String Theory and M-Theory

366 String geometry
9.3 Examples of Calabi–Yau manifolds
Calabi–Yau one-folds
The simplest examples of Calabi–Yau manifolds have one complex dimension.
Noncompact £ example:
A simple noncompact example is the complex £ plane described in terms
of the coordinates (z, ¯z). It can be described in terms of a flat metric
and the holomorphic one-form is
ds 2 = |dz| 2 , (9.19)
Ω = dz. (9.20)
Compact example: T 2
The only compact Calabi–Yau one-fold is the two-torus T 2 , which can be
described with a flat metric and can be thought of as a parallelogram with
opposite sides identified. This simple example is discussed in Sections 9.5
and 9.9 in order to introduce concepts, such as mirror symmetry, that can
be generalized to higher dimensions.
Calabi–Yau two-folds
Noncompact examples
Some simple examples of noncompact Calabi–Yau two-folds, which have
two complex dimensions, can be obtained as products of the previous two
manifolds: £ 2 = £ × £ , £ × T 2 .
Compact examples: T 4 , K3
Requiring a covariantly constant spinor is very restrictive in four real dimensions.
In fact, K3 and T 4 are the only two examples of four-dimensional
compact Kähler manifolds for which they exist. As a result, these manifolds
are the only examples of Calabi–Yau two-folds. If one requires the
holonomy to be SU(2), and not a subgroup, then only K3 survives. By contrast,
there are very many (possibly infinitely many) Calabi–Yau three-folds.
Since K3 and T 4 are Calabi–Yau manifolds, they admit a Ricci-flat Kähler
metric. Moreover, since SU(2) = Sp(1), it turns out that they are also
hyper-Kähler. 6 The explicit form of the Ricci-flat metric of a smooth K3
6 In general, a 4n-dimensional manifold of Sp(n) holonomy is called hyper-Kähler. The notation
USp(2n) is also used for the same group when one wants to emphasize that the compact form
is being used. Both notations are used in this book.