1. A Chaotic Bootstrap of AdS3 Gravity
Eric Perlmutter, Princeton University
GR21
Based on hep-th/

2. What is the space of CFTs, and of AdS quantum gravities
What is the space of CFTs, and of AdS quantum gravities? Which CFTs give rise to emergent spacetime?
CFT
Strong form of AdS/CFT says that these are the same.
At least, there is a subspace that defines what we normally mean by “holographic CFT”: large central charge with sparse spectrum, dual to weakly coupled bulk thy; in simplest cases, with Einstein grav sector.
Constrained by conformal bootstrap:
[Rattazzi, Rychkov, Vichi, Tonni; Kos, Poland, Simmons-Duffin]

3. What is the space of CFTs, and of AdS quantum gravities
What is the space of CFTs, and of AdS quantum gravities? Which CFTs give rise to emergent spacetime?
AdS quantum gravity
AdS quantum gravity
Strong form of AdS/CFT says that these are the same.
At least, there is a subspace that defines what we normally mean by “holographic CFT”: large central charge with sparse spectrum, dual to weakly coupled bulk thy; in simplest cases, with Einstein grav sector.
Constrained by high-energy behavior of scattering amplitudes, causality, …

4. What is the space of CFTs, and of AdS quantum gravities
What is the space of CFTs, and of AdS quantum gravities? Which CFTs give rise to emergent spacetime?
AdS quantum gravity
CFT
One is encouraged to take a broader approach, where we regard Einstein gravity as capturing the dynamics of a universality class of CFTs, dual to closed subsectors of AdS x M string/M-theory compactifications.
Strong form of AdS/CFT: Every CFT has an AdS dual
Which ones look like weakly coupled gravity? Like Einstein gravity? Like string theory?

5. What is the space of CFTs, and of AdS quantum gravities
What is the space of CFTs, and of AdS quantum gravities? Which CFTs give rise to emergent spacetime?
AdS quantum gravity
CFT
Large c CFT
One is encouraged to take a broader approach, where we regard Einstein gravity as capturing the stress tensor dynamics of a universality class of CFTs, dual to closed subsectors of AdS x M string/M-theory compactifications.
Classical gravity
“Sparse” large c CFT
Einstein gravity
Strong form of AdS/CFT: Every CFT has an AdS dual
Which ones look like weakly coupled gravity? Like Einstein gravity? Like string theory?

6. How is the structure of string/M-theory visible in CFT?
CFTs with string theory duals should be highly organized!
(AdS/CFT)
This is familiar from flat space QCD, bur AdS/CFT says some version of this should hold more generally.
This is a cartoon that gives the flat space picture: in AdS, the trajectories are not linear at large spin. This is consistent with known results: e.g. in gauge thys, large spin s.t. ops have anomalous dimensions growing like log(s). For short strings with small spin, we can use the flat space approximation.
We don’t actually know what the symmetry group is.
higher spin modes become massless. This casts
[Camanho, Edelstein, Maldacena, Zhiboedov; Maldacena, Zhiboedov]

7. How is the structure of string/M-theory visible in CFT?
CFTs with string theory duals should be highly organized!
(AdS/CFT)
This is familiar from flat space QCD, bur AdS/CFT says some version of this should hold more generally.
This is a cartoon that gives the flat space picture: in AdS, the trajectories are not linear at large spin. This is consistent with known results: e.g. in gauge thys, large spin s.t. ops have anomalous dimensions growing like log(s). For short strings with small spin, we can use the flat space approximation.
We don’t actually know what the symmetry group is.
higher spin modes become massless. This casts
In classical AdSD>3 gravity and in CFTd>2, finite towers of massive or massless higher spin fields either violate causality or symmetry constraints, respectively. Are there analogous constraints in AdS3/CFT2?
[Camanho, Edelstein, Maldacena, Zhiboedov; Maldacena, Zhiboedov]

8. How is the structure of string/M-theory visible in CFT?
CFTs with string theory duals should be highly organized!
(AdS/CFT)
This is familiar from flat space QCD, bur AdS/CFT says some version of this should hold more generally.
This is a cartoon that gives the flat space picture: in AdS, the trajectories are not linear at large spin. This is consistent with known results: e.g. in gauge thys, large spin s.t. ops have anomalous dimensions growing like log(s). For short strings with small spin, we can use the flat space approximation.
We don’t actually know what the symmetry group is.
higher spin modes become massless. This casts
In classical AdSD>3 gravity and in CFTd>2, finite towers of massive or massless higher spin fields either violate causality or symmetry constraints, respectively. Are there analogous constraints in AdS3/CFT2? (Large 𝛼 ′ : string theory ~ spontaneously broken higher spin gauge theory.)
[Camanho, Edelstein, Maldacena, Zhiboedov; Maldacena, Zhiboedov]

