If we are to believe that holographic principle holds over a wide number of dimensions, and gravitational theories, but specially, those that are relevant to our universe, then there must be some 3D QFT that is the dual description of our current 4D General relativity, over what it seems to be asymptotically flat de-Sitter spacetime.
What QFT theories are candidates for being the holographic dual of gravitational theory in our universe?
I thought it would be interesting to give perspective since this question was asked 11 years ago. In that span we have learned a great deal about how four-dimensional gravity might be encoded in lower-dimensional quantum field theories, even though a fully satisfactory model for our ΛCDM universe is still out of reach.
For work with a positive cosmological constant, we look to Strominger’s dS/CFT proposal, which treats future infinity of de Sitter as a three-sphere that should support a Euclidean CFT$_3$. The idea is elegant, but the would-be boundary theory seems to require an imaginary (hence non-unitary) central charge, and no explicit example reproduces pure Einstein gravity so far arXiv:hep-th/0110087.
Calculable progress came from higher-spin gravity. By analytically continuing the well-studied AdS dual, Anninos, Hartman and Strominger matched Vasiliev gravity in dS$^4$ to an Sp($N$) anticommuting vector model, successfully reproducing correlators at large $N$ arXiv:1108.5735. This is a powerful toy laboratory, but it does not yet describe the spin-2 sector of ordinary general relativity.
A different route, the dS/dS correspondence, places two coupled CFT$_3$s on a lower-dimensional de Sitter slice inside the bulk. It captures physics visible to a single observer over sub-Hubble timescales, but its microscopic completion remains unknown arXiv:hep-th/0407125. The principal challenges on the de Sitter side are reconciling unitarity with horizon entropy and extracting real-time observables from a Euclidean boundary description.
Setting the cosmological constant to zero changes the holographic screen to null infinity, topologically $S^{2}$. The past decade has therefore focused on celestial holography, in which 4-D scattering amplitudes are Mellin-transformed into correlators of a 2-D CFT living on that sphere. This celestial CFT carries the infinite-dimensional BMS/W$_\infty$ symmetry expected of asymptotically flat spacetimes and gives a natural home to soft theorems and memory effects arXiv:2303.10037.
Most recently, researchers have begun assembling the operator algebra of this celestial theory, building chiral sectors that reproduce tree-level graviton amplitudes and encompass the $w_{1+\infty}$ current algebra arXiv:2407.13558. The frontier problems are extending the construction beyond tree level, taming infrared divergences, and encoding black-hole processes.
Of course, the ideas sketched above barely scratch the surface: parallel efforts range from twistor-string and ambitwistor constructions of amplitudes, to numerical and lattice studies of quantum gravity, to bootstrapping Carrollian and BMS-invariant field theories, to exploring how quantum-information concepts like complexity and quantum error correction might constrain any would-be hologram; even lower-dimensional solvable systems such as JT gravity, SYK-type models and 2-D Liouville have become test beds for lessons that could lift to four dimensions, while cosmologists investigate whether inflationary correlators already hide a “celestial” imprint we can observe.
Either line of attack could yet yield the holographic description of gravity in a universe like ours; for now they provide complementary clues and a rich set of problems for the next decade.
