Causal Evolution of Bulk localized Excitations in AdS from CFT

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1 Causal Evolution of Bulk localized Excitations in AdS from CFT Kanato Goto The University of Tokyo, Komaba Particle Theory Group based on arxiv: KG and M.Miyaji and T.Takayanagi Kanato Goto (Tokyo Univ. Komaba) Causal Evolution of Bulk local Excitations YITP long term workshop 16 1 / 15

2 Bulk locality in AdS/CFT Main interest is the bulk locality in AdS/CFT Important for understanding the structure of quantum gravity - link with quantum error correction? (Harlow-Pastawski-Preskill-Yoshida, ) - state-dependent map between CFT and AdS? (Papadodimas-Raju, ) How does the bulk locality/causality of AdS emerge from CFT? Kanato Goto (Tokyo Univ. Komaba) Causal Evolution of Bulk local Excitations YITP long term workshop 16 2 / 15

3 Construction of bulk local states We construct CFT duals of bulk local states in AdS 3 /CFT 2 in the large c limit. ds 2 = (r 2 + 1)dt 2 + dr2 r r2 dϕ 2 We construct the bulk local states from the requirement of spacetime symmetry. Kanato Goto (Tokyo Univ. Komaba) Causal Evolution of Bulk local Excitations YITP long term workshop 16 3 / 15

4 Construction of bulk local states We construct CFT duals of bulk local states in AdS 3 /CFT 2 in the large c limit. ds 2 = (r 2 + 1)dt 2 + dr2 r r2 dϕ 2 We construct the bulk local states from the requirement of spacetime symmetry. AdS 3 has a symmetry SL(2,R) SL(2,R); global part of Virasoro symmetry. aa aa aa Kanato Goto (Tokyo Univ. Komaba) Causal Evolution of Bulk local Excitations YITP long term workshop 16 3 / 15

5 Construction of bulk local states We construct CFT duals of bulk local states in AdS 3 /CFT 2 in the large c limit. ds 2 = (r 2 + 1)dt 2 + dr2 r r2 dϕ 2 We construct the bulk local states from the requirement of spacetime symmetry. AdS 3 has a symmetry SL(2,R) SL(2,R); global part of Virasoro symmetry. Consider the bulk point at the center of AdS; r = 0 on the time slice t = 0. Kanato Goto (Tokyo Univ. Komaba) Causal Evolution of Bulk local Excitations YITP long term workshop 16 3 / 15

6 The bulk local state at r = t = 0 should be invariant under all SL(2, R) SL(2, R) transformations which keep the point invariant. (L 0 L 0 ) Ψ α = (L 1 + L 1 ) Ψ α = (L 1 + L 1 ) Ψ α = 0 Kanato Goto (Tokyo Univ. Komaba) Causal Evolution of Bulk local Excitations YITP long term workshop 16 4 / 15

7 The bulk local state at r = t = 0 should be invariant under all SL(2, R) SL(2, R) transformations which keep the point invariant. (L 0 L 0 ) Ψ α = (L 1 + L 1 ) Ψ α = (L 1 + L 1 ) Ψ α = 0 The solutions can be constructed from (analogue of) Ishibashi states for the SL(2,R) SL(2,R) symmetry Ψ α = N k=0 ( 1) k e ϵk 1 (L 1 ) k ( L 1 ) k α L α R N k α L,R are primary states with the conformal dimensions h α = h α. Kanato Goto (Tokyo Univ. Komaba) Causal Evolution of Bulk local Excitations YITP long term workshop 16 4 / 15

8 The bulk local state at r = t = 0 should be invariant under all SL(2, R) SL(2, R) transformations which keep the point invariant. (L 0 L 0 ) Ψ α = (L 1 + L 1 ) Ψ α = (L 1 + L 1 ) Ψ α = 0 The solutions can be constructed from (analogue of) Ishibashi states for the SL(2,R) SL(2,R) symmetry Ψ α = N k=0 ( 1) k e ϵk 1 (L 1 ) k ( L 1 ) k α L α R N k α L,R are primary states with the conformal dimensions h α = h α. The averaged energy is Ψα = Ψ α L 0 + L 0 Ψ α α + 1 ϵ ( α = h α + h α ) Kanato Goto (Tokyo Univ. Komaba) Causal Evolution of Bulk local Excitations YITP long term workshop 16 4 / 15

