We study the map between two descriptions of the \( T\overline{T} \) deformation of conformal field theory (CFT): one is the defining description as a deformation of CFT by the \( T\overline{T} \) -operator. The other is an alternative description as the undeformed CFT on the dynamical \( T\overline{T} \) -deformed space that backreacts to the state or operator insertions, reminiscent of the theory of gravity. Instead of adopting the topological gravity description, we develop a more literal CFT-based operator formalism that facilitates systematic and straightforward computations of the \( T\overline{T} \) -deformation of the stress tensor, operators, and their correlators, while rederiving known results in the literature. Along the way, we discuss the backreaction to the \( T\overline{T} \) -deformed space in response to local operators and exhibit the hard-disk and free-space structures in the UV-cutoff and Hagedorn phases, respectively, suggested by Cardy-Doyon and Jiang. To capitalize on the alternative description of the \( T\overline{T} \) -deformed CFT, we focus on the correlators of semi-heavy operators, i.e., the operators of large conformal dimension ∆ ≫ \( \sqrt{c} \) , and show an intuitive and simple way to obtain the \( T\overline{T} \) -deformed correlators from those of the undeformed CFT on the \( T\overline{T} \) -deformed space via dynamical coordinate transformations. This may have implications in the holographic dual description, pointing towards a working dictionary for a class of matter correlators in the cutoff AdS picture.
