Abstract
Quantum gravity in a finite region of spacetime is conjectured to be dual to a conformal field theory (CFT) deformed by the irrelevant operator TT[over ¯]. We test this conjecture with entanglement entropy, which is sensitive to ultraviolet physics on the boundary, while also probing the bulk geometry. We find that the entanglement entropy for an entangling surface consisting of two antipodal points on a sphere is finite and precisely matches the Ryu-Takayanagi formula applied to a finite region consistent with the conjecture of McGough etal. We also consider a one-parameter family of conical entropies, which are finite and verify a conjecture due to Dong. Since ultraviolet divergences are local, we conclude that the TT[over ¯] deformation acts as an ultraviolet cutoff on the entanglement entropy. Our results support the conjecture that the TT[over ¯]-deformed CFT is the holographic dual of a finite region of spacetime.
Highlights
Quantum gravity in a finite region of spacetime is conjectured to be dual to a conformal field theory (CFT) deformed by the irrelevant operator TT
Any CFT can be deformed by this operator, defining a oneparameter family of theories labeled by a deformation parameter μ with dimensions of length squared
(In [4] the metric of the CFTwas related to the induced metric of the bulk theory by a factor rc; here we set rc 1⁄4 1, which corresponds to measuring bulk and boundary distances in the same units.) This proposal passes a number of consistency checks [8]
Summary
Quantum gravity in a finite region of spacetime is conjectured to be dual to a conformal field theory (CFT) deformed by the irrelevant operator TT. We test this conjecture with entanglement entropy, which is sensitive to ultraviolet physics on the boundary, while probing the bulk geometry. Ipdeffignffi1⁄2tTifaybin−gð2th=eμÞmgaobm, eunntudmer which the flow equation and stress tensor conservation become the Arnowitt-Deser-Misner [5] Hamiltonian constraint and momentum constraint, respectively [4,6], 8pπGffiffi g ðπabπab ðπaaÞ2Þ þ pffiffi g The former is the famous Brown-Henneaux central charge [7], while the latter is the identification of μ proposed in [4]
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