- Calabi–Yau manifold
In mathematics, Calabi–Yau manifolds are compact
Kähler manifolds whosecanonical bundle is trivial. They were named "Calabi–Yau spaces" byphysicist s in 1985, [cite journal | author = Candelas, Horowitz, Strominger and Witten | 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] afterE. Calabi who first studied them in harvs|last=Calabi|year1=1954|year2=1957, and S. T. Yau who proved theCalabi conjecture that they have Ricci flat metrics in harv|Yau|1978.In theoretical physics more general definitions are often used, and they may be allowed to be singular or non-compact,Insuperstring theory the extra dimensions ofspacetime are sometimes conjectured to take the form of a 6-dimensional Calabi–Yau manifold. Physical insights about Calabi–Yau manifolds, especiallymirror 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:
*Thecanonical 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 integralChern class c1("M") of "M" vanishes.
*"M" has a Kähler metric with globalholonomy contained inSU(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 realChern class .
*"M" has a Kähler metric with with vanishing Ricci curvature.
*"M" has a Kähler metric with localholonomy contained inSU(n) .
* A positive power of thecanonical 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 realChern 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 surface s 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 properties above is proving the existence of Ricci-flat metrics. This follows from Yau's proof of the the
Calabi conjecture , which implies that a compactKähler manifold with a vanishing first realChern class has a Kähler metric in the same class with 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 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 OmegawedgearOmega - J^n/n!) must vanish asymptotically. Here, J is the Kähler form associated with the Kähler metric, g. [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] [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]
*Some definitions put restrictions on thefundamental 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.
*Some definitions require that the holonomy be exactly equal to SU(n) rather than a subgroup of it. Abelian surfaces have a (Ricci) flat metric with holonomy strictly smaller than SU(2) (in fact 0) so are not Calabi-Yau manifolds according 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".
*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 theHodge number s "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 theholonomy is thetrivial group , for which SU(1) is another name. A one-dimensional Calabi–Yau manifold is a complexelliptic curve , and in particular, algebraic.In two complex dimensions, the
K3 surface s furnish the only compact simply connected Calabi–Yau manifolds. Non simply-connected examples are given byabelian surface s.Enriques surface s andhyperelliptic 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 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 fullSU(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 thealgebraic variety consisting of all of the zeros of a homogeneous quinticpolynomial 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 bymirror symmetry .For every "n", the
zero set of a general homogeneous degree "n+2" polynomial in the homogeneous coordinates of the complex projective space CP"n"+1 is a compact Calabi–Yau "n"-fold, although it is not always adifferentiable 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 manifold s are Calabi-Yau.Applications in superstring theory
Calabi–Yau manifolds are important in
superstring theory . In the most conventional superstring models, ten conjectural dimensions instring theory are supposed to come as four of which we are aware, carrying some kind offibration with fiber dimension six.Compactification on Calabi–Yau "n"-folds are important because they leave some of the originalsupersymmetry 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 theholonomy 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 ageneralized Calabi–Yau , a notion introduced in 2002 byNigel Hitchin . [Citation | last1=Hitchin | first1=Nigel | title=Generalized Calabi-Yau manifolds | url=http://arxiv.org/abs/math.DG/0209099 | id=MathSciNet | id = 2013140 | year=2003 | journal=The Quarterly Journal of Mathematics | issn=0033-5606 | volume=54 | issue=3 | pages=281–308 | doi=10.1093/qmath/hag025] These models are known asflux compactification s.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 dimension s, which often occurs inbraneworld models, is that the Calabi–Yau is large but we are confined to a small subset on which it intersects aD-brane .Notes
References
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*Citation | last1=Calabi | first1=Eugenio | chapter=The space of Kähler metrics|title= Proc. Internat. Congress Math. Amsterdam |year=1954|pages= 206–207
*Citation | last1=Calabi | first1=Eugenio | title=Algebraic geometry and topology. A symposium in honor of S. Lefschetz | publisher=Princeton University Press | id=MathSciNet | id = 0085583 | year=1957 | chapter=On Kähler manifolds with vanishing canonical class | pages=78–89
*Citation | last1=Gross | first1=M. | last2=Huybrechts | first2=D. | last3=Joyce | first3=Dominic | author3-link=Dominic Joyce | title=Calabi-Yau manifolds and related geometries | publisher=Springer-Verlag | location=Berlin, New York | series=Universitext | isbn=978-3-540-44059-8 | id=MathSciNet | id = 1963559 | year=2003 | oclc=50695398
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*Citation | last1=Yau | first1=Shing Tung | title=On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I | doi=10.1002/cpa.3160310304 | id=MathSciNet | id = 480350 | year=1978 | journal=Communications on Pure and Applied Mathematics | issn=0010-3640 | volume=31 | issue=3 | pages=339–411External links
* [http://www.th.physik.uni-bonn.de/th/Supplements/cy.html 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.
* [http://members.wri.com/jeffb/visualization/stringtheory2.shtml Spinning Calabi–Yau Space video.]
*" [http://demonstrations.wolfram.com/CalabiYauSpace/ Calabi–Yau Space] " by Andrew J. Hanson with additional contributions by Jeff Bryant,The Wolfram Demonstrations Project .
*MathWorld |title=Calabi-Yau Space |urlname=Calabi-YauSpace
*citation|last=Yau|first=S. T. |url=http://www.scholarpedia.org/article/Calabi-Yau_manifold|title=Calabi-Yau manifold|publisher=Scholarpedia (to appear)
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