2

18 Note that K1,1 is not a K¨hler potential, since it is function of real ¬elds. Nevertheless, it has

a

some similar properties.

502 Flux compacti¬cations

where K = K3,1 + K1,1 . This potential is manifestly nonnegative, which

shows that compacti¬cations to AdS3 spaces cannot be obtained in this

way.

The radial modulus

Note that not all of the moduli need contribute to the potential Eq. (10.180).

For example, it does not depend on the radial modulus, which characterizes

the overall volume of the compact manifold M . Therefore, this modulus

is not stabilized. The reason for this is that the conditions for unbroken

supersymmetry in Eqs (10.65), (10.66) and (10.67), and also the conditions

for the existence of supersymmetry breaking solutions in Eq. (10.174), are

invariant under the rescaling of the volume by a constant. While this may

seem disappointing, it is also quite fortunate. This freedom means that the

volume can be chosen su¬ciently large to justify the approximations that

have been made. At su¬ciently large volume, most of the higher-derivative

terms of M-theory can be dropped. The situation, of course, changes once

nonperturbative e¬ects are included. It is expected that such e¬ects stabilize

the radial modulus and that the calculations made remain valid when the

¬‚ux quantum is large. This is not speci¬c to the M-theory compacti¬cations

discussed in this section, but holds for most of the ¬‚ux compacti¬cations

discussed in the literature. Very few models have been constructed in which

all moduli are stabilized without nonperturbative e¬ects.

The scalar potential for type IIB

The scalar potential for type IIB compacti¬ed on a Calabi“Yau three-fold

follows from a standard supergravity formula. In Section 10.2 the formulas

for the superpotential W and K¨hler potential K were presented. Given

a

these potentials, N = 1 supergravity determines the scalar potential in

terms of these quantities19

¯

V = eK Gab Da W D¯W ’ 3|W |2 , (10.181)

b

where Ga¯ = ‚a ‚¯K is the metric on moduli space, with a, b labelling all the

b b

¯

super¬elds, and Gab is its inverse. Moreover, Da = ‚a + ‚a K.

As it should be, this scalar potential is invariant under the K¨hler trans-

a

formation

¯z

K(z, z ) ’ K(z, z ) ’ f (z) ’ f (¯),

¯ ¯ (10.182)

19 This compacti¬cation gives N = 2 supersymmetry, but an N = 1 formalism is still applicable.

Moreover, one only has N = 1 when orientifold planes are included.

10.3 Moduli stabilization 503

since the superpotential transforms according to

W (z) ’ ef (z) W (z). (10.183)

This transformation is a consequence of the linear dependence of W on „¦ and

the behavior of the holomorphic three-form under K¨hler transformations.

a

Here z refers to the moduli ¬elds and f (z) is a holomorphic function of

these ¬elds. The four-dimensional gravitational constant (or Planck length)

κ4 has been set to one in the above formulas.

A simple calculation shows that this potential does not depend on the

radial modulus (except as an overall factor). Using the result for the K¨hler

a

potential for ρ derived in exercise 10.6, one ¬nds

Gρ¯Dρ W Dρ W ’ 3|W |2 = 0.

ρ

(10.184)

¯

As a result, the scalar potential is of the no-scale type

¯

V = eK Gij Di W D¯W , (10.185)

j

i,j=ρ

where i, j label all the ¬elds excluding ρ. At the minimum of the potential

Di W = 0, (10.186)

which implies V = 0 even though supersymmetry is broken in general, since

Dρ W = 0. (10.187)

These solutions have the interesting property that V = 0 at the minimum of

the potential, so that the cosmological constant vanishes at the same time

supersymmetry that is broken. Even though this may seem encouraging for

achieving the goal of breaking supersymmetry without generating a large

vacuum energy density, it does not constitute a solution of the cosmological

constant problem. There is no reason to believe that this result continues

to hold when ± and gs corrections are included. In the next section we will

see that nonperturbative corrections to W depending on ρ can generate a

nonvanishing cosmological constant.

Moduli stabilization by nonperturbative e¬ects

The type IIB no-go theorem excludes the possibility of compacti¬cation to

four-dimensional de Sitter (dS) space, or more generally to a space with a

positive cosmological constant. This section shows that this conclusion can

be circumvented when nonperturbative e¬ects are taken into account. This

504 Flux compacti¬cations

is of interest, since the Universe appears to have a small positive cosmological

constant.

The basic idea is to stabilize all moduli of the type IIB compacti¬cation

and to break the no-scale structure by adding nonperturbative corrections

to the superpotential. These contributions are combined in such a way

that supersymmetry is not broken. This leads to an AdS vacuum with a

negative vacuum energy density. Then one adds anti-D3-branes that break

the supersymmetry and give a positive vacuum energy density.

In the simplest case, there is only one exponential correction to the su-

perpotential, but in general there may be multiple exponentials. The cor-

rections to the K¨hler potential can be ignored in the large-volume limit.

a

The K¨hler potential for the radial modulus is then equal to its tree-level

a

expression. Assuming that all other modes are massive and can be inte-

grated out, one is left with an e¬ective theory for the radial modulus. In

the following we assume that the only K¨hler modulus is the size, while the

a

complex structure and the dilaton become massive due to the presence of

¬‚uxes.

The superpotential is assumed to be given by the tree-level result W0

together with an exponential generated by nonperturbative e¬ects

W = W0 + Aeiaρ . (10.188)

One source of nonperturbative e¬ects is instantons arising from Euclidean

D3-branes wrapping four-cycles. These give a contribution to the superpo-

tential of the form

Winst = T (z ± )e2πiρ , (10.189)

where T (z ± ) is the one-loop determinant that is a function of the complex-

structure moduli, and ρ is the radial modulus. Another possible source for

such corrections is gluino condensation in the world-volume gauge theory of

D7-branes, which might be present and wrapped around internal four-cycles.

The coe¬cient a is a constant that depends on the speci¬c source of the

nonperturbative e¬ects. For simplicity, we assume that a, A and W0 are real

and that the axion vanishes. At the supersymmetric minimum all K¨hler a

covariant derivatives of the superpotential vanish including Dρ W = 0. Using

the K¨hler potential in Eq. (10.102), Exercise 10.7 shows that the e¬ective

a

potential