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?
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.