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[[File:CalabiYau5.jpg|thumb|A 2D slice of the 6D Calabi-Yau quintic manifold.]]
A '''Calabi&ndash;Yau manifold''', also known as a '''Calabi&ndash;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 flat]]ness, also yield applications in [[theoretical physics]]. Particularly in [[superstring theory]], the extra dimensions of [[spacetime]] are sometimes conjectured to take the form of a 6-dimensional Calabi&ndash;Yau manifold, which led to the idea of  [[Mirror symmetry (string theory)|mirror symmetry]]. 

Calabi&ndash;Yau manifolds are [[complex manifolds]] that are higher-dimensional analogues of [[K3 surface]]s. 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&ndash;Yau spaces"  by  {{harvtxt|Candelas|Horowitz|Strominger|Witten|1985}}  after {{harvs|txt|first=E.|last=Calabi|authorlink=E. Calabi|year1=1954|year2=1957}} who first studied them, and {{harvs|txt|authorlink=Shing-Tung Yau|first=S. T.|last= Yau|year=1978}} who proved the [[Calabi conjecture]] that they have [[Ricci-flat manifold|Ricci flat]] metrics.

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

A Calabi&ndash;Yau ''n''-fold or Calabi&ndash;Yau manifold of (complex) 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]] c<sub>1</sub>(''M'') of ''M'' vanishes, but the converse is not true. The simplest examples where this happens are [[hyperelliptic surface]]s, 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&ndash;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 surface]]s give examples of complex manifolds  that have Ricci-flat metrics, but their canonical bundles are not trivial so they are  Calabi&ndash;Yau manifolds according to the second but not the first definition above. Their double covers are Calabi&ndash;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&ndash;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&ndash;Yau manifolds are compact, but some allow them to be non-compact. In the generalization to non-compact manifolds, the difference <math>(\Omega\wedge\bar\Omega - \omega^n/n!)</math> must vanish asymptotically. Here, <math>\omega</math> is the Kähler form associated with the Kähler metric, <math>g</math> {{harvs|last=Tian|year1=1990|year2=1991}}.
* Some definitions put restrictions on the [[fundamental group]] of a Calabi&ndash;Yau manifold, such as demanding that it be finite or trivial. Any Calabi&ndash;Yau manifold has a finite cover that is the product of a torus and a simply-connected Calabi&ndash;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 number]]s ''h''<sup>''i'',0</sup> 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&ndash;Yau manifolds according to such definitions.
* Most definitions assume that a Calabi&ndash;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&ndash;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&ndash;Yau manifold to a possibly singular Calabi&ndash;Yau variety.

==Examples==

As the most important fundamental fact [[algebraic variety]] embedded in a [[projective space]] is a Kaehler manifolds because there are natural [[Fubini-Study metric]] on a projective space and one can restrict it on the algebraic variety. By the definition if ω is the Kaehler metric on the algebraic variety X and the canonical bundle  K<sub>X</sub> is trivial then X is Calabi&ndash;Yau. And there is uniquely Kaehler metric ω on X such that [ω<sub>0</sub>]=[ω]∈H<sup>2</sup>(X,'''R'''), which was conjectured by [[Eugenio Calabi]] and proved by [[S. T. Yau]]. see [[Calabi conjecture]]

In one complex dimension, the only compact examples are [[torus|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&ndash;Yau manifold is a complex [[elliptic curve]], and in particular, [[algebraic variety|algebraic]].

In two complex dimensions, the [[K3 surface]]s furnish the only compact simply connected Calabi&ndash;Yau manifolds.  Non simply-connected examples are given by [[abelian surface]]s.  [[Enriques surface]]s and [[hyperelliptic surface]]s 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&ndash;Yau manifolds.  Abelian surfaces are sometimes excluded from the classification of being Calabi&ndash;Yau, as their holonomy (again the trivial group) is a [[subgroup|proper subgroup]] of SU(2), instead of being isomorphic to SU(2). 

