the possible string theory vacua are viewed as the local minima of a very

complicated potential function with many peaks and valleys. This function

is visualized as a landscape. This picture is based on an intuition derived

from nonrelativistic quantum mechanics. This intuition surely breaks down

if the scale of the peaks and valleys approaches the string scale or the Planck

scale, as it is based on the use of the low energy e¬ective actions that can

be derived from string theory. However, it provides a valid description if it

is smaller than those scales by a factor that can be made arbitrarily large.

The statistical approach

Motivated by the existence of this enormous number of vacua, a statisti-

cal analysis of their properties has been proposed. Consider the type IIB

¬‚ux vacua discussed in Section 10.2, where the minima of the potential

are described by isolated points. In the statistical approach, ensembles of

randomly chosen systems are picked and speci¬c quantities of interest are

studied. Rather than studying individual vacua, the overall distribution of

vacua on the moduli space is analyzed. Important examples of quantities

that can be analyzed statistically are the cosmological constant and the su-

persymmetry breaking scale. These studies are motivating string theorists

to rethink the concept of naturalness in quantum ¬eld theory. If the multi-

plicity of vacua can compensate for small numbers such as the ratio of the

weak scale to the Planck scale, then it could undermine one of the arguments

for low-energy supersymmetry breaking.

In order to study the number and distribution of type IIB ¬‚ux vacua, the

ensemble is built from the low-energy e¬ective theories with ¬‚ux described

by the superpotential of Eq. (10.101) and subject to the tadpole-cancellation

condition Eq. (10.94). It is rather important in this approach that the num-

ber of vacua that is found is ¬nite. Fortunately, this seems to be a conse-

quence of the constraints given by the tadpole-cancellation condition, which

provides a bound on the possible ¬‚uxes. Additional constraints come from

supersymmetry and duality symmetries as is discussed below. The number

of vacua, with all moduli stabilized, is ¬nite for this class of examples, but

this might not be true in general.

Counting of vacua

Let us now describe the counting of supersymmetric type IIB ¬‚ux vacua

discussed in Section 10.2. Recall that in these vacua the three-form G3 =

524 Flux compacti¬cations

F3 ’ „ H3 is nonvanishing. Since the three-forms F3 and H3 are harmonic,

they are fully characterized by their periods on a basis of three-cycles

NRR = · ±β

±

NNS = · ±β

±

F3 and H3 . (10.252)

Σβ Σβ

Here ·±β is the intersection matrix of three-cycles and · ±β is its inverse.

Recall that (for suitable normalizations) these N ™s are integers as a conse-

quence of the generalized Dirac quantization condition. In this notation the

tadpole-cancellation condition Eq. (10.94) gives the following constraint on

the ¬‚uxes

β

±

0 ¤ ·±β NRR NNS ¤ L, (10.253)

where

L = χ/24 ’ ND3 . (10.254)

Here χ is the Euler characteristic of the 3-fold and ND3 is a positive integer

describing the total R“R charge, as in Eq. (10.94).

Using Eq. (10.101), the superpotential can be written in terms of the

periods of the holomorphic three-form

Π± = „¦, (10.255)

Σ±

as

± ±

W = (NRR ’ „ NNS )Π± = N · Π. (10.256)

A supersymmetric ¬‚ux vacuum is determined by the ¬‚ux quanta N ± and

solves the equation

Di W = 0, (10.257)

where W = 0 corresponds to Minkowski space and W = 0 corresponds to

AdS space.

A simple example

The simplest examples of ¬‚ux compacti¬cations are orientifolds, such as

T 6 / 2 . As an example, let us count the ¬‚ux vacua for the simple toy model

of a rigid Calabi“Yau with no complex-structure moduli, b3 = 2 and periods

Π1 = 1 and Π2 = i. The K¨hler moduli are ignored as these moduli ¬elds

a

are ¬xed by nonperturbative e¬ects and therefore can be ignored in a pertur-

bative description. This simple example illustrates all the features of more

realistic six-dimensional examples. It has no geometric moduli at all, only

the axion“dilaton modulus „ , which can be viewed as the complex-structure

modulus of a torus.

10.6 The landscape 525

The superpotential takes the simple form

W = N · Π = A„ + B, (10.258)

with coe¬cients

1 2

A = ’(NNS + iNNS ) = a1 + ia2 , (10.259)

1 2

B = NRR + iNRR = b1 + ib2 . (10.260)

Using Eq. (10.103), the condition Eq. (10.257) gives

1 A¯ + B

„

D„ W = ‚„ W + ‚„ KW = ‚„ W ’ W =’ = 0. (10.261)

„ ’„ „ ’„

¯ ¯

This determines the „ -parameter of the axion“dilaton to be

¯¯

„ = ’B/A. (10.262)

Fig. 10.12. Values of „ in the fundamental region of SL(2, ) for a rigid Calabi“Yau

¥

manifold with L = 150.

526 Flux compacti¬cations

One ¬nal restriction on the vacua comes from the SL(2, ) duality sym-

metry of the type IIB theory. This symmetry allows one to restrict the value

of the integers appearing in the previous formula to a2 = 0 and 0 ¤ b1 < a1 ,

which then implies that a1 b2 ¤ L. For each choice of L, the values of

„ that correspond to allowed choices of the ¬‚uxes can be computed using

Eq. (10.262). A scatter plot of these values for the choice L = 150 is shown

in Fig. 10.12. This ¬gure shows that, at particular points, such as „ = ni,

there are holes. At the center of these holes there is a large degeneracy of

vacua. For example, there are 240 vacua for „ = 2i. So one concludes from

this simple toy example that the statistical analysis provides the informa-

tion where vacua with certain properties can be found in the moduli space.

With these techniques it is possible to compute the distribution function of

vacua on the moduli space of string compacti¬cations and such an analysis

can be generalized to the nonsupersymmetric case. However, this is beyond

the scope of this book. On the more speculative side, it has been proposed

that the landscape can be described in terms of a wave function of the Uni-

verse, providing an alternative way of thinking about the issue of how to

choose among the many di¬erent ¬‚ux vacua. This subject is an active area