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

TL;DR

This work shows that classical $T\bar{T}$, $J\bar{T}$, and $J T_a$ deformations of 2D CFTs preserve an infinite set of symmetries realized as field-dependent generalizations of two-dimensional conformal transformations, with an affine $U(1)$ surviving when the seed CFT has one. Using a Hamiltonian framework and dynamical coordinates, the authors derive conserved charges and compute their Poisson brackets, finding two copies of the functional Witt algebra for the $J\bar{T}$ and $J T_a$ cases and two commuting Witt algebras for $T\bar{T}$. Explicitly, free-boson examples illustrate the structure and guide the general constructions, which extend to arbitrary seed CFTs and reveal a universal charge algebra governed by the same symmetry pattern. The results provide a solid classical foundation for the continued study of non-local, integrable deformations and their potential quantum realizations, including holographic interpretations via dynamical coordinates. Taken together, the findings point to a rich, non-local but highly organized symmetry structure that survives deformations and may inform quantum representations and holographic duals in non-AdS spacetimes.

Abstract

We show that $T \bar T, J \bar T$ and $J T_a$ - deformed classical CFTs possess 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 $U(1)$ 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 $U(1)$ 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 $J \bar T$ and $J T_a$ deformations, we find two copies of a functional Witt - Kac-Moody algebra. In the case of the $T \bar T$ deformation, we show that it is also possible to obtain two commuting copies of the Witt algebra.