9. Higher spin AdS3/CFT2
In AdS3/CFT2, one can consider theories with finite towers of higher spin fields.
2d CFTs can have higher spin currents, {Js(z)}, of spins s=2,3,…,N. These generate a W-algebra.
3d higher spin gravity = matter coupled to G x G Chern-Simons theory.
: HS fields ↔ HS currents (W-algebra)
: Matter ↔ Non-current primaries

10. Higher spin AdS3/CFT2
Is higher spin gravity with matter truly consistent?
(If so, is it a subsector of string theory?)
Why do people study higher spin gravity?
Models of stringy geometry/quantum gravity
Potential insight into singularity resolution
W-algebras

11. Quantum chaos in CFT
Consider two local operators V and W, in a thermal state with inverse temperature 𝛽, separated by (x,t).
Their squared commutator has been proposed as a diagnostic of chaos in CFT:
In a chaotic system, W(t) becomes increasingly non-local in time, and the commutator grows exponentially.
This spreading can be measured by an
“out-of-time-order correlator”:
where
Scrambling
Black holes
SYK model: soluble model at strong coupling, ads2
Onset of chaos
Scrambling time
[Shenker, Stanford; Roberts, Stanford, Susskind; Roberts, Stanford]

12. A chaotic bootstrap
In unitary large c CFTs, there is a bound on the rate of onset of chaos:
“Lyapunov exponent ”
[Maldacena, Shenker, Stanford]
These are “bootstrap” results: constraints on UV completion and CFT classification from first principles
a=c, d\phi^4

13. A chaotic bootstrap
In unitary large c CFTs, there is a bound on the rate of onset of chaos:
Proposal: use the chaos bound to constrain the space of large c CFTs.
Today, I’ll apply this to the space of AdS3/CFT2 with higher spin symmetry
“Lyapunov exponent ”
[Maldacena, Shenker, Stanford]
These are “bootstrap” results: constraints on UV completion and CFT classification from first principles
a=c, d\phi^4

14. Unitary higher spin AdS3/CFT2 at large c
e.g. WN CFT

15. Unitary higher spin AdS3/CFT2 at large c
W∞ [λ] minimal models
e.g. WN CFT
[Gaberdiel, Gopakumar]

16. Unitary higher spin AdS3/CFT2 at large c
W∞ [λ] minimal models
e.g. WN CFT
[Gaberdiel, Gopakumar]

17. Unitary higher spin AdS3/CFT2 at large c
Violate chaos bound, causality.
W∞ [λ] minimal models
e.g. WN CFT
[Gaberdiel, Gopakumar]

18. Unitary higher spin AdS3/CFT2 at large c
Violate chaos bound, causality.
Obey chaos bound: amplitude Regge-izes
W∞ [λ] minimal models
e.g. WN CFT
[Gaberdiel, Gopakumar]

19. Unitary higher spin AdS3/CFT2 at large c
(Non-unitarity for 𝜆>2: imaginary scattering amplitudes)
Violate chaos bound, causality.
Obey chaos bound: amplitude Regge-izes
W∞ [λ] minimal models
e.g. WN CFT
[Gaberdiel, Gopakumar]

20. A chaotic bootstrap
Strategy: compute chaotic correlators by analytic continuation, conformal transformation of Euclidean correlators on the plane.
[Roberts, Stanford]

21. Chaotic kinematics
We want Lorentzian kinematics, 𝑡>𝑥>0,with 𝑡≫𝛽. Starting from Euclidean correlators,
the chaos regime is reached by analytic continuation:
This is also known as the Regge limit:

22. Chaotic kinematics
We want Lorentzian kinematics, 𝑡>𝑥>0,with 𝑡≫𝛽. Starting from Euclidean correlators,
the chaos regime is reached by analytic continuation:
This is also known as the Regge limit:

23. Chaotic kinematics
We want Lorentzian kinematics, 𝑡>𝑥>0,with 𝑡≫𝛽. Starting from Euclidean correlators,
the chaos regime is reached by analytic continuation:
This is also known as the Regge limit:

