Mapping relativistic to ultra/non-relativistic conformal symmetries in 2D and finite TT deformations

A bstract The conformal symmetry algebra in 2D (Diff (S 1) ⊕Diff (S 1) ) is shown to be related to its ultra/non-relativistic version (BMS 3 ≈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 T 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 TT T 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 T 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 N free bosons, which coincides with ultra-relativistic limits only for N = 1. Furthermore, Cardy formula and the S-modular transformation of the torus become mapped to their corresponding BMS 3 (or flat) versions.