scale. Thus, for a Calabi“Yau manifold that can be characterized by a

’6

single length scale, the volume satis¬es V ≈ MU . Inserting this value

into Eq. (10.245) and Eq. (10.244) one obtains a lower bound for Newton™s

520 Flux compacti¬cations

constant22

4/3

±

G4 > U2 , (10.246)

MU

which is too large by a signi¬cant factor. The lesson is that by insisting on

perturbative control, one obtains unrealistic values for the four-dimensional

Newton™s constant.

Newton™s constant from heterotic M-theory

This situation can be improved in the context of the strongly coupled het-

erotic string. At strong coupling, the corrections to the predicted value of

Newton™s constant are closer to the phenomenologically interesting regime.

If simultaneously the Calabi“Yau volume is large then the successful weak-

coupling prediction for the gauge coupling constants is not ruined. Let us

illustrate how ¬‚uxes at strong coupling can lead to the right prediction for

G4 in the example of the strongly coupled E8 — E8 heterotic string, as de-

scribed in terms of heterotic M-theory.23

The terms of interest in the action for heterotic M-theory are

√ √

1 1

d11 x gR ’ d10 x g|Fi |2 ,

L= (10.247)

2κ2 8π(4πκ2 )2/3 10

M11 Mi

11 11

i

where i = 1, 2 labels the gauge ¬elds of the two di¬erent E8 gauge groups,

and κ11 is the 11-dimensional gravitational constant as usual. If this theory

is compacti¬ed on a Calabi“Yau manifold with volume V times an interval

S1 / 2 of length πd, one can read o¬ the value of Newton™s constant and the

gauge couplings to be

κ2 (4πκ2 )2/3

11 11

G4 = and ±U = . (10.248)

2 Vd

8π 2V

These formulas show that, if ±U and MU are made small enough, then

Newton™s constant G4 can be made small by taking d to be large enough.

The length of the interval d cannot be arbitrarily large, because there is

a value of order (V/κ11 )2/3 , beyond which one of the two E8 ™s is driven to

in¬nite coupling. To derive this bound, the concrete form of the supergravity

background needs to be worked out. This was done by Witten by solving

the constraint following from the gravitino supersymmetry transformation.

22 Typical values are ±U ∼ 1/25 and MU ∼ 2—1016 GeV, whereas G4 = m’2 and mp ∼ 1019 GeV.

p

23 A similar conclusion can be drawn for the strongly coupled SO(32) heterotic string theory,

whose strong-coupling limit is given by the weakly coupled ten-dimensional type I superstring

theory.

10.5 The strongly coupled heterotic string 521

In this background the metric is warped and the ¬‚uxes are nonvanishing due

to the Bianchi identity

√

3 2 κ11 2/3 1

[trF[IJ FKL] ’ trR[IJ RKL] ]δ(x11 ). (10.249)

(dF )11IJKL = ’

2π 4π 2

The delta-function singularity on the right-hand side of this equation comes

from the boundaries or 2 -¬xed planes, and it requires the ¬‚uxes F4 to be

nonvanishing. This Bianchi identity reproduces the right Bianchi identity

for the perturbative heterotic string in the weakly coupled limit (in which

the length of the interval goes to zero). As a side remark, one can see from

Eq. (10.249) that, when higher-order corrections are taken into account,

¬‚uxes no longer obey the ordinary Dirac quantization condition. Namely, in

the appropriate normalization, the Bianchi identity implies that ¬‚uxes are

half-integer quantized,

[F4 /2π] = »(F ) ’ »(R)/2, (10.250)

where » describes the ¬rst Pontryagin class, which is an integer. Also, F

refers to the E8 bundle and R refers to the tangent bundle.

Requiring that the in¬nite coupling regime be avoided gives a lower bound

on Newton™s constant, which (up to a numerical factor) is

2

±U

G4 ≥ 2 . (10.251)

MU

This bound is about an order of magnitude weaker than what was derived

from the weakly coupled heterotic string at the beginning of this section.

Inclusion of numerical factors, such as 16π 2 , gives a bound that is close

to the correct value. Moreover, the bound can be weakened further if one

chooses a Calabi“Yau manifold that is much smaller in some directions than

in others, so that its size is not well characterized by a single scale.

Moduli stabilization

Moduli stabilization in the context of the heterotic string has not been ex-

plored in detail. It is, of course, desirable to see if a potential for the interval

length d can be generated and to make sure that the resulting value for the

interval is in agreement with the value of Newton™s constant. Without en-

tering into details, let us only mention that such a potential can be derived

from nonperturbative e¬ects in a similar manner as was done for the type

IIB theory. The nonperturbative e¬ects come from open M2-brane instan-

tons that wrap the length of the interval (as illustrated in Fig. 10.11) and

gluino condensation on the hidden boundary. Both e¬ects combine in such

522 Flux compacti¬cations

a way that the length of the interval is stabilized in a phenomenologically

interesting regime.

Fig. 10.11. Open M2-brane instantons stretching between both boundaries together

with gluino condensation generate a potential for the interval length.

10.6 The landscape

One of the goals of string theory is to derive the standard model of elemen-

tary particles from ¬rst principles and to compute as many of its parameters

as possible. The dream of a unique consistent quantum vacuum capable of

making these predictions evaporated when it was discovered that there are

several consistent superstring vacua in ten dimensions. Soon it became

evident that the situation is even more complicated, because continua of

supersymmetric vacua exist parametrized by the dilaton and other moduli.

These vacua are unrealistic because they contain massless scalars, the mod-

uli ¬elds, and they have unbroken supersymmetry. Until supersymmetry is

broken, one cannot answer the question of why the value of the cosmological

constant is incredibly small but nonzero. This problem has been addressed

in the recent string theory literature in the context of ¬‚ux compacti¬cations.

The anthropic principle

One approach proposed in the literature argues that there is a large number

of nonsupersymmetric vacua so that the typical spacing between adjacent

values for the cosmological constant is smaller than the observed value. In

this case, it is reasonable that some vacua should have approximately the

10.6 The landscape 523

observed value. Moreover, a signi¬cantly larger value than is observed would

not lead to galaxy formation and the development of life in the Universe, so