9 One can check the equivalence with the HKLL construction explicitly. Ψ α ˆϕ α 0 = dt dϕ K(t, ϕ 0, 0, ϕ)o α (t, ϕ ) 0 We can take the point r = 0 to a point (r, ϕ, t) by using SL(2,R) transformation g(r, ϕ, t). Ψ α (r, ϕ, t) = g(r, ϕ, t) Ψ α These bulk local states satisfy the bulk equations of motion and two point functions reproduce the bulk to bulk propagators. (Miyaji-Numasawa-Shiba-Takayanagi-Watanabe) Kanato Goto (Tokyo Univ. Komaba) Causal Evolution of Bulk local Excitations YITP long term workshop 16 5 / 15

10 Bulk locality from holographic CFTs The locality in AdS/CFT requires the large c limit with a large gap in the spectrum in a CFT, called holographic CFTs. (Papadodimas-Sherk, Heemskerk-Penedones-Polchinski-Sully, ) Kanato Goto (Tokyo Univ. Komaba) Causal Evolution of Bulk local Excitations YITP long term workshop 16 6 / 15

11 Bulk locality from holographic CFTs The locality in AdS/CFT requires the large c limit with a large gap in the spectrum in a CFT, called holographic CFTs. (Papadodimas-Sherk, Heemskerk-Penedones-Polchinski-Sully, ) Therefore it is intriguing to see how this condition appears when we regard Ψ α as a locally excited state in the AdS. Kanato Goto (Tokyo Univ. Komaba) Causal Evolution of Bulk local Excitations YITP long term workshop 16 6 / 15

12 Bulk locality from holographic CFTs The locality in AdS/CFT requires the large c limit with a large gap in the spectrum in a CFT, called holographic CFTs. (Papadodimas-Sherk, Heemskerk-Penedones-Polchinski-Sully, ) Therefore it is intriguing to see how this condition appears when we regard Ψ α as a locally excited state in the AdS. However, the computation of the bulk two point function is universal in any CFTs and the result reproduces the expected gravity result in AdS even for non holographic CFTs such as free CFTs. Kanato Goto (Tokyo Univ. Komaba) Causal Evolution of Bulk local Excitations YITP long term workshop 16 6 / 15

13 Bulk locality from holographic CFTs The locality in AdS/CFT requires the large c limit with a large gap in the spectrum in a CFT, called holographic CFTs. (Papadodimas-Sherk, Heemskerk-Penedones-Polchinski-Sully, ) Therefore it is intriguing to see how this condition appears when we regard Ψ α as a locally excited state in the AdS. However, the computation of the bulk two point function is universal in any CFTs and the result reproduces the expected gravity result in AdS even for non holographic CFTs such as free CFTs. Thus we study four point functions Ψ α O(x)O(y) Ψ α. Kanato Goto (Tokyo Univ. Komaba) Causal Evolution of Bulk local Excitations YITP long term workshop 16 6 / 15

14 Conformal block decomposition of the 4-pt. function The bulk local state can be written as Ψ α = k=0 ( 1) k e ϵk (L 1 ) k ( L 1 ) k α L α R N k Kanato Goto (Tokyo Univ. Komaba) Causal Evolution of Bulk local Excitations YITP long term workshop 16 7 / 15

15 Conformal block decomposition of the 4-pt. function The bulk local state can be written as Ψ α = k=0 ( 1) k e ϵk (L 1 ) k ( L 1 ) k α L α R N k Thus the four point function can be expressed as Ψ α O(1)O(e iσ ) Ψ α C p,q ( ) p ( ) p ( ) q ( ) q O α O(1)O(e iσ )O α p,q Kanato Goto (Tokyo Univ. Komaba) Causal Evolution of Bulk local Excitations YITP long term workshop 16 7 / 15

16 Conformal block decomposition of the 4-pt. function The bulk local state can be written as Ψ α = k=0 ( 1) k e ϵk (L 1 ) k ( L 1 ) k α L α R N k Thus the four point function can be expressed as Ψ α O(1)O(e iσ ) Ψ α C p,q ( ) p ( ) p ( ) q ( ) q O α O(1)O(e iσ )O α p,q Virasoro conformal block decomposition of O α O(1)O(e iσ )O α In a holographic CFT case, it is dominated by the vacuum block. We mainly calculated the following case h α = O(1), h O = O(c) in the large c limit Kanato Goto (Tokyo Univ. Komaba) Causal Evolution of Bulk local Excitations YITP long term workshop 16 7 / 15

