What are the top-down and bottom-up approaches to holography? I think I understand the main idea of top-down vs bottom-up: when trying to describe some phenomena using a particular theory, in t-d one starts from the complete theory and puts constraints so that the desired phenomena would come up, while in b-u one starts from a description of the phenomena and adds equations so that the result is a complete theory. My question is, how does this apply to holography? What is "the complete theory" (since being a duality there are two)?
 A: TL;DR: You can think of top-down approaches as starting from string theory or quantum gravity theories and imposing constraints or limits to obtain a specific holographic dual. Bottom-up approaches start with a specific, already constrained example which can be shown to be exhibit a holographic dictionary, and then try to embed these consistently into more complete theories of quantum gravity (there are many ways to do this).
For more details, we have be more precise with what we mean by "holography".
"Holography" is a rather vague term, simply meaning that degrees of fredom (DOF's) of one system can be described equivalently in a system with lesser dimensions. More precisely, the information necessary to full describe a system does not grow extensively with the voume of the system, but rather with the size of its encolsing area.
I think what you are actually interested in here is the AdS/CFT correspondence, which is a very specific, very famous example of the holographic principle. There are three main levels to this correspondence, known as the strongest, strong and weak form of AdS/CFT.

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*Strongest form: This is the statement from Maldacena in his seminal paper (arXiv: 9711200). It says that 4D super-Yang Mills theory with gauge group $SU(N)$ and coupling $g_{YM}$ is dynamically equivalent to Type IIB string theory with string coupling $g_s$ on $AdS_5\times S^5$ (this is a full-fledged quantum gravity theory).

*Strong form: Since quantum gravity is rather difficult to deal with, we can simplify things by instead working with classical strings by taking their coupling as being very small $g_s\ll 1$. This yields classical type IIB string theory on the same spacetime. On the gauge-theory side, this amounts to tháking the so-called "planar limit" of the QFT, achieved by taking the rank $N$ of the $SU(N)$ group to be very large, i.e. $N\rightarrow \infty$. It can be shown that both perturbative expansions on $1/N$ on the QFT side an $g_s$ on the string theory side coincide to all orders.

*Weak form: One can go a step further and take the limit of vanishingly short strings, which yields a 10D classical supergravity theory of pointlike particles on a now weakly curved $AdS_5\times S^5$ spacetime, since now the radius $L$ of $AdS_5$ is much larger than the string length, i.e. $L\gg l_s$. This amount to taking the strong coupling limit on the QFT side, which results in a conformal field theory (CFT) with large central charge $c$.

You can find more details on this classification in the very nice book of Ammon & Erdmenger: "Gauge/Gravity Duality: Foundations and Applications".
Having understood these three "levels", you can now think of top-down and bottom-up approaches to holography as attempts that start either on the strongest or the weakest form of the correspondence, respectively. Top-down approaches try to define a holographic dual at the level of quantum gravity in a string theory and derive other, more simple and constrained examples by taking certain limits. On the other hand, you may cook up specific CFT and find its weakly coupled holographic dual and then try to loosen up the restrictions to try and embed this pair into a larger, more complete theory. Note that both are remarkably difficult to do and current research is still trying to find a methodical way (if there is one) of defining holographically dual systems.
