and use them to form

1 κ±βγ w± wβ wγ 1

J §J §J,

G(w) = = (9.131)

w0 6w0

6

which is analogous to the prepotential for the complex-structure moduli

space. Here we have introduced one additional coordinate, namely w 0 , in

order to make G(w) a homogeneous function of degree two. Then we ¬nd

h1,1 ¯

‚G ‚G

’K1,1

wA ’ wA A

e =i ¯ , (9.132)

‚ wA

¯ ‚w

A=0

where now the new coordinate w 0 is included in the sum. In Eq. (9.132) it

is understood that the right-hand side is evaluated at w 0 = 1. A homework

problem asks you to verify that Eq. (9.132) agrees with Eq. (9.129).

The form of the prepotential

To leading order the prepotential is given by Eq. (9.131). However, note that

the size of the Calabi“Yau belongs to M1,1 (M ) and as a result ± corrections

are possible. So Eq. (9.131) is only a leading-order result. However, the

corrections are not completely arbitrary, because they are constrained by

the symmetry. First note that the real part of w ± is determined by B,

which has a gauge transformation. This leads to a Peccei“Quinn symmetry

given by shifts of the ¬elds by constants µ±

δw± = µ± . (9.133)

Together with the fact that G(w) is homogeneous of degree two, this implies

that perturbative corrections take the form

κABC wA wB wC

+ iY(w0 )2 ,

G(w) = (9.134)

0

w

where Y is a constant. Note that the coe¬cient of (w 0 )2 is taken to be

purely imaginary. Any real contribution is trivial since it does not a¬ect the

9.6 Special geometry 397

K¨hler potential. The result, which was derived by using mirror symmetry,

a

is

ζ(3)

Y= χ(M ), (9.135)

2(2π)3

where χ(M ) = 2(h1,1 ’ h2,1 ) is the Euler characteristic of the manifold.

Nonperturbatively, the situation changes, since the Peccei“Quinn symme-

tries are broken and corrections depending on w ± become possible. It turns

out that sums of exponentially suppressed contributions of the form

c± w ±

exp ’ , (9.136)

± w0

where c± are constants, are generated. These corrections arise due to in-

stantons, as is discussed in Section 9.8.

EXERCISES

EXERCISE 9.9

Use the de¬nition (9.97) to show that the metric on the complex-structure

moduli space of a two-dimensional torus is K¨hler with K¨hler potential

a a

given by

K = ’ log i „¦§„¦ and „¦ = dz. (9.137)

SOLUTION

As we saw in Exercise 7.8, a two-torus compacti¬cation, with complex-

structure modulus „ = „1 + i„2 , can be described by a metric of the form

2 2

1 „1 + „ 2 „1

g= .

„1 1

„2

√

Here we are setting B = 0 and det g = 1, since we are interested in

complex-structure deformations. The torus metric then takes the form

1

ds2 = „1 + „2 dx2 + 2„1 dxdy + dy 2 = 2gz z dzd¯,

2 2

z

¯

„2

where we have introduced a complex coordinate de¬ned by

1

dz = dy + „ dx and gz z = .

¯

2„2

398 String geometry

For these choices the K¨hler potential is

a

K = ’ log i dz § d¯ = ’ log(2„2 ).

z

This gives the metric

1

G „ „ = ‚ „ ‚„ K = 2.

¯ ¯

4„2

Under a change in complex structure „ ’ „ + d„ the metric components

change by

d„ d¯„

δgzz = 2 and δgzz = 2 .

¯¯

2„2 2„2

Using the de¬nition of the metric on moduli space (9.97) we ¬nd the moduli-

space metric

√

1 d„ d¯„

ds2 = 2G„ „ d„ d¯ = (g z z )2 δgzz δgzz gd2 x =

¯

„

¯ ¯¯ 2

2V 2„2

in agreement with the computation based on the K¨hler potential.

a 2

EXERCISE 9.10

Prove that ‚± „¦ = K± „¦ + χ± , where the χ± are the (2, 1)-forms de¬ned in

Eq. (9.99).

SOLUTION

By de¬nition

1

„¦ = „¦abc dz a § dz b § dz c ,

6

so the derivative gives

‚(dz c )

1 ‚„¦abc a 1

b c a b

dz § dz § dz + „¦abc dz § dz §

‚a „¦ = .

6 ‚t± ‚t±

2

The ¬rst term is a (3, 0)-form, while the derivative of dz c is partly a (1, 0)-

form and partly a (0, 1)-form. Since the exterior derivative d is independent

of t± , ‚„¦/‚t± is closed, and hence

‚„¦/‚t± ∈ H (3,0) • H (2,1) .

Now we are going to show that the (2, 1)-form here is exactly the χ± in