Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 354 / 2 / a / Solution

Choose a constant-time boundary interval whose endpoints differ by

π

in

ϕ

. Its Ryu–Takayanagi formula surface is the diameter through

r = 0

. Cutting it off at

r_{c} = π /2 - ϵ

, its length is

ℓ = 2 R_{AdS} \int_{0}^{r_{c}} \frac{d r}{c o s r} = 2 R_{AdS} lo g (sec r_{c} + tan r_{c}) = 2 R_{AdS} lo g \frac{2}{ϵ} + O (ϵ^{2}) .

The holographic entanglement entropy is therefore

S = \frac{ℓ}{4 G} = \frac{R _{AdS}}{2 G} lo g ϵ^{- 1} + finite = \frac{c}{3} lo g ϵ^{- 1} + finite,

where the Brown--Henneaux central charge

c = 3 R_{AdS} / (2 G)

was used.

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