J is invertible, so this implies that Rmp = 0, and thus the manifold is Ricci-

¬‚at. 2

EXERCISE 9.8

Show that „¦ § „¦ is proportional to the volume form of the Calabi“Yau

three-fold that we derived in Exercise 9.1.

9.5 Deformations of Calabi“Yau manifolds 385

SOLUTION

As in the case of Exercise 9.1, this is a nontrivial closed (3, 3)-form, so this

has to be true (up to an exact form) by uniqueness. Still, it is instruc-

tive to examine the explicit formulas and determine the normalization. By

de¬nition

1

„¦ = „¦a1 a2 a3 dz a1 § dz a2 § dz a3 ,

6

T

where „¦a1 a2 a3 = ·’ γa1 a2 a3 ·’ . Thus „¦ § „¦ becomes

1 a1 ¯ ¯ ¯

dz § dz a2 § dz a3 § d¯b1 § d¯b2 § d¯b3 „¦a1 a2 a3 „¦¯1¯2¯3

z z z bbb

36

i ¯ ¯ ¯

J § J § J(„¦a1 a2 a3 „¦¯1¯2¯3 g a1 b1 g a2 b2 g a3 b3 ).

=’ bbb

36

1

Since 6 J § J § J = dV is the volume form, we only need to prove that

the extra factor is a constant. Because of the properties m „¦abc = 0 and

a¯ = 0, we have

b

mg

2

„¦ =0

m

where

1 a1¯1 a2¯2 a3¯3

2

g b g b g b „¦a1 a2 a3 „¦¯1¯2¯3 .

„¦ = bbb

6

2 is a scalar, and thus it is a constant. It follows that „¦§„¦ is ’i „¦ 2 dV .

„¦

2

9.5 Deformations of Calabi“Yau manifolds

Calabi“Yau manifolds with speci¬ed Hodge numbers are not unique. Some

of them are smoothly related by deformations of the parameters characteriz-

ing their shapes and sizes, which are called moduli. Often the entire moduli

space of manifolds is referred to as a single Calabi“Yau space, even though

it is really a continuously in¬nite family of manifolds. This interpretation

was implicitly assumed earlier in raising the question whether or not there

is a ¬nite number of Calabi“Yau manifolds. There can also be more than

one Calabi“Yau manifold of given Hodge numbers that are topologically dis-

tinct, with disjoint moduli spaces, since the Hodge numbers do not give a full

characterization of the topology. On the other hand, when one goes beyond

the supergravity approximation, it is sometimes possible to identify smooth

topology-changing transitions, such as the conifold transition described in

Section 9.8, which can even change the Hodge numbers.

386 String geometry

This section and the next one explain how the moduli parametrize the

space of possible choices of undetermined expectation values of massless

scalar ¬elds in four dimensions. They are undetermined because the e¬ective

potential does not depend on them, at least in the leading supergravity

approximation. A very important property of the moduli space of Calabi“

Yau three-folds is that it is the product of two factors, one describing the

complex-structure moduli and one describing the K¨hler-structure moduli.

a

Let us now consider the spectrum of ¬‚uctuations about a given Calabi“

Yau manifold with ¬xed Hodge numbers. Some of these ¬‚uctuations come

from metric deformations, while others are obtained from deformations of

antisymmetric tensor ¬elds.

Antisymmetric tensor-¬eld deformations

As discussed in Chapter 8, the low-energy e¬ective actions for string theories

contain various p-form ¬elds with kinetic terms proportional to

√

d10 x ’g | Fp |2 , (9.82)

where Fp = dAp’1 . An example of such a ¬eld is the type IIA or type IIB

three-form H3 = dB2 . The equation of motion of this ¬eld is16

∆Bp’1 = d dBp’1 = 0. (9.83)

If we compactify to four dimensions on a product space M4 — M , where

M is a Calabi“Yau three-fold, then the space-time metric is a sum of a four-

dimensional piece and a six-dimensional piece. Therefore, the Laplacian is

also a sum of two pieces

∆ = ∆4 + ∆6 , (9.84)

and the number of massless four-dimensional ¬elds is given by the number of

zero modes of the internal Laplacian ∆6 . These zero modes are counted by

the Betti numbers bp . The ten-dimensional ¬eld B2 , for example, can give

rise to four-dimensional ¬elds that are two-forms, one-forms and zero-forms.

The number of these ¬elds is summarized in the following table:

BM N Bµν Bµn Bmn

p-form in 4D p=2 p=1 p=0

b2 = h1,1

# of ¬elds in 4D b0 = 1 b1 = 0

16 This assumes other terms vanish or can be neglected.

9.5 Deformations of Calabi“Yau manifolds 387

The b2 scalar ¬elds in this example are moduli originating from the B

¬eld. More generally, a p-form ¬eld gives rise to bp moduli ¬elds.

Metric deformations

The zero modes of the ten-dimensional metric (or graviton ¬eld) give rise

to the four-dimensional metric gµν and a set of massless scalar ¬elds orig-

inating from the internal components of the metric gmn . In Calabi“Yau

compacti¬cations no massless vector ¬elds are generated from the metric

since b1 = 0. A closely related fact is that Calabi“Yau three-folds have no

continuous isometry groups.

The ¬‚uctuations of the metric on the internal space are analyzed by per-

forming a small variation

gmn ’ gmn + δgmn , (9.85)

and then demanding that the new background still satis¬es the Calabi“Yau

conditions. In particular, one requires

Rmn (g + δg) = 0. (9.86)

This leads to di¬erential equations for δg, and the number of solutions counts

the number of ways the background metric can be deformed while preserving

supersymmetry and the topology. The coe¬cients of these independent

solutions are the moduli. They are the expectation values of massless scalar

¬elds, called the moduli ¬elds. These moduli parametrize changes of the size

and shape of the internal Calabi“Yau manifold but not its topology.

A simple example: the torus

Fig. 9.4. A rectangular torus can be constructed by identifying opposite sides of a

rectangle.

Consider the rectangular torus T 2 = S 1 — S 1 displayed in Fig. 9.4. This

388 String geometry