Why holographic renormalization?

Question

Why is there a need to perform holographic renormalization for the normal $AdS_5\times S^5$/CFT$_4$ correspondence if the brane theory is conformal? Since the flow along the AdS direction $r$ is related to the renormalization scale does this not explicitly introduce an "energy" parameter that breaks conformal invariance in the SYM side of the duality?

Answer 1

The fact that the boundary theory is conformal means that renormalization does not induce running of the coupling. However, there are divergences which have to be regularized and renormalized. The regularization requires the introduction of an arbitrary scale, which is not Weyl invariant and leads to a conformal anomaly (in even dimensions).

Correspondingly, also the bulk theory has to be regularized by introducing a cutoff $\epsilon$ on the radial coordinate. The supergravity fields have to be expanded close to the horizon and local counterterms have to be introduced to subtract the divergences when taking the limit of $\epsilon\to 0$. For the metric, the regularization procedure requires picking a reference metric $g_{(0)}$ from the conformal structure on the boundary. For $d$ (boundary dimension) even, the dependence of the counterterm on the chosen reference metric leads to a renormalized Lagrangian, that is not Weyl invariant. One picks up exactly the expected Weyl anomaly.

This is a very neat example of a connection of boundary UV physics (the cutoff) and bulk IR physics (divergences close to the boundary) which lead to the same Weyl anomaly.

For details, see the paper by Henningson and Skenderis. There are also these very instructive lecture notes on holographic renormalization with the example of renormalization of the action of a massive bulk scalar.

Addendum: Example of why CFT correlators need regularization/renormalization
It is well-known that conformal invariance greatly restricts the form of CFT correlation functions. For example, two-point functions of a scalar operator $\mathcal{O}$ are restricted to $$ \langle\mathcal{O}(x)\mathcal{O}(0)\rangle=\frac{C}{x^{2\Delta}} $$ where $\Delta$ is the scaling dimension of $\mathcal{O}$ and $C$ is a normalization constant. It is much less known though, that this is only a bare correlator and it is not valid at $x^{2}=0$. A correlator should be a well-defined distribution and have well-defined Fourier transforms: $$ G(p)=\int d^dx\, e^{-ipx}\frac{C}{x^{2\Delta}}=\frac{C\pi^{d/2}2^{d-2\Delta}\Gamma\left(\frac{d-2\Delta}{2}\right)}{\Gamma{\Delta}}p^{2\Delta-d}. $$ Since the $\Gamma$-function is undefined for negative integer arguments, we can see that regularization is necessary when $\Delta=\frac{d}{2}+n$, where $n$ is a positive integer. This can be done, for example, using dimensional regularization. After addition of a counterterm in the action the correlator becomes $$ G(p)=p^{2\Delta-d}\left(C_1 \log\frac{p^2}{\mu^2}+C_2\right), $$ which clearly is a scale dependent expression. Scale invariance is an anomalous symmetry in the full quantum theory. However, in $\mathcal{N}=4$ super Yang-Mills the coupling is protected from running by supersymmetry. So it is not conformal symmetry that leads to a vanishing $\beta$-function, but it is SUSY. The vanishing of the $\beta$-function for this particular theory is discussed here including some references.