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Whenever we define the OPE of a 2D CFT, we do so (at least in the literature that I have come across) on the complex plane. Similarly, the commutation relations between conformal generators $L_n$ and a primary operator are given on a cylinder (e.g. in Francesco, Mathieu and Senechal's book). I'm curious about how to deal with 2D CFT directly on the cylinder. I'm interested in this because the cylinder is more easy to visualize as the boundary of $AdS_3$ and so any CFT concepts on the cylinder can (hopefully) be directly visualized in $AdS_3$.

EDIT: A concrete example of what I'm looking for is the action of $L_n$ on the cylinder on a primary $\mathcal{O}$, i.e. the commutation relation $[L_n,\mathcal{O}]$, along with the definition of $L_n$'s themselves, because the definition through mode expansion and residue worked only on the complex plane.

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