Causal Evolutions of Bulk Local Excitations in AdS from CFT Kanato Goto The University of Tokyo Komaba Particle Theory Group Based on arXiv:1605.02835 KG, M.Miyaji and T.Takayanagi Strings and Fields 2016
AdS/CFT correspondence time AdS CFT Quantum theory of gravity Conformal field theory Quantum Isometry of AdS:SO(2,d) Conformal symmetry:SO(2,d) gravity All states come in representations of SO(2,d) in the bulk (AdS) Vacuum state Pure AdS spacetime Conformal field theory Newton const.: G N ~l Pd-1 ∝ 1/N # of degrees of freedom: N on the AdS boundary ・ CFT gives the UV completion of the quantum gravity. ・ The spacetime of AdS is emergent from CFT. I want to know about the physics in the bulk interior. Especially the BH interior!
Bulk reconstruction in AdS/CFT CFT local operator can be obtained easily from the bulk O ( x ) operator . φ ( r, x ) r →∞ r ∆ φ ( r, x ) ↔ O ( x ) ordinary font: AdS operators lim bold font : CFT operators What operator in CFT represents a bulk local operator in AdS? → ``Bulk reconstruction’’ from CFT A well-known method to construct the bulk local operators: HKLL construction. [Hamilton-Kabat-Lifschytz-Lowe ’06] Z dr 0 dx 0 K ( r, x ; x 0 ) O ( x 0 ) φ ( r, x ) = ( r, x ) Bulk local fields are represented as non-local (smeared) operator in CFT.
Bulk reconstruction in AdS/CFT HKLL construction makes explicit use of the bulk geometry to construct the smearing function . K ( r, x ; x 0 ) Z dr 0 dx 0 K ( r, x ; x 0 ) O ( x 0 ) φ ( r, x ) = We need the ``intrinsic’’ definition of bulk local operators in CFT. Consider a ``bulk local state’’ which is dual with the spacetime locally excited by free scalar fields in the bulk . φ ( r, x ) | 0 i CF T = | φ ( r, x ) i
AdS CFT Quantum theory of gravity Conformal field theory Isometry of AdS:SO(2,d) Conformal symmetry:SO(2,d) All states come in representations of SO(2,d) ・ For simplicity we consider AdS 3 /CFT 2 . ・ Bulk local states should come in rep. of G=SO(2,2)~SL(2,R) 2 , the global part of Virasoro symmetry generated by { L 0 , L 1 , L − 1 } which acts on AdS as isometry. ・ A bulk local state at the center of AdS is invariant under | φ i SL(2, R) symmetry H which keep that point invariant. ( L 0 � ˜ L 0 ) | φ i = ( L ± 1 + ˜ L ⌥ 1 ) | φ i = 0
Bulk reconstruction in AdS/CFT The solution can be constructed from Ishibashi state for SL(2,R) 2 algebra. [Miyaji-Numasawa-Shiba-Takayanagi-Watanabe,Nakayama-Ooguri ’15] p ∞ e − ✏ k ( � 1) k ( L − 1 ) k (˜ X L − 1 ) k |O i | φ i = N N k k =0 The bulk local state at the origin can be moved to an arbitrary point in AdS by g ∈ G/H=SO(2,d)/SO(1,d). i.e, coset construction of AdS One can check that it satisfies the E.O.M in AdS and reproduces the two point function in the bulk i.e, bulk-to-bulk propagator. One can also check the equivalence with the HKLL construction. [KG-Miyaji-Takayanagi ’16]
Bulk locality in AdS/CFT Question: How the (sub-AdS) locality/causality in the bulk spacetime emerge and how they can be described in CFT ? In general, the bulk ``local’’ state we constructed is not local or causal; It is determined only by the symmetry and any dynamical CFT information are not used. Not all CFTs are dual with local effective field theories in the bulk! Condition for the bulk locality; (i) Many degrees of freedom: large N (ii) The low energy spectrum is sparse We call such class of CFTs ``holographic CFTs’’
What kind of quantities should we use to observe locality/causality in the bulk? The bulk 2-pt. functions → determined only by the symmetry. → Need different quantities depend on more details of CFTs! Consider 2-pt. functions of primary fields for the bulk excited state O O | φ i O ( x ) ∆ i.e, four point functions X h φ | O ( x ) O ( y ) | φ i ∆ O O 1 , T, ∂ T, · · · For Holographic CFTs,these correlators are exponentially dominated by the ∼ Holographic CFT vacuum conformal block! Vacuum block → Exchange of operators built from the energy momentum tensor (boundary ``graviton’’)
Short distance behavior The spacetime is modified due to the gravitational back reactions of the bulk φ (0) excitation. O ( e i σ ) h φ | O (1) O ( e i σ ) | φ i h 0 | O (1) O ( e i σ ) | 0 i ' 1 + ∆ φ ∆ O A σ 2 O (1) c pure AdS g µ ν = g + O ( ∆ φ /c ) µ ν ∆ φ = h φ | L 0 + ˜ L 0 | φ i This short distance behavior can capture the information of the bulk geometry. One can see that it is related to the first law of entanglement. If we take to be a twist operator in the replica method, we get O ( x ) ∆ S A ' ∆ φ σ 2 6 Increased amount of Increased amount of energy entanglement entropy due to the excitation by φ
Causality in the bulk spacetime excitation (Outside the light cone it can be well approximated inside the light cone Excitation spreads back-reacted geometry by BTZ black hole (deficit angle spacetime) ) Bulk localized Causality from holographic CFTs Question: How the causality in the bulk spacetime can be described in CFT ? Consider the time evolution of the excited bulk spacetime which is locally excited by the bulk local field at t=0; . | φ ( t ) i ⌘ e − iHt | φ i Expectation from the bulk picture: t (i)At t=0 only the origin of the AdS 3 is locally excited. However, the metric t = π will be modified everywhere due to 2 the gravitational back reactions. (ii)Under the time evolution, the excitation will spread relativistically t = 0 inside a light cone. The geometry is perturbed only inside the light cone.
