(2,2)

As a result, F ∈ Hprimitive . 2

EXERCISE 10.4

Show that a harmonic (3, 1)-form on a Calabi“Yau four-fold is anti-self-dual.

SOLUTION

A harmonic (3, 1)-form

1 ¯

F 3,1 = Fabcd dz a § dz b § dz c § dz d

¯

6

satis¬es

¯

Fabcd J cd = 0.

¯

If this did not vanish, it would give a harmonic (2, 0)-form, but this does

not exist on a Calabi“Yau four-fold. Using this equation and the explicit

representation of the µ symbol,

µabcd¯qrs = (ga¯gb¯gc¯gd¯ ± permutations),

pqrs

p ¯¯¯

it is easy to verify that F 3,1 = ’F 3,1 . Note that this argument can be

easily generalized to show that a primitive (p, 4 ’ p)-form satis¬es

F (p,4’p) = (’1)p F (p,4’p) .

2

EXERCISE 10.5

Show that the supersymmetry constraints in Eqs (10.63) and (10.69) lead

to the ¬‚ux constraints in Eqs (10.65)“(10.67).

10.1 Flux compacti¬cations and Calabi“Yau four-folds 479

SOLUTION

In analogy to the three-fold case discussed in Chapter 9, the following for-

mulas hold for four-folds:

I = 1, ..., h3,1

‚I „¦ = K I „¦ + χ I ,

and

J = K A eA , A = 1, ..., h1,1,

where χI and eA describe bases of harmonic (3, 1)-forms and (1, 1)-forms,

respectively. Since „¦ is a (4, 0)-form one obtains from Eq. (10.63)

„¦ § F 0,4 = 0 χI § F 1,3 = 0.

and

M M

Since h0,4 = 1, the ¬rst constraint leads to F 0,4 = 0. Since χI describes a

3,1

basis of harmonic (3, 1)-forms, F 3,1 = h AI χI , which leads to

I=1

√

(F 1,3 )— § F 1,3 =

F 3,1 § F 1,3 = |F 1,3 |2 g d8 x = 0,

M M M

as F is real. This leads to the ¬‚ux constraint

F 1,3 = F 3,1 = F 0,4 = F 4,0 = 0.

Using ‚A W 1,1 = 0 and Eq. (10.68), one gets

eA § J § F 2,2 = 0.

Since (J § F 2,2 ) is a harmonic (1, 1)-form, we have

h1,1

(J § F 2,2 ) = U A eA .

A=1

So the above constraint results in

√

(J § F 2,2 ) § (J § F 2,2 ) = |J § F 2,2 |2 g d8 x = 0,

M M

which leads to the primitivity condition Eq. (10.67). Notice that the condi-

tion W 1,1 = 0 is then satis¬ed, too. 2

480 Flux compacti¬cations

10.2 Flux compacti¬cations of the type IIB theory

No-go theorems for warped compacti¬cations of perturbative string theory

date back as far as the 1980s. The arguments used then, based on low-

energy supergravity approximations to string theory, were claimed to rule

out warped compacti¬cations to a Minkowski or a de Sitter space-time. If

the internal spaces are compact and nonsingular, and no brane sources are

included, the warp factor and ¬‚uxes are necessarily trivial in the leading

supergravity approximation. These theorems were revisited in the 1990s

when the contributions of branes and higher-order corrections to low-energy

supergravity actions were understood better. These ingredients made it pos-

sible to evade the no-go theorems and to construct warped compacti¬cations

of the type IIB theory, which we will describe in detail below.

The no-go theorem

The no-go theorem states that if the type IIB theory is compacti¬ed on

internal spaces that are compact and nonsingular, and no brane sources are

included, the warp factor and ¬‚uxes are necessarily trivial in the leading su-

pergravity approximation. This subsection shows how this result is derived

and then it shows how sources invalidate the no-go theorem. A similar no-go

theorem shows that compacti¬cations to D = 4 de Sitter space-time do not

solve the equations of motion (see Problem 10.8).

Type IIB action in the Einstein frame

For illustrative purposes, as well as concreteness, let us consider warped

compacti¬cations of the type IIB theory to four-dimensional Minkowski

space-time M4 on a compact manifold M . The ten-dimensional low-energy

e¬ective action for the type IIB theory was presented in Chapter 8. In the

Einstein frame it takes the form9

√ |‚„ |2 |G3 |2 |F5 |2

1 10

d x ’G R ’ ’ ’

S= 2

2(Im „ )2 2 Im „

2κ 4

C4 § G3 § G3

1

+ , (10.75)

8iκ2 Im „

where

G3 = F 3 ’ „ H 3 , (10.76)

9 Recall that the Einstein-frame and string-frame metrics are related by gM N = e’¦/2 gM N .

E S

10.2 Flux compacti¬cations of the type IIB theory 481

and F3 = dC2 , H3 = dB2 . The R“R scalar C0 , which is sometimes called an

axion, and the dilaton ¦ are combined in the complex axion“dilaton ¬eld

„ = C0 + ie’¦ . (10.77)

The only change in notation from that described in Section 8.1 is the use of

M, N (rather than µ, ν) for ten-dimensional vector indices. As explained in

that section,

F5 = 10 F5 (10.78)

has to be imposed as a constraint. Here is the Hodge-star operator in

10

ten dimensions. |G3 |2 is de¬ned by

1 M 1 N1 M 2 N2 M 3 N3

|G3 |2 = G G G G M 1 M 2 M 3 G N1 N2 N3 . (10.79)

3!

The equations of motion and their solution