A 2D slice of the 6D CalabiYau quintic manifold.
A Calabi–Yau manifold, also known as a Calabi–Yau space, is a special type of manifold that is described in certain branches of mathematics such as algebraic geometry. The Calabi–Yau manifold's properties, such as Ricci flatness, also yield applications in theoretical physics. Particularly in superstring theory, the extra dimensions of spacetime are sometimes conjectured to take the form of a 6dimensional Calabi–Yau manifold, which led to the idea of mirror symmetry.
Calabi–Yau manifolds are complex manifolds that are higherdimensional 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.
Contents

Definitions 1

Examples 2

Applications in superstring theory 3

See also 4

References 5

External links 6
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 nfold or Calabi–Yau manifold of (complex) dimension n is sometimes defined as a compact ndimensional Kähler manifold M satisfying one of the following equivalent conditions:
These conditions imply that the first integral Chern class c_{1}(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 nontrivial canonical bundle.
For a compact ndimensional 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 Ricciflat 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 Ricciflat 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 2form.) 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 noncompact. In the generalization to noncompact manifolds, the difference (\Omega\wedge\bar\Omega  \omega^n/n!) must vanish asymptotically. Here, \omega is the Kähler form associated with the Kähler metric, g (Gang Tian;ShingTung Yau 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 simplyconnected 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 h^{i,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 nonsingular, but some allow mild singularities. While the Chern class fails to be welldefined 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
The most important fundamental fact is that any smooth algebraic variety embedded in a projective space is a Kahler manifold, because there are natural Fubini–Study metric on a projective space which one can restrict to the algebraic variety. By definition, if ω is the Kahler metric on the algebraic variety X and the canonical bundle K_{X} is trivial, then X is Calabi–Yau. Moreover, there is unique Kahler metric ω on X such that [ω_{0}]=[ω]∈H^{2}(X,R), a fact which was conjectured by Eugenio Calabi and proved by S. T. Yau (see Calabi conjecture).
In one complex dimension, the only compact examples are tori, which form a oneparameter family. The Ricciflat metric on a torus is actually a flat metric, so that the holonomy is the trivial group SU(1). A onedimensional 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 simplyconnected 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 Ricciflat 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). In turn, it has also been conjectured by Miles Reid that the number of topological types of CalabiYau 3folds is infinite, and that they can all be transformed continuously (through certain mild singularizations such as conifolds) one into another—much as Riemann surfaces can.^{[1]} One example of a threedimensional Calabi–Yau manifold is a nonsingular quintic threefold in CP^{4}, which is the algebraic variety consisting of all of the zeros of a homogeneous quintic polynomial in the homogeneous coordinates of the CP^{4}. Another example is a smooth model of the Barth–Nieto quintic. Some discrete quotients of the quintic by various Z_{5} 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 nonsingular homogeneous degree n+2 polynomial in the homogeneous coordinates of the complex projective space CP^{n+1} is a compact Calabi–Yau nfold. The case n=1 describes an elliptic curve, while for n=2 one obtains a K3 surface.
All hyperKähler 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 nfolds are important because they leave some of the original supersymmetry unbroken. More precisely, in the absence of fluxes, compactification on a Calabi–Yau 3fold (real dimension 6) leaves one quarter of the original supersymmetry unbroken if the holonomy is the full SU(3).
More generally, a fluxfree compactification on an nmanifold with holonomy SU(n) leaves 2^{1−n} of the original supersymmetry unbroken, corresponding to 2^{6−n} supercharges in a compactification of type II supergravity or 2^{5−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 Dbrane.
Ftheory compactifications on various Calabi–Yau fourfolds provide physicists with a method to find a large number of classical solution in the socalled string theory landscape.
Connected with each hole in the CalabiYau space is a group of lowenergy string vibrational patterns. Since string theory states that our familiar elementary particles correspond to lowenergy string vibrations, the presence of multiple holes causes the string patterns to fall into multiple groups, or families. Although the following statement has been simplified, it conveys the logic of the argument: if the CalabiYau has three holes, then three families of vibrational patterns and thus three families of particles will be observed experimentally.
Logically, since strings vibrate through all the dimensions, the shape of the curledup ones will affect their vibrations and thus the properties of the elementary particles observed. For example, Andrew Strominger and Edward Witten have shown that the masses of particles depend on the manner of the intersection of the various holes in a CalabiYau. In other words, the positions of the holes relative to one another and to the substance of the CalabiYau space was found by Strominger and Witten to affect the masses of particles in a certain way. This, of course, is true of all particle properties.^{[2]}
See also
References

^ Reid, Miles (1987), "The Moduli Space of 3Folds with K = 0 May Nevertheless be Irreducible", Math. Ann., 278, 329

^ "The Shape of CurledUp Dimensions". Archived from the original on Sep 13, 2006. Retrieved 2006.

Besse, Arthur L. (1987), Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 10, Berlin, New York:

Chan,YatMing (2004)"Desingularization Of Calabi–Yau 3Folds With A Conical Singularity"

Calabi, Eugenio (1954), "The space of Kähler metrics", Proc. Internat. Congress Math. Amsterdam 2, pp. 206–207

Calabi, Eugenio (1957), "On Kähler manifolds with vanishing canonical class", in Fox, Ralph H.; Spencer, D. C.; Tucker, A. W., Algebraic geometry and topology. A symposium in honor of S. Lefschetz, Princeton Mathematical Series 12,

Greene, Brian "String Theory On Calabi–Yau Manifolds"

Candelas, Philip; Horowitz, Gary; Strominger, Andrew; Witten, Edward (1985), "Vacuum configurations for superstrings", Nuclear Physics B 258: 46–74,

Gross, M.; Huybrechts, D.;


Hübsch, Tristan (1994), Calabi–Yau Manifolds: a Bestiary for Physicists, Singapore, New York:

Im, Mee Seong (2008) "SingularitiesinCalabiYauvarieties.pdf Singularities in Calabi–Yau varieties"


Tian, Gang; Yau, ShingTung (1990), "Complete Kähler manifolds with zero Ricci curvature, I", Amer. Math. Soc. 3 (3): 579–609,

Tian, Gang; Yau, ShingTung (1991), "Complete Kähler manifolds with zero Ricci curvature, II", Invent. Math. 106 (1): 27–60,

Yau, Shing Tung (1978), "On the Ricci curvature of a compact Kähler manifold and the complex MongeAmpère equation. I",

Yau, ShingTung (2009), "Surveys in differential geometry. Vol. XIII. Geometry, analysis, and algebraic geometry: forty years of the Journal of Differential Geometry", Scholarpedia, Surv. Differ. Geom. (Somerville, Massachusetts: Int. Press) 4 (8): 277–318,
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, Wolfram Demonstrations Project.

Weisstein, Eric W., "Calabi–Yau Space", MathWorld.

Yau, S. T., Calabi–Yau manifold, Scholarpedia (similar to (Yau 2009))
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