Mapping relativistic to ultra/non-relativistic conformal symmetries in 2D and finite $\sqrt{T\bar{T}}$ deformations

The conformal symmetry algebra in 2D (Diff($S^{1}$)$\oplus$Diff($S^{1}$)) is shown to be related to its ultra/non-relativistic version (BMS$_{3}$$\approx$GCA$_{2}$) through a nonlinear map of the generators, without any sort of limiting process. For a generic classical CFT$_{2}$, the BMS$_{3}$ generators then emerge as composites built out from the chiral (holomorphic) components of the stress-energy tensor, $T$ and $\bar{T}$, closing in the Poisson brackets at equal time slices. Nevertheless, supertranslation generators do not span Noetherian symmetries. BMS$_{3}$ becomes a bona fide symmetry once the CFT$_{2}$ is marginally deformed by the addition of a $\sqrt{T\bar{T}}$ term to the Hamiltonian. The generic deformed theory is manifestly invariant under diffeomorphisms and local scalings, but it is no longer a CFT$_{2}$ because its energy and momentum densities fulfill the BMS$_{3}$ algebra. The deformation can also be described through the original CFT$_{2}$ on a curved metric whose Beltrami differentials are determined by the variation of the deformed Hamiltonian with respect to $T$ and $\bar{T}$. BMS$_{3}$ symmetries then arise from deformed conformal Killing equations, corresponding to diffeomorphisms that preserve the deformed metric and stress-energy tensor up to local scalings. As an example, we briefly address the deformation of $\mathrm{N}$ free bosons, which coincides with ultra-relativistic limits only for $\mathrm{N}=1$. Furthermore, Cardy formula and the S-modular transformation of the torus become mapped to their corresponding BMS$_{3}$ (or flat) versions.