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The Gubser-Klebanov-Polyakov-Witten (GKPW) prescription relates the partition function of a CFT to that of a bulk theory of quantum gravity. Since the CFT partition function is fixed, does that mean that there's only one consistent theory of quantum gravity with all the bulk fields and the action fixed?

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The GKPW prescription relates the bulk partition function to the CFT partition function,

$$Z_{bulk}[\Phi_{\partial} = \phi] = Z_{CFT}[O]$$

such that the non-normalizable modes of bulk fields at the boundary act as sources for the dual CFT's operators $O$. (An underlying assumption here is that the CFT states in the partition function are chosen such that they admit a dual semiclassical bulk description.) As you mentioned, one can then calculate the CFT partition function using standard QFT methods.

One part of the answer to your question is that the AdS-CFT conjecture does stipulate that the unique dual description of the large-$N$ CFT is a string theory living in the bulk, with the semiclassical bulk fields' boundary values fixed according to GKPW. Thus the semiclassical description of bulk operators can be precisely mapped to the CFT.

However that does not solve many bulk gravity's problems, as fixing the partition function is akin to a precise enumeration of the number of the semiclassical bulk microstates, not the exact nature of bulk microstates. The dual CFT description does tell you that the bulk quantum gravity in a certain limit is a CFT, but so far has not shed sufficient light on the precise nature of bulk microstates.

As an example, one can calculate the entropy of black holes in the bulk using CFT matrix models, understand Hawking-Page transitions of black holes via confinement-deconfinement etc (See this and references therein). All these involve calculating the CFT partition function. However, the precise nature of bulk microstates is yet unknown. This has led to people conjecturing whether the CFT actually contains all black hole microstates at all, the resolution of bags of gold paradox, whether the interior operators are state-dependent, whether stringy solutions like fuzzballs describe black holes etc. There is a plethora of unknown wealth regarding the duality regarding how bulk states map to CFT microstates.

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