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

Abstract

The conformal algebra in 2D (Diff(S1)⨁Diff(S1)) is shown to be preserved under a nonlinear map that mixes both chiral (holomorphic) generators T and $$ \overline{T} $$ 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 $$ \sqrt{T\overline{T}} $$ T 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 CFT2 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 itself 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 (BMS3≈CCA2≈GCA2), including the corresponding finite $$ \sqrt{T\overline{T}} $$ T 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 BMS3 are also briefly discussed.

Carroll black holes

“…which holds only in 2d. Using additionally the definition of the Carrollian curvature scalar R = 2e µ e ν R µν γ=0 we find that the Lagrange-2-form (72) matches with (23).…”

“…Another condensed matter application of Carroll symmetries is flat bands [18] (the Fourier-transformed statement of collapsed lightcones). Additionally, Carroll symmetries may be relevant for cosmology [19], black hole microstates [20], CFT deformations [21][22][23][24], and govern Bjorken flow [25]. Finally, the conformal Carroll algebra is isomorphic to BMS in one higher dimension [4], which led to numerous developments in flat space holography, especially in 2+1 bulk dimensions [26][27][28][29][30][31][32][33][34][35][36] and in 3+1 bulk dimensions [37][38][39][40].…”

Asymptotic $$ \mathcal{O} $$(r) gauge symmetries and gauge-invariant Poincaré generators in higher spacetime dimensions

“…1 By nonlinear automorphism we refer to a nonlinear map of the generators that preserves a given Lie algebra. In the case of infinite-dimensional Lie algebras this sort of redefinitions has shown to play a central role in the mapping between different two-dimensional models [33][34][35]. See also section 5.1 in [27] for a recent discussion on nonlinear redefinitions (of Lie algebra generators) and Poisson manifolds.…”

Kafala System and Precariat Exploitation at Qatar World Cup Title

“…Along with international demands and condemnation of the kafala system, Qatar is faced with pressure to make changes and ensure migrant workers' rights are protected. FIFA, as the governing body of international football, along with other international organizations, encourages Qatar to adopt international work standards regulated by the International Labour Organization (ILO) (Tempo et al, 2022).…”