Carrollian Approach to $1+3$D Flat Holography

The isomorphism between the (extended) BMS$_4$ algebra and the $1+2$D Carrollian conformal algebra hints towards a co-dimension one formalism of flat holography with the field theory residing on the null-boundary of the asymptotically flat space-time enjoying a $1+2$D Carrollian conformal symmetry. Motivated by this fact, we study the general symmetry properties of a source-less $1+2$D Carrollian CFT, adopting a purely field-theoretic approach. After deriving the position-space Ward identities, we show how the $1+3$D bulk super-translation and the super-rotation memory effects emerge from them, manifested by the presence of a temporal step-function factor in the same. Temporal-Fourier transforming these memory effect equations, we directly reach the bulk null-momentum-space leading and sub-leading soft graviton theorems. Along the way, we construct six Carrollian fields $S^\pm_0$, $S^\pm_1$, $T$ and $\bar{T}$ corresponding to these soft graviton fields and the Celestial stress-tensors, purely in terms of the Carrollian stress-tensor components. The 2D Celestial shadow-relations and the null-state conditions arise as two natural byproducts of these constructions. We then show that those six fields consist of the modes that implement the super-rotations and a subset of the super-translations on the quantum fields. The temporal step-function allows us to relate the operator product expansions (OPEs) with the operator commutation relations via a complex contour integral prescription. We deduce that not all of those six fields can be taken together to form consistent OPEs. So choosing $S^+_0$, $S^+_1$ and $T$ as the local fields, we form their mutual OPEs using only the OPE-commutativity property, under two general assumptions. The symmetry algebra manifest in these holomorphic-sector OPEs is then shown to be $\text{Vir}\ltimes\hat{\overline{\text{sl}(2,\mathbb{R})}}$ with an abelian ideal.