Comments on large central charge $T\bar{T}$ deformed conformal field theory and cutoff AdS holography

In this article we study large central charge partition function and entanglement entropy of $T\bar{T}$ deformed two dimensional conformal field theory, following the approach to $T\bar{T}$ deformation as integrated infinitesimal double trace deformation used by Guica et al.. For sphere partition function and entanglement entropy of half great circle with antipodal points being the entangling surface, we obtain different results compared to previous works, with more reasonable CFT limits and qualitatively different behaviour as the deformation parameter $\mu$ goes to infinity, which contradicts the simple version of the cutoff AdS holography proposal. For a modified version of cutoff AdS holography which is supposed to work only in the sector of classical pure gravity, we show that the flow equation of the metric and one point function of energy-momentum tensor in $T\bar{T}$ deformation corresponds to the flow equation of the boundary metric and Brown-York tensor on a cutoff surface in AdS space as the cutoff surface moves in the direction of normal geodesics. In addition the flow equation of gravity on-shell action takes the form of $T\bar{T}$ deformation, with straightforward generalization to higher dimensions. As an example we give a holographic computation of the sphere partition function of $T\bar{T}$ CFT.