Entanglement Dynamics in 2D CFT with Boundary: Entropic origin of JT gravity and Schwarzian QM

We study the dynamics of the geometric entanglement entropy of a 2D CFT in the presence of a boundary. We show that this dynamics is governed by local equations of motion, that take the same form as 2D Jackiw-Teitelboim gravity coupled to the CFT. If we assume that the boundary has a small thickness $\epsilon$ and constant boundary entropy, we derive that its location satisfies the equations of motion of Schwarzian quantum mechanics with coupling constant $C = {c \epsilon}/{12\pi}$. We rederive this result via energy-momentum conservation.