Uplifting AdS$_{3}$/CFT$_{2}$ to flat space holography

Four-dimensional (4D) flat Minkowski space admits a foliation by hyperbolicslices. Euclidean AdS$_{3}$ slices fill the past and future lightcones of the origin, while dS$_{3}$ slices fill the region outside the lightcone. The resulting link between 4D asymptotically flat quantum gravity and AdS$_{3}$/CFT$_{2}$ is explored in this paper. The 4D superrotations in the extended BMS$_{4}$ group are found to act as the familiar conformal transformations on the 3D hyperbolic slices, mapping each slice to itself. The associated 4D superrotation charge is constructed in the covariant phase space formalism. The soft part gives the 2D stress tensor, which acts on the celestial sphere at the boundary of the hyperbolic slices, and is shown to be an uplift to 4D of the familiar 3D holographic AdS$_{3}$ stress tensor. Finally, we find that 4D quantum gravity contains an unexpected second, conformally soft, dimension (2, 0) mode that is symplectically paired with the celestial stress tensor.