Is there a consistent celestial CFT for four-dimensional gravity
In plain words
Scattering of particles in flat space can be rewritten as correlations on a sphere at infinity. Whether these correlations form a self-consistent quantum theory on that sphere is unknown.
Precise statement
Celestial holography maps the 4D asymptotically flat S-matrix, via a Mellin transform in energy, to correlators on the celestial sphere. Determine whether for Einstein gravity beyond tree level these correlators form a CFT with a local operator algebra, OPE and state-operator map, and find an intrinsic definition independent of the bulk.
What would settle it
An intrinsic 2D construction reproducing loop-level gravitational amplitudes, or a proof that the required CFT axioms fail.
Status in the literature
A top-down dual, a 2D chiral algebra, exists for a self-dual-type theory (Mabuchi gravity coupled to a 4D WZW model) on Burns space (Costello, Paquette, Sharma 2022-2023); full Einstein gravity remains open.