„¦ § ‚I „¦ = 0, (9.111)

which implies

‚FJ 1‚

FI = X J X J FJ ,

= (9.112)

I I

‚X 2 ‚X

or, equivalently,

‚F 1

F = X I FI .

FI = where (9.113)

‚X I 2

As a result, all of the B periods are expressed as derivatives of a single

function F called the prepotential. Moreover, since

‚F

2F = X I

, (9.114)

‚X I

F is homogeneous of degree two, which means that if we rescale the coordi-

nates by a factor »

F (»X) = »2 F (X). (9.115)

19 As usual in this type of construction, these coordinates parametrize the open set X 0 = 0.

394 String geometry

Since the prepotential is de¬ned only up to an overall scaling, strictly speak-

ing it is not a function but rather a section of a line bundle over the moduli

space.

The prepotential determines the metric on moduli space. Using the gen-

eral rule for closed three-forms ± and β

±§β =’ β’

± β ±, (9.116)

AI AI

M BI BI

I

the K¨hler potential (9.104) can be rewritten in the form

a

h2,1

I

2,1

¯

e’K X I FI ’ X FI ,

= ’i (9.117)

I=0

as you are asked to verify in a homework problem. As a result, the K¨hler

a

potential is completely determined by the prepotential F , which is a holo-

morphic homogeneous function of degree two. This type of geometry is

called special geometry.

An important consequence of the product structure (9.98) of the moduli

space is that the complex-structure prepotential F is exact in ± . Indeed,

the ± expansion is an expansion in terms of the Calabi“Yau volume V ,

which belongs to M1,1 (M ), and it is independent of position in M2,1 (M ),

that is, the complex structure.20 When combined with mirror symmetry,

this important fact provides insight into an in¬nite series of stringy ± cor-

rections involving the K¨hler-structure moduli using a classical geometric

a

computation involving the complex-structure moduli space only.

The K¨hler transformations

a

The holomorphic three-form „¦ is only determined up to a function f , which

can depend on the moduli space coordinates X I but not on the Calabi“Yau

coordinates, that is, the transformation

„¦ ’ ef (X) „¦ (9.118)

should not lead to new physics. This transformation does not change the

K¨hler metric, since under Eq. (9.118)

a

¯

K2,1 ’ K2,1 ’ f (X) ’ f (X), (9.119)

which is a K¨hler transformation that leaves the K¨hler metric invariant.

a a

20 Since V and ± are the only scales in the problem, the only dimensionless quantity containing

± is (± )3 /V . So if one knows the full V dependence, one also knows the full ± dependence.

9.6 Special geometry 395

Equations (9.103) and (9.104) determine K± to be

K± = ’‚± K2,1 . (9.120)

One can then introduce the covariant derivatives

D± = ‚± + ‚± K2,1 , (9.121)

and write

χ± = D± „¦, (9.122)

which now transforms as χ± ’ ef (X) χ± .

The K¨hler-structure moduli space

a

The K¨hler potential

a

An inner product on the space of (1, 1) cohomology classes is de¬ned by

¯ ¯√

1 1

ρad σ¯ g ab g cd gd6 x = ρ § σ,

G(ρ, σ) = (9.123)

¯ bc

2V 2V

M M

where denotes the Hodge-star operator on the Calabi“Yau, and ρ and σ

are real (1, 1)-forms. Let us now de¬ne the cubic form

ρ § σ § „,

κ(ρ, σ, „ ) = (9.124)

M

and recall from Exercise 9.1 that κ(J, J, J) = 6V . Using the identity

1

σ = ’J § σ + κ(σ, J, J)J § J (9.125)

4V

the metric can be rewritten in the form

1 1

G(ρ, σ) = ’ κ(ρ, σ, J) + κ(ρ, J, J)κ(σ, J, J). (9.126)

8V 2

2V

If we denote by e± a real basis of harmonic (1, 1)-forms, then we can

expand

J = B + iJ = w ± e± ± = 1, . . . , h1,1 .

with (9.127)

The metric on moduli space is then

1 ‚ ‚ 1,1

¯K ,

G ±β = G(e± , eβ ) = (9.128)

¯ ±

2 ‚w ‚ w β

¯

where

4

1,1

e’K J § J § J = 8V.

= (9.129)

3

396 String geometry

A change in the normalization of the right-hand side would correspond to

shifting the K¨hler potential by an inconsequential constant. These equa-

a

tions show that the space spanned by w ± is a K¨hler manifold and the

a

K¨hler potential is given by the logarithm of the volume of the Calabi“Yau.

a

We also de¬ne the intersection numbers

e± § e β § e γ