Setting $T^2$ free for braneworld holography

We identify what has been referred to as 'cut-off CFT' in holographic braneworld with $T^2$ or $T\bar T$ theory (depending on the dimension of the bulk), so that the holographic dual of AdS-gravity with Neumann boundary conditions is a $T^2$-deformed CFT that is set free. After making statements that apply for general dimensions higher than three, we focus on the case of a three-dimensional bulk. We find from bulk arguments that the effective theory on the brane is governed by a $T\bar T$-like flow equation, such that under certain assumptions the effective gravity theory on the brane is given by a $T\bar T$-like deformed timelike Liouville theory, which limits to the description of the holographic Weyl anomaly for branes that approach the asymptotic boundary.