SLIDE 1
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
SLIDE 2 Quantum gravity
in the bulk (AdS)
time
Conformal field theory
AdS/CFT correspondence
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) Vacuum state Pure AdS spacetime # of degrees of freedom:N Newton const.:GN~lPd-1∝1/N
・CFT gives the UV completion of the quantum gravity. I want to know about the physics in the bulk interior.
Especially the BH interior!
・The spacetime of AdS is emergent from CFT.
SLIDE 3 Bulk reconstruction in AdS/CFT
What operator in CFT represents a bulk local operator in AdS? →``Bulk reconstruction’’ from CFT
(r, x)
A well-known method to construct the bulk local operators: HKLL construction.
[Hamilton-Kabat-Lifschytz-Lowe ’06]
Bulk local fields are represented as non-local (smeared) operator in CFT. CFT local operator can be obtained easily from the bulk
φ(r, x) O(x)
- rdinary font: AdS operators
bold font: CFT operators
φ(r, x) = Z dr0dx0K(r, x; x0)O(x0) lim
r→∞ r∆φ(r, x) ↔ O(x)
SLIDE 4
Bulk reconstruction in AdS/CFT
HKLL construction makes explicit use of the bulk geometry to construct the smearing function . K(r, x; x0) 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) = Z dr0dx0K(r, x; x0)O(x0)
φ(r, x)|0iCF T = |φ(r, x)i
SLIDE 5 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 AdS3/CFT2. ・Bulk local states should come in rep. of G=SO(2,2)~SL(2,R)2, the global part of Virasoro symmetry generated by which acts on AdS as isometry. {L0, L1, L−1} ・A bulk local state at the center of AdS is invariant under SL(2, R) symmetry H which keep that point invariant.
|φi
(L0 ˜ L0)|φi = (L±1 + ˜ L⌥1)|φi = 0
SLIDE 6 |φi = p N
∞
X
k=0
e−✏k Nk (1)k(L−1)k(˜ L−1)k|Oi
The solution can be constructed from Ishibashi state for SL(2,R)2
- algebra. [Miyaji-Numasawa-Shiba-Takayanagi-Watanabe,Nakayama-Ooguri ’15]
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.
Bulk reconstruction in AdS/CFT
One can also check the equivalence with the HKLL construction.
[KG-Miyaji-Takayanagi ’16]
SLIDE 7
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.
Bulk locality in AdS/CFT
How the (sub-AdS) locality/causality in the bulk spacetime emerge and how they can be described in CFT ? Question:
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’’
SLIDE 8 What kind of quantities should we use to observe locality/causality in the bulk?
The bulk 2-pt. functions→determined only by the symmetry.
hφ|O(x)O(y)|φi
→Need different quantities depend on more details of CFTs!
Consider 2-pt. functions of primary fields for the bulk excited state i.e, four point functions O(x) |φi
X
∆
∼
Holographic CFT
1, T, ∂T, · · ·
Vacuum block
∆ O O O O
For Holographic CFTs,these correlators are exponentially dominated by the vacuum conformal block!
→Exchange of operators built from the energy momentum tensor (boundary ``graviton’’)
SLIDE 9 Short distance behavior
The spacetime is modified due to the gravitational back reactions of the bulk excitation.
∆φ = hφ|L0 + ˜ L0|φi
φ(0)
O(1)
O(eiσ)
A
gµν = g
pure AdS
µν
+ O(∆φ/c)
hφ|O(1)O(eiσ)|φi h0|O(1)O(eiσ)|0i ' 1 + ∆φ∆O c σ2 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)
∆SA ' ∆φσ2 6
Increased amount of energy due to the excitation byφ Increased amount of entanglement entropy
SLIDE 10 Causality from holographic CFTs
How the causality in the bulk spacetime can be described in CFT ? Question:
t = 0
t = π 2
Bulk localized excitation
t
back-reacted geometry Excitation spreads inside the light cone
(Outside the light cone it can be well approximated by BTZ black hole (deficit angle spacetime) )
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|φi (i)At t=0 only the origin of the AdS3 is locally excited. However, the metric will be modified everywhere due to the gravitational back reactions. (ii)Under the time evolution, the excitation will spread relativistically inside a light cone. The geometry is perturbed only inside the light cone.
Expectation from the bulk picture:
Causality in the bulk spacetime
SLIDE 11 Causality from holographic CFTs
hφ(t)|O(1)O(eiσ)|φ(t)i h0|O(1)O(eiσ)|0i
hφ(t)|O(1)O(eiσ)|φ(t)i hφ(t = 0)|O(1)O(eiσ)|φ(t = 0)i h0|O(1)O(eiσ)|0i
σ σ
Numerical calculation for holographic CFTs
Indeed we observed such an evolution for holographic CFTs when the conformal dimension of is light. On the other hand, we find that this does not happen for a free CFTs.
O(x)
SLIDE 12 The expected bulk picture
t = 0 t = π 2 Bulk localized excitation t back-reacted geometry Excitation spreads inside the light cone (Outside the light cone it can be well approximated by BTZ black hole (deficit angle space time) )
|Ψα(t)i
|φ(t)i
SLIDE 13 Suppression by the back-reacted geometry Two point function
O(1)
hφ(t = 0)|O(1)O(eiσ)|φ(t = 0)i h0|O(1)O(eiσ)|0i
O(eiσ)
|φ(t = 0)i
The 4-pt. correlators can be evaluated in the bulk by the length of the geodesics on the spacetime excited by .
φ
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.
SLIDE 14 t = 0 t = π 2 Bulk localized excitation t back-reacted geometry Excitation spreads inside the light cone (Outside the light cone it can be well approximated by BTZ black hole (deficit angle space time) )
|Ψα(t)i
The expected bulk picture
|φ(t)i
SLIDE 15 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.
t = 0
t = π
t = π/2 π/4
σ∗
σ∗
For large σ where the bulk geodesics enter in the light cone,the correlators behave differently from t=0.
Under the time evolution, the excitation will spread relativistically inside a light cone. The geometry is perturbed only inside the light cone.
SLIDE 16 t = 0
t = π
t = π/2
π/4
σ∗
σ∗
Under the time evolution, the excitation will spread relativistically inside a light cone. The geometry is perturbed only inside the light cone.
SLIDE 17 t = 0
t = π
t = π/2
π/4
σ∗
σ∗
Under the time evolution, the excitation will spread relativistically inside a light cone. The geometry is perturbed only inside the light cone.
SLIDE 18 hφ(t)|O(1)O(eiσ)|φ(t)i h0|O(1)O(eiσ)|0i
hφ(t)|O(1)O(eiσ)|φ(t)i hφ(t = 0)|O(1)O(eiσ)|φ(t = 0)i h0|O(1)O(eiσ)|0i
σ σ
Numerical calculation for holographic CFTs
We also evaluated the point σ✳ where the geodesic enter in the light cone by the calculation in the bulk gravity theory.
Emergence of bulk causal structure purely from CFT computations!
We conclude that this shows a light-like spread of the localized bulk excitation in AdS as expected.
SLIDE 19
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.
SLIDE 20 Summary and future work
Application for the dS/CFT? Reconstruction in the Rindler-AdS? Study the black hole interior
Using the bulk local states of holographic CFTs we can see the bulk causal structure. Short distance behavior of the four point functions
- beys the first law like relation of entanglement.
Analytical study in the lower dimension?
hφ|O(x)O(y)|φi
