Abstract
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 ϵ and constant boundary entropy, we derive that its location satisfies the equations of motion of Schwarzian quantum mechanics with coupling constant C = c ϵ/12π. We rederive this result via energy-momentum conservation.
Highlights
Let ρ(u,v) denote the density matrix of the CFT on the red half line, obtained by tracing out the green segment
We study the dynamics of the geometric entanglement entropy of a 2D CFT in the presence of a boundary
In a 2D CFT with a boundary, the entanglement entropy S(u, v) and modular Hamiltonian K(u, v) associated with a point (u, v) in the bulk, as shown in figure 1, satisfy local equations of motion that take the same form as those of Jackiw-Teitelboim dilaton gravity coupled to the CFT
Summary
For a general class of states defined below, the entropy S(u, v) and modular Hamiltonian K(u, v) satisfy the same local equations of motion as JackiwTeitelboim gravity. In a 2D CFT, the modular Hamiltonian of a vacuum state |0 x is given by the generator of time evolution along the Killing vector of the constant curvature metric (2.6) that leaves the point (u, v) fixed. This modular flow is indicated by the blue lines in figure 1. In the expression (2.4) for the vacuum entanglement entropy S0 produces an extra term of the same form as the zero mode in (2.17) with c δC = δ This constant is finite for large c. Note that JT gravity has no physical degrees of freedom: its equations of motion (2.13) are imposed as gauge constraints that define S1 as a collective mode of the CFT
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