54

Using this and

ω2 § ω3 = ’dβ § sin θ1 dθ1 § dφ1 § sin θ2 dθ2 § dφ2 , (10.143)

one obtains

(ω2 § ω3 ) = 54r’5 e8A dr § dx0 § dx1 § dx2 § dx3 . (10.144)

The warp factor is determined in terms of the ¬ve-form ¬‚ux by Eq. (10.81),

or equivalently

= d± § dx0 § dx1 § dx2 § dx3 ,

10 F (10.145)

while ± = exp(4A) according to Eq. (10.98). Using the expression for the

¬ve-form ¬‚ux in Eq. (10.138) this leads to the equation

d± = 27π(± )2 ±2 r’5 Ne¬ (r)dr. (10.146)

10.2 Flux compacti¬cations of the type IIB theory 493

Integration then gives the warp factor

27π(± )2 3 r 3

’4A(r)

(gs M )2 log (gs M )2 ,

e = gs N + + (10.147)

4

4r 2π r0 8π

where r0 is a constant of integration.

Problem 10.13 asks you to show that G3 is primitive. This result implies

that this is a supersymmetric background. Note that in this section we have

used the constraints in Eqs (10.98) and (10.115), which were derived for

compact spaces. However, these constraints can also be derived from the

Killing spinor equations for type IIB, which are local. As a result, they also

hold for noncompact spaces.

Warped space-times and the gauge hierarchy

The observation that Poincar´ invariance allows space-times with extra di-

e

mensions that are warped products has interesting consequences for phe-

nomenology. Brane-world scenarios are toy models based on the proposal

that the observed four-dimensional world is con¬ned to a brane embedded

in a ¬ve-dimensional space-time.14 In one version of this proposal, the ¬fth

dimension is not curled up. Instead, it is in¬nitely extended. If we live on

such a brane, why is there a four-dimensional Newtonian inverse-square law

for gravity instead of a ¬ve-dimensional inverse-cube law? The answer is

that the space-time is warped. Let™s explore how this works.

Localizing gravity with warp factors

The action governing ¬ve-dimensional gravity with a cosmological constant

Λ in the presence of a 3-brane is

√ √

5

d4 x ’g,

S∼ d x ’G (R ’ 12Λ) ’ T (10.148)

where T is the 3-brane tension, GM N is the ¬ve-dimensional metric, and gµν

is the induced four-dimensional metric of the brane. This action admits a

solution of the equations of motion of the form

ds2 = e’2A(x5 ) ·µν dxµ dxν + dx2 , (10.149)

5

with

√

’Λ|x5 |.

A(x5 ) = (10.150)

14 There could be an additional compact ¬ve-dimensional space that is ignored in this discussion.

494 Flux compacti¬cations

Fig. 10.5. Gravity is localized on the Planck brane due to the presence of a warp

factor in the metric.

Here ’∞ ¤ x5 ¤ ∞ is in¬nite, and the brane is at x5 = 0. Moreover,

for a static solution it is necessary that the brane tension is related to the

space-time cosmological constant Λ by

√

T = 12 ’Λ, (10.151)

which requires that the cosmological constant is negative. This geometry is

locally anti-de Sitter (AdS5 ), except that there is a discontinuity in deriva-

tives of the metric at x5 = 0. This discontinuity is determined by the delta

function brane source using standard matching formulas of general relativity.

The metric (10.149) contains a warp factor, which has the interesting

consequence that, even though the ¬fth dimension is in¬nitely extended,

four-dimensional gravity is observed on the brane. This way of concealing

an extra dimension is an alternative to compacti¬cation. Computing the

normal modes of the ¬ve-dimensional graviton in this geometry, one ¬nds

that the zero mode, which is interpreted as the four-dimensional graviton,

is localized in the vicinity of the brane and that G4 controls its interactions.

The e¬ective four-dimensional Planck mass on the brane is given by

√

’2 ’Λ|x5 |

2 3

M4 = M5 dx5 e , (10.152)

or in terms of Newton™s constant

’1

√

’2 ’Λ|x5 |

G4 = G 5 dx5 e . (10.153)

10.2 Flux compacti¬cations of the type IIB theory 495

Fig. 10.6. On the SM brane the energy scales are redshifted due to the presence of

the warp factor in the metric.

Large hierarchies from warp factors

If instead of one 3-brane, two parallel 3-branes are considered, the implica-

tions for phenomenology are even more interesting. In this construction the

background geometry is again a warped product, but now the warp factor

provides a natural way to solve the hierarchy problem.

Imagine that the 3-branes are again embedded in a ¬ve-dimensional space-

time as shown Fig. 10.6. One brane is located at x5 = πr, and called the

standard-model brane (SM), while the other brane is located at x5 = 0 and

called the Planck brane (P). The action governing ¬ve-dimensional gravity

coupled to the two branes is

√

S = d5 x ’G (R ’ 12Λ) ’ TSM d4 x ’g SM ’ TP d4 x ’g P ,

(10.154)

where TSM and TP are the tensions of the two branes. The metric is again

assumed to be a warped product

ds2 = e’2A(x5 ) ·µν dxµ dxν + dx2 (10.155)

5

in the interval 0 ¤ x5 ¤ πr.

The equations of motion are solved by a warp factor of the form

√

A(x5 ) = ’Λ|x5 |, (10.156)

as before, and

√

TP = ’TSM = 12 ’Λ. (10.157)

496 Flux compacti¬cations

Negative tension may seem disturbing. However, negative-tension branes

can be realized in orientifold models and in F-theory compacti¬cations. In

this solution the metric is normalized so that it takes the form

P

gµν = ·µν . (10.158)

on the Planck brane. Then, because of the warp factor, the SM brane metric

is

√

’2πr ’Λ

SM

gµν =e ·µν . (10.159)

This scale factor means that objects with energy E at the Planck brane are

√

’πr ’Λ E.

red-shifted on the SM brane, and appear as objects with energy e