I am wondering about the space-time dual of a CFT vertex operator in the context of AdS3/CFT2-correspondence. In particular, a boundary CFT2 with vertex insertion should be dual to some AdS3-space, possibly with excitations propagating through it. Does a precise description of these AdS3-spaces exist?
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$\begingroup$ Vertex operators are features of CFT in 2 dimensions. Is your question specifically about 3d gravity in AdS space, then? A quick googling returned for instance this paper, maybe you can find answers there? $\endgroup$– M.JoJan 22, 2021 at 10:17
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$\begingroup$ My question is indeed specifically about AdS3/CFT2, thank you for pointing that out. The paper you link to describes a relation between AdS3 and LFT. Do you know if this LFT is the CFT that is dual to AdS3, or is this a particular limit of the full CFT? $\endgroup$– WLVJan 26, 2021 at 12:25
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The "duality" between $AdS_3$ gravity and Liouville theory goes back to the brilliant work of Brown and Henneaux which predates AdS / CFT. By https://arxiv.org/abs/hep-th/9910013 this idea had become quite well developed. It unifies an impressive range of research directions by using the Chern-Simons formulation of $AdS_3$ gravity to argue for a WZW model on the boundary. Then the imposition of extra boundary conditions results in a gauge fixing of this WZW known as Drinfeld Sokolov reduction. However, it's important to remember that this is all still semi-classical. At minimum, the exact dual would need to depend on the full $AdS_3 \times M_7$ background which provides a UV completion.
So Liouville theory is only an approximation to holographic duals in two dimensions. Recently, https://arxiv.org/abs/1912.00222 provided an explanation for its universality. If you want to use this, you are necessarily describing heavy operators. And then it doesn't look like vertex operators will have a special property anymore. Most heavy operators will just be dual to black hole microstates.
There could be something interesting about holographic theories with light vertex operators but finding an example might be hard. First, the theory needs to have free fields otherwise what is meant by "vertex operator"? Second, it needs to have many free fields because a weakly coupled gravity dual only occurs for large central charge. The symmetric orbifold theory is something like this which gets discussed in the AdS / CFT context. But the dual of string theory on $AdS_3 \times S^3 \times K3$ is actually not this theory but something more strongly coupled which lives on the same moduli space.
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$\begingroup$ This is a really useful and extremely illuminating answer. Thanks a lot for sharing this with us! A little question about your last paragraph. What about just pure gravity in $AdS_{3}$? Your argument that a macroscopic gravity dual only exists at large central charge seems to be correct, but how can it be that that pure gravity in $AdS_{3}$ has a dual when by definition there are no other bulk fields, apart from the metric? $\endgroup$ Jun 12, 2021 at 14:57
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1$\begingroup$ Pure gravity is where Brown and Henneaux's $c = \frac{3L}{2G}$ formula was first derived. If it exists as a quantum theory (which I'm still not convinced of) its dual needs to have a huge gap with no primary operators for the reason you mention. The first would be at weight $\frac{c}{12}$ and correspond to the massless BTZ black hole. Recent modular bootstrap work seems to indicate that this constraint rules out a dual which is a single CFT and forces us to expand our search into ensembles of CFTs. $\endgroup$ Jun 12, 2021 at 15:30
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$\begingroup$ Awesome! I've taken the existence for a CFT dual to pure gravity in this context (and its completeness) for guaranteed. Thanks again for your kind answer. $\endgroup$ Jun 12, 2021 at 15:45
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$\begingroup$ Thank you for your answer, this clarifies it a lot. $\endgroup$– WLVJul 6, 2021 at 13:08