Causality from holographic CFTs Indeed we observed such an evolution for holographic CFTs when the conformal dimension of is light. O ( x ) On the other hand, we find that this does not happen for a free CFTs. Numerical calculation for holographic CFTs σ σ h φ ( t ) | O (1) O ( e i σ ) | φ ( t ) i h φ ( t ) | O (1) O ( e i σ ) | φ ( t ) i � h φ ( t = 0) | O (1) O ( e i σ ) | φ ( t = 0) i h 0 | O (1) O ( e i σ ) | 0 i h 0 | O (1) O ( e i σ ) | 0 i
The expected bulk picture t | Ψ α ( t ) i | φ ( t ) i t = π 2 Excitation spreads inside the light cone Bulk localized excitation t = 0 back-reacted geometry (Outside the light cone it can be well approximated by BTZ black hole (deficit angle space time) )
At t=0 the bulk excitation is localized at the center of the AdS. On the other hand, the metric will be modified everywhere due to the gravitational back reaction of the bulk excitation. h φ ( t = 0) | O (1) O ( e i σ ) | φ ( t = 0) i Suppression by the back-reacted geometry h 0 | O (1) O ( e i σ ) | 0 i O ( e i σ ) Two point function on the pure AdS O (1) The 4-pt. correlators can be evaluated in the bulk by the length of the geodesics on the spacetime excited by . φ | φ ( t = 0) i
The expected bulk picture t | Ψ α ( t ) i | φ ( t ) i t = π 2 Excitation spreads inside the light cone Bulk localized excitation t = 0 back-reacted geometry (Outside the light cone it can be well approximated by BTZ black hole (deficit angle space time) )
Under the time evolution, the excitation will spread relativistically inside a light cone. The geometry is perturbed only inside the light cone. For small σ ,the bulk geodesics are outside of the light cone where the geometry is not perturbed. Correlators behave just the same as t=0. σ ∗ For large σ where the bulk geodesics enter in the light cone,the correlators behave t = π differently from t=0. π / 4 t = π / 2 t = 0 σ ∗
Under the time evolution, the excitation will spread relativistically inside a light cone. The geometry is perturbed only inside the light cone. σ ∗ t = π π / 4 t = π / 2 t = 0 σ ∗
Under the time evolution, the excitation will spread relativistically inside a light cone. The geometry is perturbed only inside the light cone. σ ∗ t = π π / 4 t = π / 2 t = 0 σ ∗
Numerical calculation for holographic CFTs σ σ h φ ( t ) | O (1) O ( e i σ ) | φ ( t ) i h φ ( t ) | O (1) O ( e i σ ) | φ ( t ) i � h φ ( t = 0) | O (1) O ( e i σ ) | φ ( t = 0) i h 0 | O (1) O ( e i σ ) | 0 i h 0 | O (1) O ( e i σ ) | 0 i We also evaluated the point σ ✳ where the geodesic enter in the light cone by the calculation in the bulk gravity theory. We conclude that this shows a light-like spread of the localized bulk excitation in AdS as expected. Emergence of bulk causal structure purely from CFT computations!
Comments Numerical calculation for c=1 CFTs We also calculated the 4-pt. functions for the c=1 CFTs (free Dirac fermion or free scalar). Such correlators do not satisfy the same short distance behavior as the holographic CFT case, and do not show the causal propagations in the ``bulk’’. This is consistent with the fact that there is no classical gravity dual to a c=1 free CFT.
Summary and future work Using the bulk local states of holographic CFTs we can see the bulk causal structure. Short distance behavior of the four point functions obeys the first law like relation of entanglement. h φ | O ( x ) O ( y ) | φ i Analytical study in the lower dimension? Application for the dS/CFT? Reconstruction in the Rindler-AdS? Study the black hole interior cf. firewall paradox
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