Calabi-Yau manifold
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In mathematics, a Calabi-Yau manifold is a compact Kähler manifold with a vanishing first Chern class. A Calabi-Yau manifold of complex dimension n is also called a Calabi-Yau n-fold. The mathematician Eugenio Calabi conjectured in 1957 that all such manifolds admit a Ricci-flat metric (one in each Kähler class), and this conjecture was proved by Shing-Tung Yau in 1977 and became Yau's theorem. Consequently, a Calabi-Yau manifold can also be defined as a compact Ricci-flat Kähler manifold.
It is also possible to define a Calabi-Yau n-fold as a manifold with an SU(n) holonomy. Yet another equivalent condition is that the manifold admit a global nowhere vanishing holomorphic (n,0)-form.
Examples
In one complex dimension, the only examples are family of tori. Note that the Ricci-flat metric on the torus is actually a flat metric, so that the holonomy is the trivial group, for which SU(1) is another name.
In two complex dimensions, the torus T4 and the K3 manifolds furnish the only examples. T4 is sometimes excluded from the classification of being a Calabi-Yau, as its holonomy (again the trivial group) is a proper subgroup of SU(2), instead of being isomorphic to SU(2). On the other hand, the holonomy group of a K3 surface is the full SU(2), so it may properly be called a Calabi-Yau in 2 dimensions.
In three complex dimensions, classification of the possible Calabi-Yaus is an open problem. One example of a 3 dimensional Calabi-Yau is the quintic threefold in CP4. add more on the quintic
Application in string theory
Calabi-Yau manifolds are important in superstring theory. In the most conventional superstring models, ten conjectural dimensions in string theory are supposed to come as four of which we are aware, carrying some kind of fibration with fiber dimension six. Compactification on Calabi-Yau n-folds are important because they leave some of the original supersymmetry unbroken. More precisely, compactification on a Calabi-Yau 3-fold (real dimension 6) leaves one quarter of the original supersymmetry unbroken.
See also: HyperKähler manifold