Cutoff $\mathrm{A}\mathrm{d}{\mathrm{S}}_3$ versus $\mathrm{T}\bar{\mathrm{T}}$ $\mathrm{C}\mathrm{F}{\mathrm{T}}_2$ in the large central charge sector: correlators of energy-momentum tensor

In this article we probe the proposed holographic duality between $T\bar{T}$ deformed two dimensional conformal field theory and the gravity theory of $\rm AdS_3$ with a Dirichlet cutoff by computing correlators of energy-momentum tensor. We focus on the large central charge sector of the $T\bar{T}$ CFT in a Euclidean plane and in a sphere, and compute the correlators of energy-momentum tensor using an operator identity promoted from the classical trace relation. The result agrees with a computation of classical pure gravity in $\mathrm{A}\mathrm{d}{\mathrm{S}}_3$ with the corresponding cutoff surface, given a holographic dictionary which identifies gravity parameters with $T\bar{T}$ CFT parameters.