Calabi-Yau
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Calabi-Yau manifold (3D projection)

In mathematics, Calabi–Yau manifolds are compact Kähler manifolds whose canonical bundle is trivial. They were named "Calabi–Yau spaces" by physicists in 1985,^[1] after E. Calabi who first studied them in (Calabi 1954, 1957), and S. T. Yau who proved the Calabi conjecture that they have Ricci flat metrics in (Yau 1978). In theoretical physics more general definitions are often used, and they may be allowed to be singular or non-compact, In superstring theory the extra dimensions of spacetime are sometimes conjectured to take the form of a 6-dimensional Calabi–Yau manifold. Physical insights about Calabi–Yau manifolds, especially mirror symmetry, led to progress in pure mathematics.

Formal definition

A Calabi–Yau manifold is a compact Kähler manifold M satisfying any of the following equivalent conditions:

• The canonical bundle of M is trivial.
• M has a holomorphic n-form that vanishes nowhere (where n=dim(M)).
• The structure group of M can be reduced from U(n) to SU(n).
• The first integral Chern class c₁(M) of M vanishes.
• M has a Kähler metric with global holonomy contained in SU(n).

A Calabi–Yau manifold of complex dimension n is also called a Calabi–Yau n-fold.

For a compact Kähler manifold M the following conditions are equivalent to each other, but are weaker than the conditions defining a Calabi-Yau manifold (though are sometimes used as the definition of a Calabi-Yau manifold):

• M has vanishing first real Chern class.
• M has a Kähler metric with 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.

The relation between the two groups of properties above is as follows. A compact Kähler manifold has a vanishing first real Chern class if and only if it has a finite cover that is a Calabi-Yau manifold. Moreover this finite cover can be taken to be the product of a torus and a simply connected Calabi-Yau manifold. In particular if a compact Kähler manifold is simply connected (or more generally has torsion-free first homology group) then all the properties above become equivalent. Enriques surfaces give examples of complex manifolds that have Ricci-flat metrics, but their canonical bundles are not trivial so they are not Calabi-Yau manifolds. Thier double covers are Calabi-Yau manifolds (in fact K3 surfaces).

By far the hardest part of proving the equivalences between the various propertiess above is proving the existence of Ricci-flat metrics. This follows from Yau's proof of the the Calabi conjecture, which states that a compact Kähler manifold with a vanishing first real Chern class has a Kähler metric with with vanishing Ricci curvature. Calabi showed that any such metric is unique.

Abelian surfaces have a (Ricci) flat metric with holonomy strictly smaller than SU(2) (in fact 0) so are not Calabi-Yau manifolds according to some definitions that require the holonomy to be exactly equal to SU(n).

There are also many other inequivalent definitions of Calabi-Yau manifolds that are sometimes used. Moreover the literature on them contains many incorrect assertions claiming that inequivalent definitions are equivalent. The various definitions in the literature differ in the following ways (among others):

• 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 must vanish asymptotically. Here, J is the Kähler form associated with the Kähler metric, g.^[2][3]
• Some definitions put restrictions on the fundamental group of a Calabi-Yau manifold, such as demanding that it be trivial. Any Calabi-Yau manifold has a finite cover that is the product of a torus and a simply-connected Calabi-Yau manifold.
• 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.
• Several definitions confuse the real and integral Chern classes (or the local and global holonomy groups), and incorrectly assume that if the first real Chern class vanishes then so does the first integral Chern class.
• Algebraic geometers sometimes add the condition that the Hodge numbers h^i,0 vanish for 0 < i < dim(M).

Examples

In one complex dimension, the only compact examples are tori, which form a one-parameter family. Note that the Ricci-flat metric on a 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 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 not usually 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). 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–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. 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 zero set 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.

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 in 2002 by Nigel Hitchin.^[4] 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.

References

1. ^ Candelas, Horowitz, Strominger and Witten (1985). "Vacuum configurations for superstrings". Nuclear Physics B 258: 46–74. doi:10.1016/0550-3213(85)90602-9. 
2. ^ Tian, Gang & Yau, Shing-Tung (1990), "Complete Kähler manifolds with zero Ricci curvature, I", Amer. Math. Soc. 3(3): 579–609, doi:10.2307/1990928 
3. ^ Tian, Gang & Yau, Shing-Tung (1991), "Complete Kähler manifolds with zero Ricci curvature, II", Invent. Math. 106(1): 27–60, doi:10.1007/BF01243902 
4. ^ Hitchin, Nigel (2003), "Generalized Calabi-Yau manifolds", The Quarterly Journal of Mathematics 54(3): 281–308, MR2013140, ISSN 0033-5606, <http://arxiv.org/abs/math.DG/0209099> 

• Besse, Arthur L. (1987), Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 10, Berlin, New York: Springer-Verlag, ISBN 978-3-540-15279-8 
• Calabi, Eugenio (1954), "The space of Kähler metrics", Proc. Internat. Congress Math. Amsterdam, pp. 206–207 
• Calabi, Eugenio (1957), "On Kähler manifolds with vanishing canonical class", Algebraic geometry and topology. A symposium in honor of S. Lefschetz, Princeton University Press, pp. 78–89, MR0085583 
• Gross, M.; Huybrechts, D. & Joyce, Dominic (2003), Calabi-Yau manifolds and related geometries, Universitext, Berlin, New York: Springer-Verlag, MR1963559, ISBN 978-3-540-44059-8 
• Hübsch, Tristan (1994), Calabi-Yau Manifolds: a Bestiary for Physicists, Singapore, New York: World Scientific, ISBN 981-02-1927-X 
• Joyce, Dominic (2000), Compact Manifolds with Special Holonomy, Oxford University Press, ISBN 978-0-19-850601-0 
• Yau, Shing Tung (1978), "On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I", Communications on Pure and Applied Mathematics 31(3): 339–411, doi:10.1002/cpa.3160310304, MR480350, ISSN 0010-3640 

External links

• 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.
• Spinning Calabi–Yau Space video.
• Calabi–Yau Space by Andrew J. Hanson with additional contributions by Jeff Bryant, The Wolfram Demonstrations Project.
• Eric W. Weisstein, Calabi-Yau Space at MathWorld.
• Yau, S. T., Calabi-Yau manifold, Scholarpedia, <http://www.scholarpedia.org/article/Calabi-Yau_manifold>  (to appear)