From Wikipedia, the free encyclopedia

Calabi-Yau manifolds are a special class of manifolds used in some branches of mathematics (such as algebraic geometry) as well as in theoretical physics. For instance, in superstring theory the extra dimensions of spacetime are sometimes conjectured to take the form of a 6-dimensional Calabi-Yau manifold. The precise definition of a Calabi-Yau manifold, given below, builds on a considerable mathematical background. The designation "Calabi-Yau space" for a member of this class was coined by physicists in the 1980s,^[1] but mathematicians have been studying such manifolds since at least the 1950s. Physical insights about Calabi-Yau manifolds, especially mirror symmetry, led to tremendous progress in pure mathematics.

Contents 1 Formal definition 2 Examples 3 Applications in Superstring Theory 4 Calabi-Yau Manifold in Popular Culture 5 See also 6 References

A Calabi-Yau manifold is a Kähler manifold with a vanishing first Chern class. A Calabi-Yau manifold of complex dimension n is also called a Calabi-Yau n-fold. The mathematician Eugenio Calabi conjectured in 1957 that all such manifolds admit a Ricci-flat metric (one in each Kähler class), and this conjecture was proved by Shing-Tung Yau (丘成桐) in 1977 and became Yau's theorem. Consequently, a Calabi-Yau manifold is a compact Ricci-flat Kähler manifold. It is possible for the first Chern class to vanish as an element of the real cohomology group, so that the manifold has a Ricci-flat metric, yet not as an element of the integral cohomology group, so that the manifold does not have a global nowhere vanishing holomorphic (n,0)-form.

Equivalently one may define a Calabi-Yau n-fold as a manifold with an SU(n) holonomy. Yet another equivalent condition is that the manifold admit a global nowhere vanishing holomorphic (n,0)-form.

The first Chern class vanishes if and only if the canonical bundle is trivial, which in turn is the case if and only if the canonical class is the zero class. While the Chern class fails to be well-defined for singular Calabi-Yau's, the canonical bundle and canonical class may still be defined and so may be used to extend to definition of a smooth Calabi-Yau manifold to a possibly singular Calabi-Yau variety.

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.

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 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 in 2002 by Nigel Hitchin.^[2] 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.

See also: hyper-Kähler manifold

In the best-selling computer game Half-Life 2, the human "teleportation device" employs the Calabi-Yau model of space to teleport users to different points instantaneously, without passing through the intervening space. At the time of the game, the technology hadn't been perfected, and the character is transported to several different locations as a result of the holonomy of the model.

Calabi-Yau space is also mentioned in the Doctor Who novel, Quantum Archangel, and is said to be the home of creatures and entities that live outside of space/time such as the chronovores, the Eternals, and the Guardians (the Black Guardian, the White Guardian, et al).

In Dan Simmons' Illium and Olympus series the Calabi-Yau Manifold is said to be used in order to allow worm hole transportation between points in the universe.

• Holonomy

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. ^ Hitchin, Nigel (2002). "Generalized Calabi-Yau Manifolds". 

• 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.
• Arthur Besse Einstein manifolds, Springer-Verlag, Berlin, Heidelberg, 1987 ISBN 3-540-15279-2
• Dominic D. Joyce Compact Manifolds with Special Holonomy (Oxford Mathematical Monographs) ISBN 0-19-850601-5