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I have a question regarding an equation in the book "Gauge/Gravity Duality" by Martin Ammon and Johanna Erdmenger which however can also be found in other AdS/CFT books or lecture notes.

The authors motivate that the "classical" AdS/CFT correspondence (${\cal N}=4$ SYM <-> IIB superstring theory on $AdS_5 \times S^5$) equates the partition function of the string theory with the generating functional of the CFT correlation functions. In the weak limit, i.e. for $N \rightarrow \infty$ and large $\lambda$, $N$ being the rank of the $SU(N)$ gauge group and $\lambda$ being the 't Hooft coupling, the explicit relation is given in terms of a saddle point approximation

$$\langle \text{exp} \big( \int d^d x \, \mathcal{O} \phi_{(0)} \big) \rangle_{CFT} = e^{-S_{sugra}} \vert_{\lim \limits_{z \to 0}(\tilde{\phi}(z,x) z^{\Delta-d})= \phi_{(0)}(x)} \tag{1}.$$

It is stated that $\tilde{\phi}$ denotes the solution of type IIB supergravity with leading asymptotic behaviour $z^{d−\Delta}\phi_{(0)}$ near the conformal boundary at $z = 0$.

I am a novice in supergravity and AdS/CFT which is why I might lack the required background to fully understand equation (1) and which gets revealed in the following questions:

Is $\tilde{\phi}$ an abbreviation for the collection of field solutions of the respective equations of motions obtained from the variation of the action w.r.t each field that occurs in SUGRA, $\frac{\partial S_{sugra}}{\partial \phi_{i}} = 0$? I.e. $\tilde{\phi} = \{ \phi_1,...,\}$ (by the way, what is the field content of SUGRA?) And what use does it have to plug back the solution of the equation of motions into the action that has been used to retrieve it?

And if this is the case: Can we always find solutions that satisfy the boundary behaviour, i.e. that have the property $$\lim \limits_{z \to 0} \tilde{\phi}(z,x)= z^{d-\Delta} \phi_{(0)}(x)~?$$

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Suppose the classical bulk theory is described in terms of a set of fields $\phi = ( \phi_1 , \phi_2 , \cdots )$. These fields could be scalars or vectors or fermions - I'm just using short-hand notation. These fields satisfy their equation of motions $$ E(\phi) = 0. $$ It is known in AdS that regular solutions to equations are motion are parameterized by a function on the boundary of AdS. For instance, for a scalar field $\Phi$ of mass $m$, the full bulk solution $\Phi(z,x)$ is completely fixed in terms of $\phi_0(x)$ where $\phi_0(x) = \lim_{z \to 0} z^{\Delta-d} \Phi(z,x)$, $\Delta(\Delta-d)=m^2L^2$. In other words we can write $\Phi(z,x) = {\tilde \Phi}[\phi_0(x)]$ where ${\tilde \Phi}$ is completely determinable from the equations of motion. In essence $\phi_0(x)$ is the "integration constant" you get when solving the differential equation $E(\phi)=0$. A similar property holds for other fields as well.

OK, so on the RHS of your equation (1), we are evaluating the action of the bulk classical theory on-shell, i.e. we substitute $\Phi(z,x)\to {\tilde \Phi}[\phi_0(x)]$ into $S[\Phi]$ so we end up with a quantity that's a function(al) of $\phi_0(x)$. The LHS of (1) also depends only on $\phi_0(x)$. Here, $\phi_0(x)$ plays the role of a source for some operator ${\cal O}(x)$ in the CFT. The operator has a scaling dimension $\Delta$.

The field content of SUGRA depends on which supergravity theory you are considering. The SUGRA that is dual to ${\cal N}=4$ $SU(N)$ SYM is $D=5$, ${\cal N}=8$ gauged supergravity which contains 1 metric field, 8 gravitini (spin 3/2), 15 vector fields (adjoint of $SO(6)$), 12 antisymmetric $B$ fields, 48 dilatini (spin 1/2) and 42 scalar fields.

The fall-offs $z^{d-\Delta} \phi_0(x)$ for the field $\Phi(z,x)$ holds only for scalar fields. Vector fields and metric, etc. have slightly different fall-offs. However, every single field does have a fall-off of the type $z^s$ near the boundary. $s$ is fixed by analyzing the equations of the motion of the theory.

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