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Calabi-Yau Space


Calabi-Yau spaces are important in string theory, where one model posits the geometry of the universe to consist of a ten-dimensional space of the form M×V, where M is a four dimensional manifold (space-time) and V is a six dimensional compact Calabi-Yau space. They are related to Kummer surfaces. Although the main application of Calabi-Yau spaces is in theoretical physics, they are also interesting from a purely mathematical standpoint. Consequently, they go by slightly different names, depending mostly on context, such as Calabi-Yau manifolds or Calabi-Yau varieties.

Although the definition can be generalized to any dimension, they are usually considered to have three complex dimensions. Since their complex structure may vary, it is convenient to think of them as having six real dimensions and a fixed smooth structure.

A Calabi-Yau space is characterized by the existence of a nonvanishing harmonic spinor phi. This condition implies that its canonical bundle is trivial.

Consider the local situation using coordinates. In R^6, pick coordinates x_1,x_2,x_3 and y_1,y_2,y_3 so that

 z_j=x_j+iy_j
(1)

gives it the structure of C^3. Then

 phi_z=dz_1 ^ dz_2 ^ dz_3
(2)

is a local section of the canonical bundle. A unitary change of coordinates w=Az, where A is a unitary matrix, transforms phi by detA, i.e.,

 phi_w=detAphi_z.
(3)

If the linear transformation A has determinant 1, that is, it is a special unitary transformation, then phi is consistently defined as phi_z or as phi_w.

On a Calabi-Yau manifold V, such a phi can be defined globally, and the Lie group SU(3) is very important in the theory. In fact, one of the many equivalent definitions, coming from Riemannian geometry, says that a Calabi-Yau manifold is a 2n-dimensional manifold whose holonomy group reduces to SU(n). Another is that it is a calibrated manifold with a calibration form psi, which is algebraically the same as the real part of

 dz_1 ^ ... ^ dz_n.
(4)

Often, the extra assumptions that V is simply connected and/or compact are made.

Whatever definition is used, Calabi-Yau manifolds, as well as their moduli spaces, have interesting properties. One is the symmetries in the numbers forming the Hodge diamond of a compact Calabi-Yau manifold. It is surprising that these symmetries, called mirror symmetry, can be realized by another Calabi-Yau manifold, the so-called mirror of the original Calabi-Yau manifold. The two manifolds together form a mirror pair. Some of the symmetries of the geometry of mirror pairs have been the object of recent research.


See also

Calibrated Manifold, Canonical Bundle, Complex Manifold, Harmonic, Kähler Form, Lie Group, Mirror Pair, Moduli Space, Spinor, Variety

This entry contributed by Todd Rowland

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Cite this as:

Rowland, Todd. "Calabi-Yau Space." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/Calabi-YauSpace.html

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