Infinite pseudo-conformal symmetries of classical $T \bar T$, $J \bar T $ and $J T_a$ - deformed CFTs

We show that T\bar{T}, J\bar{T}TT‾,JT‾ and JT_aJTa - deformed classical CFTs posses an infinite set of symmetries that take the form of a field-dependent generalization of two-dimensional conformal transformations. If, in addition, the seed CFTs possess an affine U(1)U(1) symmetry, we show that it also survives in the deformed theories, again in a field-dependent form. These symmetries can be understood as the infinitely-extended conformal and U(1)U(1) symmetries of the underlying two-dimensional CFT, seen through the prism of the ``dynamical coordinates’’ that characterise each of these deformations. We also compute the Poisson bracket algebra of the associated conserved charges, using the Hamiltonian formalism. In the case of the J\bar{T}JT‾ and JT_{a}JTa deformations, we find two copies of a functional Witt - Kac-Moody algebra. In the case of the T\bar{T}TT‾ deformation, we show that it is also possible to obtain two commuting copies of the Witt algebra