String Theory and M-Theory

9.3 Examples of Calabi–Yau manifolds 369
Calabi–Yau n-folds
The complete classification of Calabi–Yau n-folds for n > 2 is an unsolved
problem, and it is not even clear that the number of compact Calabi–Yau
three-folds is finite. Many examples have been constructed. Here we mention
a few of them.
Submanifolds of complex projective spaces
Examples of a Calabi–Yau n-folds can be constructed as a submanifold of
£ P n+1 for all n > 1. Complex projective space, £ P n , sometimes just denoted
P n , is a compact manifold with n complex dimensions. It can be
constructed by taking £ n+1 /{0}, that is the set of (z 1 , z 2 , . . . , z n+1 ) where
the z i are not all zero and making the identifications
(z 1 , z 2 , . . . , z n+1 ) ∼ (λz 1 , λz 2 , . . . , λz n+1 ), (9.30)
for any nonzero complex λ = 0. Thus, lines 7 in £ n+1 correspond to points
in £ P n .
£ P n is a Kähler manifold, but it is not a Calabi–Yau manifold. The simplest
example is £ P 1 , which is topologically the two-sphere S 2 . Obviously,
it does not admit a Ricci-flat metric. The standard metric of £ P n , called the
Fubini–Study metric, is constructed as follows. First one covers the manifold
by n + 1 open sets given by z a = 0. Then on each open set one introduces
local coordinates. For example, on the open set with z n+1 = 0, one defines
w a = z a /z n+1 , with a = 1, . . . , n. Then one introduces the Kähler potential
(for this open set)
K = log
1 +
n
|w a | 2
. (9.31)
This determines the metric by formulas given in the appendix. A crucial
requirement is that the analogous formulas for the Kähler potential on the
other open sets differ from this one by Kähler transformations. You are
asked to verify this in a homework problem.
Examples of Calabi–Yau manifolds can be obtained as subspaces of complex
projective spaces. Specifically, let G be a homogenous polynomial of
degree k in the coordinates z a of £ n+2 , that is,
a=1
G(λz 1 , . . . , λz n+2 ) = λ k G(z 1 , . . . , z n+2 ). (9.32)
The submanifold of £ P n+1 defined by
7 A line in a complex manifold has one complex dimension.
G(z 1 , . . . , z n+2 ) = 0 (9.33)