Nonlinear automorphism of the conformal algebra in 2D and continuous $\sqrt{T\bar{T}}$ deformations

Abstract

The conformal algebra in 2D (Diff($S^1$)$\oplus$Diff($S^1$)) is shown to be preserved under a nonlinear map that mixes both chiral (holomorphic) generators $T$ and $\bar{T}$. It depends on a single real parameter and it can be regarded as a ``nonlinear $SO(1,1)$ automorphism.'' The map preserves the form of the momentum density and naturally induces a flow on the energy density by a marginal $\sqrt{T\bar{T}}$ deformation. In turn, the general solution of the corresponding flow equation of the deformed action can be analytically solved in closed form, recovering the nonlinear automorphism. The deformed CFT${}_2$ can also be described through the original theory on a field-dependent curved metric whose lapse and shift functions are given by the variation of the deformed Hamiltonian with respect to the energy and momentum densities, respectively. The conformal symmetries of the deformed theories can then also be seen to arise from diffeomorphisms that fulfill suitably deformed conformal Killing equations. Besides, Cardy formula is shown to map to itseft under the nonlinear automorphism. As a simple example, the deformation of $N$ free bosons is briefly addressed, making contact with recent related results and the dimensional reduction of the ModMax theory. Furthermore, the nonlinear map between the conformal algebra in 2D and its ultra/non-relativistic versions (BMS${}_3$$\approx$CCA${}_2$$\approx$GCA${}_2$), including the corresponding finite $\sqrt{T\bar{T}}$ deformation, are recovered from a limiting case of the nonlinear automorphism. The extension to a three-parameter nonlinear $ISO(1,1)$ automorphism of the conformal algebra, and a discrete nonlinear automorphism of BMS${}_3$ are also briefly discussed.

14 pages, no figures

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Cited by in corpus (19)

• Carroll black holes
• Duality-Invariant Non-linear Electrodynamics and Stress Tensor Flows
• Interacting Chiral Form Field Theories and $T\overline{T}$-like Flows in Six and Higher Dimensions
• Geometric formulation of generalized root-$T\bar{T}$ deformations
• Stress Tensor Flows, Birefringence in Non-Linear Electrodynamics, and Supersymmetry
• Root-$T\overline{T}$ Deformed Boundary Conditions in Holography
• A BMS-invariant free fermion model
• 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
• Deforming the ODE/IM correspondence with $\mathrm{T}\bar{\mathrm{T}}$
• Asymptotic $\mathcal{O}(r)$ gauge symmetries and gauge-invariant Poincaré generators in higher spacetime dimensions
• 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
• On $\sqrt{T\overline{T}}$ deformed pathways: CFT to CCFT