Abstract
The conformal symmetry algebra in 2D (Diff(S1)⊕Diff(S1)) is shown to be related to its ultra/non-relativistic version (BMS3≈GCA2) through a nonlinear map of the generators, without any sort of limiting process. For a generic classical CFT2, the BMS3 generators then emerge as composites built out from the chiral (holomorphic) components of the stress-energy tensor, T and overline{T} , closing in the Poisson brackets at equal time slices. Nevertheless, supertranslation generators do not span Noetherian symmetries. BMS3 becomes a bona fide symmetry once the CFT2 is marginally deformed by the addition of a sqrt{Toverline{T}} term to the Hamiltonian. The generic deformed theory is manifestly invariant under diffeomorphisms and local scalings, but it is no longer a CFT2 because its energy and momentum densities fulfill the BMS3 algebra. The deformation can also be described through the original CFT2 on a curved metric whose Beltrami differentials are determined by the variation of the deformed Hamiltonian with respect to T and overline{T} . BMS3 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 BMS3 (or flat) versions.
Highlights
BMS3 bec√omes a bona fide symmetry once the CFT2 is marginally deformed by the addition of a
The generic deformed theory is manifestly invariant under diffeomorphisms and local scalings, but it is no longer a CFT2 because its energy and momentum densities fulfill the BMS3 algebra
The algebra (1.2) and its centrally-extended version appeared long ago in the context of the tensionless limit of string theory [5,6,7], and more recently in the “flat” analog of Liouville theory [8, 9] as well as in fluid dynamics and integrable systems in 2D [10,11,12]. It plays a leading role for nonrelativistic and flat holography [13, 14], and it emerges from the spacetime structure near generic horizons [15,16,17,18,19]
Summary
The mapping in (2.2) naturally makes one wondering about how precisely the BMS3 algebra manifests itself for a generic (nonanomalous) classical CFT2. Supertranslation generators are not conserved, as it can bee seen from the time evolution of P , that can be obtained from that of the (anti-)chiral T and Tby virtue of the map (2.2), given by. Where it is implicitly assumed that ICFT is written in Hamiltonian form, and Tμν stands for the stress-energy tensor of the undeformed CFT2. The action (4.2) keeps being invariant under diffeomorphisms and local scalings, but it is no longer a CFT2 because the energy and momentum densities of the deformed theory yield to generators that fulfill the BMS3 algebra (2.3) instead of the conformal one in (2.1). Supertranslation and superrotation densities evolution is spanned by the deformed Hamiltonian H, which by virtue of (2.3) reads. The transformation law of supertranslation and superrotation densities is given by (3.4), corresponding to Noetherian BMS3 symmetries
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