Classical and quantum symmetries of $T\bar T$-deformed CFTs

It has previously been proven that $T\bar T$ - deformed CFTs possess Virasoro $\times$ Virasoro symmetry at the full quantum level, whose generators are obtained by simply transporting the original CFT generators along the $T\bar T$ flow. In this article, we explicitly solve the corresponding flow equation in the classical limit, obtaining an infinite set of conserved charges whose action on phase space is well-defined even when the theory is on a compact space. The field-dependent coordinates that are characteristic of the $T\bar T$ deformation are shown to emerge unambiguously from the flow. Translating the symmetry transformations from the canonical to the covariant formalism, we find that they are different from those that have been previously proposed in the Lagrangian context, but that they agree precisely with those obtained from holography. We also comment on different possible bases for the symmetry generators and on how our explicit classical results could be extended perturbatively to the quantum level.