The mathematics that is needed to describe Calabi“Yau moduli spaces,

known as special geometry, is described in this section.

The metric on moduli space

The moduli space has a natural metric de¬ned on it18 , which is given as

a sum of two pieces. The ¬rst piece corresponds to deformations of the

complex structure and the second to deformations of the complexi¬ed K¨hler

a

form

√

1 ¯¯

ds2 = g ab g cd [δgac δg¯d + (δgad δgc¯ ’ δBad δBc¯)] g d6 x, (9.97)

¯ ¯

¯

b b b

2V

where V is the volume of the Calabi“Yau manifold M . The fact that the

metric splits into two pieces in this way implies that the geometry of the

moduli space has a product structure (at least locally)

M(M ) = M2,1 (M ) — M1,1 (M ). (9.98)

Each of these factors has an interesting geometric structure of its own de-

scribed below.

The complex-structure moduli space

The K¨hler potential

a

Let us begin with the space of complex-structure deformations of the metric.

First we de¬ne a set of (2, 1)-forms according to

1 1 ¯ ‚g¯ ¯

(χ± )ab¯dz a § dz b § d¯c

z¯ (χ± )ab¯ = ’ „¦ab d cd , (9.99)

χ± = with

c c

‚t±

2 2

where t± , with ± = 1, . . . , h2,1 are local coordinates for the complex-structure

moduli space. Indices are raised and lowered with the hermitian metric,

¯ ¯

so that „¦ab d = g cd „¦abc , for example. As in Eq. (9.96), these forms are

harmonic. These relations can be inverted to show that under a deformation

of the complex structure the metric components change according to

1 1

cd abc

(χ± )cd¯δt± , 2

δga¯ = ’ „¦¯ where „¦ = „¦abc „¦ .

2a

¯b b

„¦ 6

(9.100)

18 The metric on the moduli space, which is a metric on the parameter space describing deforma-

tions of Calabi“Yau manifolds, should not be confused with the Calabi“Yau metric.

392 String geometry

Writing the metric on moduli space as

¯¯

ds2 = 2G±β δt± δ tβ , (9.101)

¯

and using Eqs (9.97) and Eq. (9.100) for δga¯, one ¬nds that the metric on

¯b

moduli space is

i χ± § χ β

¯¯

¯¯ ¯¯

G±β δt δ tβ = ’

±

δt± δ tβ . (9.102)

¯

i „¦§„¦

Under a change in complex structure the holomorphic (3, 0)-form „¦ be-

comes a linear combination of a (3, 0)-form and (2, 1)-forms, since dz be-

comes a linear combination of dz and d¯. More precisely,

z

‚± „¦ = K ± „¦ + χ ± , (9.103)

where ‚± = ‚/‚t± and K± depends on the coordinates t± but not on the

coordinates of the Calabi“Yau manifold M . The concrete form of K± is

determined below. Moreover, the χ± are precisely the (2, 1)-forms de¬ned

in (9.99). Exercise 9.10 veri¬es Eq. (9.103).

Combining Eqs (9.102) and (9.103) and recalling that G±β = ‚± ‚β K, one

¯ ¯

sees that the metric on the complex-structure moduli space is K¨hler with

a

K¨hler potential given by

a

K2,1 = ’ log i „¦§„¦ . (9.104)

Exercise 9.9 considers the simple example of a two-dimensional torus and

shows that the K¨hler potential is given by Eq. (9.104) for „¦ = dz.

a

Special coordinates

In order to describe the complex-structure moduli space in more detail, let

us introduce a basis of three-cycles AI , BJ , with I, J = 0, . . . , h2,1 , chosen

such that their intersection numbers are

AI © BJ = ’BJ © AI = δJ

I

AI © AJ = BI © BJ = 0. (9.105)

and

The dual cohomology basis is denoted by (±I , β I ). Then

±I § β J = δ I

J

βI = β I § ±J = ’δJ . (9.106)

I

±I = and

AJ BJ

The group of transformations that preserves these properties is the symplec-

tic modular group Sp(2h2,1 + 2; ).

9.6 Special geometry 393

In analogy with the torus example, we can de¬ne coordinates X I on the

moduli space by using the A periods of the holomorphic three-form

XI = I = 0, . . . , h2,1 .

„¦ with (9.107)

AI

The number of coordinates de¬ned this way is one more than the number of

moduli ¬elds. However, the coordinates X I are only de¬ned up to a complex

rescaling, since the holomorphic three-form has this much nonuniqueness.

To take account of this factor consider the quotient19

X±

±

± = 1, . . . , h2,1 ,

t= 0 with (9.108)

X

where the index ± now excludes the value 0. This gives the right number

of coordinates to describe the complex-structure moduli. Since the X I give

the right number of coordinates to span the moduli space, the B periods

FI = „¦ (9.109)

BI

must be functions of the X, that is, FI = FI (X). It follows that

„¦ = X I ±I ’ FI (X)β I . (9.110)

A simple consequence of Eq. (9.103) is