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

Abstract: The light-cone gauge approach to $T{\overline T}$ deformed models is generalised to models deformed by U(1) conserved currents $J^\alpha$, $\widetilde J^\alpha$, stress-energy tensor $T^\alpha{}_\beta$, and their various quadratic combinations of the form $\epsilon_{\alpha\beta} K_1^\alpha K_2^\beta$. 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 $J\Theta$ and $\bar J\overline \Theta$ deformations. This leads to the existence of two extra relations which generalise the $J\Theta=0$ and $\bar J\overline \Theta=0$ relations of the undeformed CFT. The spectrum of the deformed CFT is found and shown to satisfy the flow equations.