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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.

Calabi–Yau manifolds are complex manifolds that are higher-dimensional analogues of K3 surfaces. They are sometimes defined as compact Kähler manifolds whose canonical bundle is trivial, though many other similar but inequivalent definitions are sometimes used. They were named "Calabi–Yau spaces" by Candelas et al. (1985) after E. Calabi (1954, 1957) who first studied them, and S. T. Yau (1978) who proved the Calabi conjecture that they have Ricci flat metrics. 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.

Definitions

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

A Calabi–Yau n-fold or Calabi–Yau manifold of dimension n is sometimes defined as a compact n-dimensional Kähler manifold M satisfying one of the following equivalent conditions:

The canonical bundle of M is trivial.
M has a holomorphic n-form that vanishes nowhere.
The structure group of M can be reduced from U(n) to SU(n).
M has a Kähler metric with global holonomy contained in SU(n).

These conditions imply that the first integral Chern class c1(M) of M vanishes, but 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 the canonical bundle is not trivial.

For a compact n-dimensional Kähler manifold M the following conditions are equivalent to each other, but are weaker than the conditions above, and are sometimes used as the definition of a Calabi–Yau manifold:

M has vanishing first real Chern class.
M has a Kähler metric with vanishing Ricci curvature.
M has a Kähler metric with local holonomy contained in SU(n).
A positive power of the canonical bundle of M is trivial.
M has a finite cover that has trivial canonical bundle.
M has a finite cover that is a product of a torus and a simply connected manifold with trivial canonical bundle.

In particular 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. 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 that a compact Kähler manifold with a vanishing first real Chern class has a Kähler metric in the same class with vanishing Ricci curvature. (The class of a Kähler metric is the cohomology class of its associated 2-form.) Calabi showed such a metric is unique.

There are many other inequivalent definitions of Calabi–Yau manifolds that are sometimes used, which differ in the following ways (among others):

The first Chern class may vanish as an integral class or as a real class.
Most definitions assert that Calabi–Yau manifolds are compact, but some allow them to be non-compact. In the generalization to non-compact manifolds, the difference $$(\Omega\wedge\bar\Omega - J^n/n!)$$ must vanish asymptotically. Here, J is the Kähler form associated with the Kähler metric, g (Tian 1990, 1991).
Some definitions put restrictions on the fundamental group of a Calabi–Yau manifold, such as demanding that it be finite or trivial. Any Calabi–Yau manifold has a finite cover that is the product of a torus and a simply-connected Calabi–Yau manifold.
Some definitions require that the holonomy be exactly equal to SU(n) rather than a subgroup of it, which implies that the Hodge numbers hi,0 vanish for 0 < i < dim(M). Abelian surfaces have a Ricci flat metric with holonomy strictly smaller than SU(2) (in fact trivial) so are not Calabi–Yau manifolds according to such definitions.
Most definitions assume that a Calabi–Yau manifold has a Riemannian metric, but some treat them as complex manifolds without a metric.
Most definitions assume the manifold is non-singular, but some allow mild singularities. While the Chern class fails to be well-defined for singular Calabi–Yau's, the canonical bundle and canonical class may still be defined if all the singularities are Gorenstein, and so may be used to extend the definition of a smooth Calabi–Yau manifold to a possibly singular Calabi–Yau variety.

Examples

In one complex dimension, the only compact examples are tori, which form a one-parameter family. The Ricci-flat metric on a torus is actually a flat metric, so that the holonomy is the trivial group SU(1). A one-dimensional Calabi–Yau manifold is a complex elliptic curve, and in particular, algebraic.

In two complex dimensions, the K3 surfaces furnish the only compact simply connected Calabi–Yau manifolds. Non simply-connected examples are given by abelian surfaces. Enriques surfaces and hyperelliptic surfaces have first Chern class that vanishes as an element of the real cohomology group, but not as an element of the integral cohomology group, so Yau's theorem about the existence of a Ricci-flat metric still applies to them but they are sometimes not considered to be Calabi–Yau manifolds. Abelian surfaces are sometimes excluded from the classification of being Calabi–Yau, as their holonomy (again the trivial group) is a proper subgroup of SU(2), instead of being isomorphic to SU(2).

In three complex dimensions, classification of the possible Calabi–Yau manifolds is an open problem, although Yau suspects that there is a finite number of families (albeit a much bigger number than his estimate from 20 years ago). One example of a three-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. Another example is a smooth model of the Barth–Nieto quintic. 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 positive integer n, the zero set of a non-singular homogeneous degree n+2 polynomial in the homogeneous coordinates of the complex projective space CPn+1 is a compact Calabi–Yau n-fold. The case n=1 describes an elliptic curve, while for n=2 one obtains a K3 surface.

All hyper-Kähler manifolds are Calabi–Yau.
Applications in superstring 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 by Hitchin (2003). These models are known as flux compactifications.

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.

F-theory compactifications on various Calabi–Yau four-folds provide physicists with a method to find a large number of classical solution in the so-called string theory landscape.
References

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