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

TL;DR

This paper reveals a nonlinear automorphism of the 2D conformal algebra that induces a $ ext{sqrt}(Tar{T})$ deformation, preserving symmetries and allowing analytical solutions, with implications for various related theories.

Contribution

It introduces a novel nonlinear automorphism of the 2D conformal algebra that generates and characterizes $ ext{sqrt}(Tar{T})$ deformations, connecting to multiple symmetry structures.

Findings

The conformal algebra is preserved under a nonlinear map depending on a real parameter.

The flow of the deformed action can be solved analytically, recovering the automorphism.

The deformation can be described via a field-dependent curved metric and relates to diffeomorphisms satisfying deformed conformal Killing equations.

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.