Mapping relativistic to ultra/non-relativistic conformal symmetries in 2D and finite $$ \sqrt{T\overline{T}} $$ deformations
Pablo Rodríguez, David Tempo, Ricardo Troncoso
Abstract
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 $$ \overline{T} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>T</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> , 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 $$ \sqrt{T\overline{T}} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msqrt> <mml:mrow> <mml:mi>T</mml:mi> <mml:mover> <mml:mi>T</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:mrow> </mml:msqrt> </mml:math> 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 $$ \overline{T} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>T</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> . 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.