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What are the differences (if any) between Generalized gravitational entropy (Lewkowycz-Maldacena) and holographic entanglement entropy (Ryu-Takayanagi)?

More specifically, I was wondering following the same procedure in the appendix A of Lewkowycz-Maldacena paper shouldn't one reproduce the usual correction to EE due to a "heavy operator" (whose dimension is fixed by the mass of the bulk scalar field) in the dual 2D CFT?

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