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Calabi–Yau manifold

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From Wikipedia, the free encyclopedia

Riemannian manifold with SU(n) holonomy

"Calabi–Yau" redirects here. For the play by Susanna Speier, see Calabi-Yau (play).

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A 2D slice of a 6D Calabi–Yau quintic manifold.<br>In algebraic and differential geometry, a Calabi–Yau manifold , also known as a Calabi–Yau space , is a particular type of manifold which has certain properties, such as Ricci flatness, yielding applications in theoretical physics. Particularly in superstring theory, the extra dimensions of spacetime are sometimes conjectured to take the form of a 6-dimensional Calabi–Yau manifold, which led to the idea of mirror symmetry. Their name was coined by Candelas et al. (1985), after Eugenio Calabi (1954, 1957),[1] who first conjectured that compact complex manifolds of Kähler type with vanishing first Chern class always admit Ricci-flat Kähler metrics, and Shing-Tung Yau (1978), who proved the Calabi conjecture.

Calabi–Yau manifolds are complex manifolds that are generalizations of K3 surfaces in any number of complex dimensions (i.e. any even number of real dimensions). They were originally defined as compact Kähler manifolds with a vanishing first Chern class and a Ricci-flat metric, though many other similar but inequivalent definitions are sometimes used.

Definitions<br>[edit]

The definition was motivated by the work of Shing-Tung Yau, who proved Eugenio Calabi's conjecture that any compact Kähler manifold with vanishing first Chern class also admits a (typically different) Kähler metric with vanishing Ricci tensor.[2]

There are many other definitions of a Calabi–Yau manifold used by different authors, some inequivalent. This section summarizes some of the more common definitions and the relations between them.

A Calabi–Yau

{\displaystyle n}

-fold or Calabi–Yau manifold of (complex) dimension

{\displaystyle n}

is sometimes defined as a compact

{\displaystyle n}

-dimensional Kähler manifold

{\displaystyle M}

satisfying one of the following equivalent conditions:

The canonical line bundle of

{\displaystyle M}

is trivial.

{\displaystyle M}

has a holomorphic

{\displaystyle n}

-form that vanishes nowhere.

The structure group of the tangent bundle of

{\displaystyle M}

can be reduced from

{\displaystyle \mathrm {U} (n)}

, the unitary group, to

{\displaystyle \mathrm {SU} (n)}

, the special unitary group, in a manner that is compatible with the complex structure.

{\displaystyle M}

has a Kähler metric with global holonomy contained in

{\displaystyle \mathrm {SU} (n)}

These conditions imply that the first integral Chern class

{\displaystyle c_{1}(M)}

of

{\displaystyle M}

vanishes. Nevertheless, the converse is not true. The simplest examples where this happens are hyperelliptic surfaces, finite quotients of a complex torus of complex dimension 2, which have vanishing first integral Chern class but non-trivial canonical bundle.

For a compact complex

{\displaystyle n}

-dimensional manifold

{\displaystyle M}

that admits Kähler metrics, the following conditions are equivalent to each other, but are weaker than the conditions above, though they are sometimes used as the definition of a Calabi–Yau manifold:

{\displaystyle M}

has vanishing first real Chern class.

{\displaystyle M}

has a Kähler metric with vanishing Ricci curvature.

{\displaystyle M}

has a Kähler metric with local holonomy contained in

{\displaystyle \mathrm {SU} (n)}

A positive power of the canonical bundle of

{\displaystyle M}

is trivial.

{\displaystyle M}

has a finite cover that has trivial canonical bundle.

{\displaystyle M}

has a finite cover that is a product of a torus and a simply connected manifold with trivial canonical bundle.

If a compact Kähler manifold is simply connected, then the weak definition above is equivalent to the stronger definition. Enriques surfaces give examples of complex manifolds that have Ricci-flat metrics, but their canonical bundles are not trivial, so they are Calabi–Yau manifolds according to the second but not the first definition above. On the other hand, their double covers are Calabi–Yau manifolds for both definitions (in fact, K3 surfaces).

By far the hardest part of proving the equivalences between the various properties above is proving the existence of Ricci-flat metrics. This follows from Yau's proof of the Calabi conjecture, which implies...