On the Flavor Problem in Strongly Coupled Theories - THEP Mainz

2.2. AdS/CFT 41
It tells us, that if the string length ℓs ≪ R, the Yang-Mills gauge coupling times
the number of colors N (the t’Hooft coupling) becomes large and the Yang-Mills
theory is strongly coupled. In addition, nonperturbative string states with masses
proportional to the inverse string coupling m ∼ 1/gs, are required to decouple as
well [107, p.22]. This means gs → 0 and therefore we have necessarily a large N
boundary theory, because gs ∼ 1/N. The picture becomes much more intuitive once
one examines the spacetime symmetries on both sides. Consider the AdS metric in
conformal coordinates (2.2). It is evident, that this metric is invariant under the
rescaling
x µ → λx µ , z → λz. (2.18)
Fields transform under such a rescaling according to their scaling dimension ∆φ,
φ(xµ, z) → λ −∆φφ(λxµ, λz). In contrast to flat space, every Lorentz invariant action
will turn out to be invariant under rescalings as well due to this property of the metric.
For example
�
SAdS = d 4 � �5 �
R 1
x dz
z 2 GMN M 2
∂Mφ ∂Nφ −
2 φ2 − λ
4! φ4
�
�
→ λd 4 � �5 �
R 1
x dλz
λz 2 λ−2G MN (λ∂M)λ −∆φφ (λ∂N)λ −∆φφ �
M 2
− , (2.19)
2 λ−2∆φ φ 2 − λ
4! λ−4∆φ φ 4
which is invariant upon absorption of a factor λ −∆φ in each field, in contrast to flat
space, where additional factors from the metric appear. This implies, that regardless
of the mass dimension of whatever operator we write down in the bulk Lagrangian, the
action will remain scale invariant. In the dual boundary theory the fifth coordinate is
identified with the inverse of an energy scale and thus we will have a scale invariant
boundary action. In general, it is still an open question whether scale invariance in
4D already implies conformal invariance (see e.g.[108]), however this is a strong hint
that we are dealing with a CFT on the boundary.
In fact it can be shown, that the 15 generators of the spacetime symmetry (isometry)
group of AdS5 correspond to the 15 generators of the conformal group of the dual
boundary theory, whereas the SO(6) isometry of the additional S5 corresponds to
the R-symmetry group of the N = 4 supersymmetry [109, p.187-188]. Changing the
non–AdS part of the manifold will therefore only cut down the amount of supersymmetry
on the CFT side and we can consider the duality on AdS5 alone without loosing
essential ingredients. At this point it should be pointed out, that there is no rigorous
mathematical proof of the original conjecture as of this writing and for the stripped
down versions we are considering here it could be that the corresponding CFT does
not even exist. However, we will see that it is very insightful to switch to the CFT
view in order to get a deeper understanding of the RS model.
In a more precise formulation, the conjecture states that the 5D partition function of
bulk fields φ(x µ , z) with boundary conditions at the AdS5 boundary φ(x µ , z) � � z=0 =