metric

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.

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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 NERFINISHED ⓘ
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 NERFINISHED ⓘ
hasAssociatedObject	Kähler form NERFINISHED ⓘ holomorphic volume form ⓘ
hasConsequence	vanishing of the beta function in certain sigma models ⓘ
hasHolonomy	SU(n) NERFINISHED ⓘ
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 NERFINISHED ⓘ 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 NERFINISHED ⓘ 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 NERFINISHED ⓘ complex structure moduli space ⓘ moduli space of Ricci-flat Kähler metrics ⓘ

Referenced by (1)

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

Calabi–Yau manifold → hasStructure → Calabi–Yau metric ⓘ