In three complex dimensions, classification of the possible Calabi&ndash;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&ndash;Yau manifold is a non-singular [[quintic threefold]] in [[complex projective space|'''CP'''<sup>4</sup>]], which is the [[algebraic variety]] consisting of all of the zeros of a homogeneous quintic [[polynomial]] in the homogeneous coordinates of the '''CP'''<sup>4</sup>.  Another example is a smooth model of the [[Barth–Nieto quintic]]. Some discrete quotients of the quintic by various '''Z'''<sub>5</sub> actions are also Calabi&ndash;Yau and have received a lot of attention in the literature.  One of these is related to the original quintic by [[mirror symmetry (string theory)|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 '''CP'''<sup>''n''+1</sup> is a compact Calabi&ndash;Yau ''n''-fold.  The case ''n=1'' describes an elliptic curve, while for ''n=2'' one obtains a K3 surface.

All [[hyper-Kähler manifold]]s are Calabi&ndash;Yau.

==Applications in superstring theory==

Calabi&ndash;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 (physics)|Compactification]] on Calabi&ndash;Yau ''n''-folds are important because they leave some of the original [[supersymmetry]] unbroken. More precisely, in the absence of [[Ramond&ndash;Ramond field|fluxes]], compactification on a Calabi&ndash;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 2<sup>1−''n''</sup> of the original supersymmetry unbroken, corresponding to 2<sup>6−''n''</sup> supercharges in a compactification of type II [[supergravity]] or 2<sup>5−''n''</sup> supercharges in a compactification of type I.  When fluxes are included the supersymmetry condition instead implies that the compactification manifold be a [[generalized Calabi&ndash;Yau]], a notion introduced by {{harvtxt|Hitchin|2003}}.  These models are known as [[Compactification (physics)#Flux compactification|flux compactification]]s.

Essentially, Calabi&ndash;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 dimension]]s, which often occurs in [[braneworld]] models, is that the Calabi&ndash;Yau is large but we are confined to a small subset on which it intersects a [[D-brane]].

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

Connected with each hole in the Calabi-Yau space is a group of low-energy string vibrational patterns. Since string theory states that our familiar elementary particles correspond to low-energy string vibrations, the presence of multiple holes causes the string patterns to fall into multiple groups, or [[Generation (particle physics)|families]]. Although the following statement has been simplified, it conveys the logic of the argument: if the Calabi-Yau 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 curled-up 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 Calabi-Yau. In other words, the positions of the holes relative to one another and to the substance of the Calabi-Yau 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.<ref>{{cite web|url=http://library.thinkquest.org/27930/stringtheory5.htm|title=The Shape of Curled-Up Dimensions|accessdate=2006|archiveurl=http://web.archive.org/web/20060913014709/http://library.thinkquest.org/27930/stringtheory5.htm|archivedate=Sep 13, 2006}}</ref>

==See also==
*[[G2 manifold]]

