Calabi-Yau manifold
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A Calabi-Yau manifold is a special type of manifold that shows up in certain branches of mathematics such as algebraic geometry, as well as 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. The definition of a Calabi-Yau manifold, given below, is somewhat technical.
Formal definition
A Calabi-Yau manifold is a 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.
Equivalently one may 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.
The first Chern class vanishes if and only if the canonical bundle is trivial, which in turn is the case if and only if the canonical class is the zero class. While the Chern class fails to be well-defined for singular Calabi-Yau's, the canonical bundle and canonical class may still be defined and so may be used to extend to definition of a smooth Calabi-Yau manifold to a possibly singular Calabi-Yau variety.
Examples
In one complex dimension, the only compact 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. A one-dimensional Calabi-Yau manifold is also called an elliptic curve over the complex numbers.
In two complex dimensions, the torus T4 and the K3 manifolds furnish the only compact 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, although Yau suspects that there are a finite number of families (albeit a much bigger number than his estimate from 20 years ago). One example of a 3 dimensional Calabi-Yau manifold is a non-singular quintic threefold in CP4, which is the algebraic variety consisting of all of the zeros of a homogeneous quintic polynomial in the homogeneous coordinates of the CP4. Some discrete quotients of the quintic by various Z5 actions are also Calabi-Yau and have received a lot of attention in the literature. One of these is related to the original quintic by mirror symmetry.
For every n, the set of zeros of a general homogeneous degree n+2 polynomial in the homogeneous coordinates of the complex projective space CPn+1 is a compact Calabi-Yau n-fold, although it is not always a differentiable manifold. The case n=1 describes an elliptic curve, while for n=2 one obtains a K3 surface, one of which is a singular Z2 quotient of the 4-torus.
Applications 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, in the absence of fluxes, compactification on a Calabi-Yau 3-fold (real dimension 6) leaves one quarter of the original supersymmetry unbroken if the holonomy is the full SU(3).
More generally, a flux-free compactification on an N-manifold with holonomy SU(N) leaves 21-N of the original supersymmetry unbroken, corresponding to 26-N supercharges in a compactification of type II supergravity or 25-N supercharges in a compactification of type I. When fluxes are included the supersymmetry condition instead implies that the compactification manifold be a generalized Calabi-Yau, a notion introduced in 2002 by Nigel Hitchin in his paper [Generalized Calabi-Yau Manifolds].
Essentially, Calabi-Yau manifolds are shapes that satisfy the requirement of space for the six "unseen" spatial dimensions of string theory, which may be smaller than our currently observable lengths as they have not yet been detected. A popular alternative known as large extra dimensions, which often occurs in braneworld models, is that the Calabi-Yau is large but we are confined to a small subset on which it intersects a D-brane.
See also: hyper-Kähler manifold
References
[Calabi-Yau Homepage] is an interactive reference which describes many examples and classes of Calabi-Yau manifolds and also the physical theories in which they appear.
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