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

Abstract

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.

18 pages. Minor changes, typos corrected and references added

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Cited by in corpus (34)

• Root-$T\overline{T}$ Deformations in Two-Dimensional Quantum Field Theories
• Carroll/fracton particles and their correspondence
• A BMS-invariant free scalar model
• Carroll black holes
• Boosting to BMS
• Duality-Invariant Non-linear Electrodynamics and Stress Tensor Flows
• From CFTs to theories with Bondi-Metzner-Sachs symmetries: Complexity and out-of-time-ordered correlators
• Nonlinear automorphism of the conformal algebra in 2D and continuous $\sqrt{T\bar{T}}$ deformations
• Interacting Chiral Form Field Theories and $T\overline{T}$-like Flows in Six and Higher Dimensions
• Carroll fermions in two dimensions
• Marginal $T\bar{T}$-Like Deformation and ModMax Theories in Two Dimensions
• Geometric formulation of generalized root-$T\bar{T}$ deformations
• Root-$T\overline{T}$ Deformed Boundary Conditions in Holography
• Stress Tensor Flows, Birefringence in Non-Linear Electrodynamics, and Supersymmetry
• A BMS-invariant free fermion model
• Asymptotic structure of the gravitational field in five spacetime dimensions: Hamiltonian analysis
• $T\bar{T}$ flow as characteristic flows
• The Shadow Formalism of Galilean CFT${}_2$
• Quantization of the ModMax Oscillator
• ModMax Oscillators and Root-$T\overline{T}$-Like Flows in Supersymmetric Quantum Mechanics
• $T\overline{T}$-Like Flows and $3d$ Nonlinear Supersymmetry
• Flows in the Space of Interacting Chiral Boson Theories
• Asymptotic $\mathcal{O}(r)$ gauge symmetries and gauge-invariant Poincaré generators in higher spacetime dimensions
• $T\bar{T}$ and root-$T\bar{T}$ deformations in four-dimensional Chern-Simons theory
• Irrelevant and marginal deformed BMS field theories
• Waves and strings in an interacting conformal chiral 2-form theory in six dimensions
• The Courant-Hilbert construction in 4D Chern-Simons theory
• Root-$T\overline{T}$ deformed CFT partition functions at large central charge
• AdS${}_3$ Carroll gravity: asymptotic symmetries and C-thermal configurations
• Self-dual instantons and gravitating dyons in non-Abelian ModMax theory
• Field-Dependent Metrics and Higher-Form Symmetries in Duality-Invariant Theories of Non-Linear Electrodynamics
• Asymptotic symmetries in Carrollian theories of gravity with a negative cosmological constant
• Metric approach to a $\mathrm{T}\bar{\mathrm{T}}-$like deformation in arbitrary dimensions
• On $\sqrt{T\overline{T}}$ deformed pathways: CFT to CCFT