Which conformal field theories have a semiclassical gravity dual
In plain words
Only special quantum theories behave as if they described a smooth curved space with ordinary gravity. The exact conditions that single them out are unknown.
Precise statement
Heemskerk, Penedones, Polchinski and Sully (2009) conjectured that a CFT with large central charge $c$ and a large gap $\Delta_{\mathrm{gap}}$ to single-trace operators of spin greater than 2 has a local Einstein-gravity dual. Determine necessary and sufficient conditions on CFT data for a semiclassical local bulk, and prove or refute the conjecture.
What would settle it
A derivation from CFT axioms showing that large c and a large gap imply bulk locality and Einstein gravity at low energy, or a counterexample.
Status in the literature
Causality and bootstrap bounds (2016-2023) derived suppression of higher-derivative couplings by $\Delta_{\mathrm{gap}}$; a full proof is missing.