A Calabi-Yau space, in its metric formulation, is a Kähler manifold whose Riemannian metric has holonomy group contained in 
, where 
is its complex dimension.
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 
, where 
is a four dimensional manifold
(space-time) and 
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 metric has a nonvanishing spinor whose covariant derivative vanishes. The existence of a nonvanishing Dirac-harmonic spinor alone does not imply Calabi-Yau holonomy. In the complex-geometric description, the canonical bundle is trivial.
Consider the local situation using coordinates. In 
, pick coordinates 
and 
so that
 |
(1)
|
gives it the structure of 
. Then
 |
(2)
|
is a local bundle section of the canonical bundle. A unitary change of coordinates 
, where 
is a unitary matrix, transforms
by 
, i.e.,
 |
(3)
|
If the linear transformation 
has determinant 1, that is,
it is a special unitary transformation,
then 
is consistently defined as 
or as 
.
On a Calabi-Yau manifold 
, such a 
can be defined globally, and the Lie
group 
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 
-dimensional manifold
whose holonomy group reduces to 
. Such a manifold has a calibration form given locally by the real
part of its parallel complex volume form, whose
algebraic form is
 |
(4)
|
A closed calibration form of this algebraic type alone does not imply Calabi-Yau holonomy. Often,
the extra assumptions that 
is simply connected and/or
compact are made.
Calabi-Yau manifolds and their moduli spaces have interesting geometric properties. Compact
Calabi-Yau manifolds have Hodge
diamonds with intrinsic symmetries. Mirror
symmetry instead relates the Hodge numbers of
different Calabi-Yau manifolds. For a mirror
pair 
of complex dimension 
, it predicts the relation
 |
(5)
|
This exchange is distinct from the intrinsic symmetries of either Hodge diamond.
