I have a question about replica wormholes and the CFT ensembles in AdS/CFT. To make sure that my question isn't coming from a simple misunderstanding, I'll first sketch out my current understanding on the topic.
The lore I have internalized is that replica wormholes exist from an AdS/CFT perspective because the bulk theory of gravity we usually use is actually an EFT. One perspective on this is that a fixed EFT of quantum gravity is actually dual to an ensemble of CFTs, rather than any fixed CFT. An argument for this is the fact that in the bulk, $G_N$ is a continuous parameter so it can shift continuously as we coarse grain (which we must as we renormalize, for we are treating gravity as an EFT). But on the boundary, $G_N \sim 1/N^2$ is quantized, and so this continuity needs some average over CFTs to make sense, at minimum over the central charge. Another argument is that multi-boundary wormholes imply that there is no factorization of the boundary partition functions, impossible if there was only a single CFT dual to the bulk theory. A different perspective is that in the UV, the bulk theory really is dual to just a single, fixed CFT, and the "ensemble" is just a convenient way of averaging out the fine-grained information of the boundary which is irrelevant for the IR behavior of the bulk. Wormholes simply reflect this coarse-graining. At the level of EFT, I think these perspectives are compatible. To distinguish between these two perspectives in the UV, one would need to demonstrate "discreteness" (or not) of $G_N$ directly in the bulk, probably through some sort of topological argument but I'm not totally sure.
Either way, given this family of ideas, now consider the classic example of holography, type IIB strings on AdS$_5 \times S^5$ and $\mathcal{N}=4$ SU(N) Super Yang-Mills. This duality is usually presented as an exact equivalence of theories in the large N limit, but I usually hear people talk about this equivalence as being non-perturbative and true in principle for finite N as well. Furthermore, in the bulk, the AdS/$S^5 $ radius is usually taken to be finite (but large). Since the $S^5$ radius is supported with N units of 5-form flux, that means that N is fixed on the boundary in principle.
On the one hand, this example does not involve any averaging over boundary theories. Under RG, N is fixed on the boundary and does not flow even under a perturbation, but maybe the tHooft coupling could flow. On the other hand, computations of e.g. Reyni entropies in AdS$_5 \times S^5$ involve replica wormholes, implying a lack of boundary factorization. An exact equivalence of the partition functions between AdS gravity and the CFT therefore seems impossible, even if we fix the $S^5$ radius. Even if you tried to say some words about the wormholes being a saddlepoint solution and 1/N or finite N corrections saving the day, it's hard for me to see how this could be extended to a more complete argument. One possible solution is that I don't understand the corrections to the $S^5$ radius beyond large N (seems likely now that I've typed this out), but it's not clear to me how these corrections destroy replica wormholes in the bulk. Another possibility is that there is some answer involving D-branes I don't understand. Does anyone have any insights?
TL;DR: Replica wormholes imply a lack of factorization of the boundary partition function for disconnected components. This is impossible for a single finite N theory. On the other hand, $\mathcal{N}=4$ SYM is a fixed finite N theory and its bulk dual, type IIB strings on AdS$_5 \times S^5$, seems to have replica wormholes. What's the resolution?