==References==
{{more footnotes|date=January 2013}}
<references />
* {{Citation | last1=Besse | first1=Arthur L. | title=Einstein manifolds | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Ergebnisse der Mathematik und ihrer Grenzgebiete (3)| volume=10 | isbn=978-3-540-15279-8 | year=1987 | oclc=13793300}}
*Chan,Yat-Ming (2004)"[http://arxiv.org/abs/math/0410260v1 Desingularization Of Calabi&ndash;Yau 3-Folds With A Conical Singularity]"
*{{Citation | last1=Calabi | first1=Eugenio | chapter=The space of Kähler metrics|title= Proc. Internat. Congress Math. Amsterdam |year=1954|volume=2|pages= 206&ndash;207|url=http://mathunion.org/ICM/ICM1954.2/}}
*{{Citation | last1=Calabi | first1=Eugenio | editor1-last=Fox | editor1-first=Ralph H. | editor2-last=Spencer | editor2-first=D. C. | editor3-last=Tucker | editor3-first=A. W. | title=Algebraic geometry and topology. A symposium in honor of S. Lefschetz | url=http://books.google.com/books?id=n_ZQAAAAMAAJ | publisher=[[Princeton University Press]] | series=Princeton Mathematical Series | mr=0085583 | year=1957 | volume=12 | chapter=On Kähler manifolds with vanishing canonical class | pages=78–89}}
*Greene, Brian "[http://arxiv.org/abs/hep-th/9702155v1 String Theory On Calabi&ndash;Yau Manifolds]"
*{{citation| last1=Candelas|first1=Philip|last2=Horowitz|first2=Gary|last3=Strominger|first3=Andrew|last4=Witten|first4=Edward | year = 1985 | title = Vacuum configurations for superstrings | journal = Nuclear Physics B | volume = 258 | pages = 46–74 | doi = 10.1016/0550-3213(85)90602-9|url=http://www-lib.kek.jp/cgi-bin/img_index?8504007|bibcode = 1985NuPhB.258...46C }}
*{{Citation | last1=Gross | first1=M. | last2=Huybrechts | first2=D. | last3=Joyce | first3=Dominic | author3-link=Dominic Joyce | title=Calabi&ndash;Yau manifolds and related geometries | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Universitext | isbn=978-3-540-44059-8 | mr=1963559 | year=2003 | oclc=50695398}}
*{{Citation | authorlink=Nigel Hitchin | last1=Hitchin | first1=Nigel | title=Generalized Calabi&ndash;Yau manifolds | arxiv=math.DG/0209099 | mr=2013140 | year=2003 | journal=The Quarterly Journal of Mathematics  | volume=54 | issue=3 | pages=281–308 | doi=10.1093/qmath/hag025}}
* {{Citation | last1=Hübsch | first1=Tristan | title=Calabi&ndash;Yau Manifolds: a Bestiary for Physicists | publisher=[[World Scientific]] | location=Singapore, New York | isbn=981-02-1927-X | year=1994 | oclc=34989218|url=http://www.worldscibooks.com/physics/1410.html}}
*Im, Mee Seong (2008) "[http://silver.ima.umn.edu/2007-2008/SP7.14-25.08/activities/Im-Mee%20Seong/ Singularities-in-Calabi-Yau-varieties.pdf Singularities in Calabi&ndash;Yau varieties]"
* {{Citation | last1=Joyce | first1=Dominic | author1-link=Dominic Joyce | title=Compact Manifolds with Special Holonomy | publisher=[[Oxford University Press]] | isbn=978-0-19-850601-0 | year=2000 | oclc=43864470}}
*{{Citation| first1=Gang| last1=Tian| first2=Shing-Tung| last2=Yau|title=Complete Kähler manifolds with zero Ricci curvature, I| journal=Amer. Math. Soc.| volume=3| issue=3| pages=579–609|  year=1990|doi=10.2307/1990928| jstor=1990928}}
*{{Citation| first1=Gang| last1=Tian| first2=Shing-Tung| last2=Yau| title=Complete Kähler manifolds with zero Ricci curvature, II| journal=Invent. Math.| volume=106| issue=1| pages=27–60|  year=1991| doi=10.1007/BF01243902|bibcode = 1991InMat.106...27T }}
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*{{Citation | last1=Yau | first1=Shing-Tung | title=Surveys in differential geometry. Vol. XIII. Geometry, analysis, and algebraic geometry: forty years of the Journal of Differential Geometry | url=http://www.scholarpedia.org/article/Calabi-Yau_manifold | publisher=Int. Press, Somerville, MA | series=Surv. Differ. Geom. | mr=2537089 | year=2009 | volume=4 | chapter=A survey of Calabi-Yau manifolds | pages=277–318 | doi=10.4249/scholarpedia.6524 | journal=Scholarpedia | issue=8}}

==External links==
*[http://www.th.physik.uni-bonn.de/th/Supplements/cy.html Calabi&ndash;Yau Homepage] is an interactive reference which describes many examples and classes of Calabi&ndash;Yau manifolds and also the physical theories in which they appear.
*[http://members.wri.com/jeffb/visualization/stringtheory2.shtml Spinning Calabi&ndash;Yau Space video.]
*''[http://demonstrations.wolfram.com/CalabiYauSpace/ Calabi&ndash;Yau Space]'' by Andrew J. Hanson with additional contributions by Jeff Bryant, [[Wolfram Demonstrations Project]].
*{{MathWorld |title=Calabi&ndash;Yau Space |urlname=Calabi-YauSpace}}
*{{citation|last=Yau|first=S. T. |url=http://www.scholarpedia.org/article/Calabi-Yau_manifold|title=Calabi&ndash;Yau manifold|publisher=Scholarpedia}} (similar to {{harv|Yau|2009}})

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[[Category:Algebraic geometry]]
[[Category:Mathematical physics]]
[[Category:String theory]]
[[Category:Complex manifolds]]