17 Short distance behavior and the first law of entanglement Behavior near σ 0 the two point function is expanded as Ψ α O(1)O(e iσ ) Ψ α 1 e iσ 4h O 1 Ψ α O σ 2 c Kanato Goto (Tokyo Univ. Komaba) Causal Evolution of Bulk local Excitations YITP long term workshop 16 8 / 15

18 Short distance behavior and the first law of entanglement Behavior near σ 0 the two point function is expanded as Ψ α O(1)O(e iσ ) Ψ α 1 e iσ 4h O 1 Ψ α O σ 2 c If we take the operator O to be a twist operator for the replica method with h O = c(n 1/n), we can evaluate the increased amount of the entanglement entropy. S A Ψ α σ 2 6 Kanato Goto (Tokyo Univ. Komaba) Causal Evolution of Bulk local Excitations YITP long term workshop 16 8 / 15

19 Short distance behavior and the first law of entanglement Behavior near σ 0 the two point function is expanded as Ψ α O(1)O(e iσ ) Ψ α 1 e iσ 4h O 1 Ψ α O σ 2 c If we take the operator O to be a twist operator for the replica method with h O = c(n 1/n), we can evaluate the increased amount of the entanglement entropy. S A Ψ α σ 2 = E A /T ent 6 The first law like relation of the entanglement entropy Kanato Goto (Tokyo Univ. Komaba) Causal Evolution of Bulk local Excitations YITP long term workshop 16 8 / 15

20 We also observed the time evolution of the four point functions. Especially when the mass of dual bulk scalar field is close to the BF bound,... Kanato Goto (Tokyo Univ. Komaba) Causal Evolution of Bulk local Excitations YITP long term workshop 16 9 / 15

21 We also observed the time evolution of the four point functions. Especially when the mass of dual bulk scalar field is close to the BF bound,... we observed the causal (light-like) propagation of the bulk excitations. Kanato Goto (Tokyo Univ. Komaba) Causal Evolution of Bulk local Excitations YITP long term workshop 16 9 / 15

22 Causal evolution of bulk localized states The primary operators O(1) and O(e iσ ) can probe a light-like spread of the localized bulk excitation in AdS. Kanato Goto (Tokyo Univ. Komaba) Causal Evolution of Bulk local Excitations YITP long term workshop / 15

23 Causal evolution of bulk localized states The primary operators O(1) and O(e iσ ) can probe a light-like spread of the localized bulk excitation in AdS. Kanato Goto (Tokyo Univ. Komaba) Causal Evolution of Bulk local Excitations YITP long term workshop / 15

24 Causal evolution of bulk localized states The primary operators O(1) and O(e iσ ) can probe a light-like spread of the localized bulk excitation in AdS. Kanato Goto (Tokyo Univ. Komaba) Causal Evolution of Bulk local Excitations YITP long term workshop / 15

25 Causal evolution of bulk localized states When the mass of dual bulk scalar field is close to the BF bound, we observed the causal (light-like) propagation of the bulk excitations. Such behavior cannot be seen in the four point functions of free fermion CFTs; non-holographic CFTs Kanato Goto (Tokyo Univ. Komaba) Causal Evolution of Bulk local Excitations YITP long term workshop / 15

26 Summary We studied four point functions Ψ α O(x)O(y) Ψ α. The short distance behavior is universal, and the increased amount of the entanglement obeys the first law like relation. In holographic CFTs we can see the light-like propagations of the bulk excitations, while we cannot see such behavior in non holographic CFTs. Kanato Goto (Tokyo Univ. Komaba) Causal Evolution of Bulk local Excitations YITP long term workshop / 15

27 When the mass of dual bulk scalar field is much heavier than the BF bound, we cannot observe the light-like propagation of the bulk excitations. Also, each two point function is greatly suppressed. This is because the scalar field is very massive, the excitations does not reach the AdS boundary, and also the primary field in Ψ α gives a back reacted geometry (deficit angle spacetime). Kanato Goto (Tokyo Univ. Komaba) Causal Evolution of Bulk local Excitations YITP long term workshop / 15