String Theory and M-Theory

10.2 Flux compactifications of the type IIB theory 493
Integration then gives the warp factor
e −4A(r) = 27π(α′ ) 2
4r 4
gsN + 3
2π (gsM) 2
r
log
r0
+ 3
2
(gsM) , (10.147)
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 Poincaré invariance allows space-times with extra dimensions
that are warped products has interesting consequences for phenomenology.
Brane-world scenarios are toy models based on the proposal
that the observed four-dimensional world is confined to a brane embedded
in a five-dimensional space-time. 14 In one version of this proposal, the fifth
dimension is not curled up. Instead, it is infinitely extended. If we live on
such a brane, why is there a four-dimensional Newtonian inverse-square law
for gravity instead of a five-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 five-dimensional gravity with a cosmological constant
Λ in the presence of a 3-brane is
S ∼ d 5 x √
−G (R − 12Λ) − T d 4 x √ −g, (10.148)
where T is the 3-brane tension, GMN is the five-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
with
ds 2 = e −2A(x5) ηµνdx µ dx ν + dx 2 5, (10.149)
A(x5) = √ −Λ|x5|. (10.150)
14 There could be an additional compact five-dimensional space that is ignored in this discussion.