a

We will motivate the above theorem by showing that the existence of a

covariantly constant spinor implies that the background is K¨hler and has

a

c1 = 0. A fundamental theorem states that a compact K¨hler manifold has

a

c1 = 0 if and only if the manifold admits a nowhere vanishing holomorphic

n-form „¦. In local coordinates

„¦(z 1 , z 2 , . . . , z n ) = f (z 1 , z 2 , . . . , z n )dz 1 § dz 2 · · · § dz n . (9.8)

In section 9.5 we will establish the vanishing of c1 by explicitly constructing

„¦ in backgrounds of SU (n) holonomy.

Hodge numbers of a Calabi“Yau n-fold

Betti numbers are fundamental topological numbers associated with a mani-

fold.4 The Betti number bp is the dimension of the pth de Rham cohomology

4 There is more discussion of this background material in the appendix of this chapter.

364 String geometry

of the manifold M , H p (M ), which is de¬ned in the appendix. When the

manifold has a metric, Betti numbers count the number of linearly inde-

pendent harmonic p-forms on the manifold. For K¨hler manifolds the Betti

a

numbers can be decomposed in terms of Hodge numbers hp,q , which count

the number of harmonic (p, q)-forms on the manifold

k

hp,k’p .

bk = (9.9)

p=0

Hodge diamond

A Calabi“Yau n-fold is characterized by the values of its Hodge numbers.

This is not a complete characterization, since inequivalent Calabi“Yau man-

ifolds sometimes have the same Hodge numbers. There are symmetries and

dualities relating di¬erent Hodge numbers, so only a small subset of these

numbers is independent. The Hodge numbers of a Calabi“Yau n-fold satisfy

the relation

hp,0 = hn’p,0 . (9.10)

This follows from the fact that the spaces H p (M ) and H n’p (M ) are isomor-

phic, as can be proved by contracting a closed (p, 0)-form with the complex

conjugate of the holomorphic (n, 0)-form and using the metric to make a

closed (0, n ’ p)-form. Complex conjugation gives the relation

hp,q = hq,p , (9.11)

and Poincar´ duality gives the additional relation

e

hp,q = hn’q,n’p . (9.12)

Any compact connected K¨hler complex manifold has h0,0 = 1, correspond-

a

ing to constant functions. A simply-connected manifold has vanishing funda-

mental group (¬rst homotopy group), and therefore vanishing ¬rst homology

group. As a result,5

h1,0 = h0,1 = 0. (9.13)

In the important case of n = 3 the complete cohomological description of

Calabi“Yau manifolds only requires specifying h1,1 and h2,1 . The full set of

Hodge numbers can be displayed in the Hodge diamond

5 Aside from tori, the Calabi“Yau manifolds that are considered here are simply connected.

Calabi“Yau manifolds that are not simply connected can then be constructed by modding out

by discrete freely acting isometry groups. In all cases of interest, these groups are ¬nite, and

thus the resulting Calabi“Yau manifold still satis¬es Eq. (9.13).

9.2 Calabi“Yau manifolds: mathematical properties 365

h3,3 1

h3,2 h2,3 0 0

h3,1 h2,2 h1,3 h1,1

0 0

h3,0 h2,1 h1,2 h0,3 = 2,1 2,1 (9.14)

1 h h 1

h2,0 h1,1 h0,2 1,1

0 h 0

h1,0 h0,1 0 0

h0,0 1

Using the relations discussed above, one ¬nds that the Euler characteristic

of the Calabi“Yau three-fold is given by

6

(’1)p bp = 2(h1,1 ’ h2,1 ).

χ= (9.15)

p=0

In Chapter 10 compacti¬cations of M-theory on Calabi“Yau four-folds

are discussed. This corresponds to the case n = 4. These manifolds are

characterized in terms of three independent Hodge numbers h1,1 , h1,3 , h1,2 .

The Hodge diamond takes the form

1

0 0

h1,1

0 0

2,1 2,1

0 h h 0

3,1 2,2 3,1 (9.16)

1 h h h 1

2,1 2,1

0 h h 0

h1,1

0 0

0 0

1

For Calabi“Yau four-folds there is an additional relation between the

Hodge numbers, which will not be derived here, namely

h2,2 = 2(22 + 2h1,1 + 2h1,3 ’ h1,2 ). (9.17)

As a result, only three of the Hodge numbers can be varied independently.

The Euler number can therefore be written as

8

(’1)p bp = 6(8 + h1,1 + h3,1 ’ h2,1 ).

χ= (9.18)

p=0

366 String geometry

9.3 Examples of Calabi“Yau manifolds

Calabi“Yau one-folds

The simplest examples of Calabi“Yau manifolds have one complex dimen-

sion.

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 ¬‚at metric

¯

ds2 = |dz|2 , (9.19)

and the holomorphic one-form is

„¦ = 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 ¬‚at metric and can be thought of as a parallelogram with

opposite sides identi¬ed. 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