$T{\overline T}$, $\widetilde JJ$, $JT$ and $\widetilde JT$ deformations

TL;DR

The paper extends the light-cone gauge method for ${T\overline{T}}$ deformations to a comprehensive ten-parameter family built from conserved currents $J^\alpha$, $\widetilde{J}^\alpha$ and the stress tensor $T^\alpha{}_\beta$, including all quadratic combinations. It derives a gauge-fixed deformed Hamiltonian ${\cal H}_A$, the corresponding flow equations for both the Hamiltonian density and the energy, and analyzes the spectrum for deformed CFTs with left- and right-moving currents, revealing two extra relations generalizing ${J\Theta}=0$ and ${\bar J}\overline{\Theta}=0$ that constrain the spectrum. The study shows the flow equations differ from prior proposals due to parameter-dependent coefficients and demonstrates consistency with known TTbar and JTe deformations in various limits, including a matching with results in the literature for specific choices of parameters. The results illuminate how multi-parameter deformations influence spectra, level-matching constraints, and Burgers-type evolution, with potential implications for holography, higher-spin deformations, and non-Lorentz-invariant models.

Abstract

The light-cone gauge approach to $T{\overline T}$ deformed models is generalised to models deformed by U(1) conserved currents $J^α$, $\widetilde J^α$, stress-energy tensor $T^α_β$, and their various quadratic combinations of the form $ε_{αβ} K_1^αK_2^β$. It is then applied to derive a ten-parameter deformed Hamiltonian for a system of scalars with an arbitrary potential, the flow equations for the Hamiltonian density, and the flow equations for the energy of the deformed model. The flow equations disagree with the ones recently proposed in arXiv:1903.07606. The results obtained are applied to analyse a CFT with left- and right-moving conserved currents deformed by these operators. It is shown that with a proper choice of the parameter of the $T{\overline T}$ deformation the deformed CFT Hamiltonian density is independent of the parameters of the $JΘ$ and $\bar J\overline Θ$ deformations. This leads to the existence of two extra relations which generalise the $JΘ=0$ and $\bar J\overline Θ=0$ relations of the undeformed CFT. The spectrum of the deformed CFT is found and shown to satisfy the flow equations.