Brane effective actions and their island rule from $T\overline T$ flows

Abstract: We derive a $2d$ effective action for semi-classical AdS$_3$ gravity with an EOW brane of arbitrary tension by solving the radial Hamiltonian flow in a derivative expansion. Interpreting the radial flow as an RG flow, the effective theory is identified with a $T\bar T$-like deformed CFT. This leads us to propose that the holographic dual for semi-classical AdS$_3$ gravity with Neumann or conformal boundary conditions is obtained by setting the $T\bar T$-like deformed CFT free. We verify our proposal explicitly for saddles corresponding to $(A)dS_2$ branes in empty AdS$_3$. We discuss three useful ways to decompose the effective brane theory into a matter sector ($T\bar T$-deformed, $T\bar T$-like deformed, and undeformed CFT, respectively) and a corresponding braneworld gravity sector. In particular, the split into $T\bar T$-deformed CFT coupled to $T\bar T$-like deformed Liouville gravity directly extends the Liouville description of the integrated Weyl anomaly into the bulk. The effective action is then used to perform an island-rule calculation for a non-asymptotic brane in an AdS/bCFT set-up, finding exact agreement with the Ryu-Takayanagi result. This extends previous holographic checks on the island rule to non-asymptotic regimes of application, as well as provides a Wald entropy interpretation for $T\bar T$ entanglement. Finally, we relate our effective theory and its higher derivative corrections to AdS$_3$ gravity with a fluctuating radial cutoff by explicit calculation of the on-shell action.