Question

AdS/CFT at D = 3 (on the AdS side) seems to have some special issues which I bundled into a single question

• The CFT is 2D hence it has an infinite-dimensional group of symmetries (locally). The global (Mobius) conformal transformations correspond to isometries of AdS. What is the meaning of the other conformal transformations on the AdS side?
• Is it possible to apply the duality to CFTs with non-zero central charge?
• The CFT can be regarded as a string theory on its own. Hence we get a duality between different string sectors. Is there another way to describe/interpret this duality?

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Answer 1

I recommend you Chapter 5 (page 150+) of the AdS Bible,

http://arxiv.org/abs/hep-th/9905111

Concerning your individual questions, which are mostly answered at the beginning of that Chapter,

1. the additional Virasoro generators correspond to bulk coordinate reparametrizations that preserve the metric at infinity, but they do map the ground state to excited states

2. yes, the CFTs in AdS/CFT typically have a nonzero central charge which is directly related to the $AdS_3$ curvature radius in the Planck units; there is no reason for $c=0$ here because the boundary CFT isn't really coupled to gravity (which is what the world sheet CFT is doing)

3. for the same reason, you can't directly interpret the CFT as string theory; the full string theory needs $c=0$ in total, so extra ghosts must be added; also, the interpretation of "winding/twisted" sectors is different in boundary CFTs and string CFTs. Of course, this doesn't eliminate the fact that similar "building blocks of CFTs" are used in both kinds of CFTs...

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