24. CFTs with Einstein gravity duals
For CFTd’s with Einstein gravity duals, bulk calculations saturate the bound.
Reggeon = spin-2 = gravity couples universally, gives leading long-wavelength behavior
Let me briefly explain how this works. Recall that we want to examine the planar correlator. In a conformal block expansion, xx. In general, the spin sum is infinite; the Regge limit is dominated by higher spin contributions, so this makes it tricky. However, we use the following two facts: 1. Regge conf blocks behave like spin-s exchange: xx. This means that when the spin sum over pirmaries is bounded, we can take the limit block-by-block. And 2. In a causal hol CFT with a local bulk dual, L=2. The currents are sufficient to read off exponent, the rest just gives x-dep.
Einstein gravity
Famous numbers in AdS/CFT
[Shenker, Stanford; Roberts, Stanford, Susskind]

25. Chaotic destruction of higher spin theories
Upshot: In a sparse, large c CFT, the Regge scaling (i.e. λ 𝐿 ) is determined by the vacuum conformal block, dual to massless exchanges in the bulk.
For a tower of HS currents of spins s ≤ N,
Unitary, causal holographic CFTs with a finite tower of higher spin currents – and their would-be AdS higher spin gravity duals – do not exist.
The details of the rest of the spectrum – the non-currents – only affect the x-dependence.
A more precise argument can be made by taking W to be a heavy operator. Then the correlator is dominated by the semiclassical W vacuum block, up to exp suppressed corrections in c. In earlier work I derived these using bulk Wilson lines. One can then show the same result for the Lyapunov exponent (and the scrambling time).
It’s been shown, on the basis of G-gauge invariance, that bulk correlators in heavy backgrounds = Wilson lines of G.
Correlator = Wilson line = vacuum block
We showed that these violate causality. Therefore, our calculation is equally a bulk calculation of a full correlator.
HS shock wave acausality (?)
Calls into question whether HS BH can form from collapse of HS-charged matter.

26. Acausality will be manifest in “higher spin shock wave”
Chaotic destruction of higher spin theories
Acausality will be manifest in “higher spin shock wave”
Start with eternal BTZ black hole, dual to thermofield double state

27. Acausality will be manifest in “higher spin shock wave”
Chaotic destruction of higher spin theories
Acausality will be manifest in “higher spin shock wave”
Start with eternal BTZ black hole, dual to thermofield double state
Create a “higher spin shock wave”

28. Acausality will be manifest in “higher spin shock wave”
Chaotic destruction of higher spin theories
Acausality will be manifest in “higher spin shock wave”
Start with eternal BTZ black hole, dual to thermofield double state
Create a “higher spin shock wave”
This can lead to acausality!
[Camanho, Edelstein, Maldacena, Zhiboedov]

29. Aside: SL(N) higher spin gravity vs. Gauss-Bonnet
Both theories are acausal, require ∞ towers of higher spin fields for completion
Gauss-Bonnet in higher D has a coupling which sets scale of new massive higher spin fields:
SL(N) theories are in even worse shape: all fields are massless, only one scale in the problem: LAdS
This requires infinite tower of massless higher spin fields.
[Camanho, Edelstein, Maldacena, Zhiboedov]

30. Regge-ization of Vasiliev theory
Now consider theories with one HS gauge field at every spin s=2,3,…
Summing over an infinite set of current exchanges,
Regge-ization as proxy for stringiness: encourages tensionless string theory interpretation of Vasiliev theory
Suggests that all theories with W∞ symmetry have integrable dynamics (like the minimal models).

31. Some future directions
Can Vasiliev theory be embedded into string theory?
What is the space of irrational, unitary higher spin 2d CFTs?
Conformal bootstrap in the Regge limit – can we constrain towers of non- conserved higher spin operators in CFTd?

32. Unitary, large c W∞[λ>2] CFTs do not exist
Putting chaos aside, one can easily establish the above result.
This follows from the following fact: W∞[λ>2] is complex.
e.g. the OPE coefficient is
Straightforward to prove that
3D Vasiliev + pure hs[λ] gravity have imaginary scattering amplitudes.

33. Note: causality limits j,k = 0 or 1!
λ 𝐿 = 2𝜋 𝛽 from CFT
In a somewhat simplified model of holography, only s ≤2 exchanges in VV-WW channel.
Equivalently, the only Witten diagrams are
The Regge limit of spin-s conformal block behaves like spin-s exchange, so spin-2 exchanges dominate:
Note: causality limits j,k = 0 or 1!
[EP]