A Chaotic Bootstrap of AdS3 Gravity Eric Perlmutter, Princeton University GR21 Based on hep-th/1602.08272

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]

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, …

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?

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?

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]

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]

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]

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

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

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]

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

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

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

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

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

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

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]

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]

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

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:

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:

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:

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]

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.

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

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”

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]

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]

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).

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?

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 3-3-4 OPE coefficient is Straightforward to prove that 3D Vasiliev + pure hs[λ] gravity have imaginary scattering amplitudes.

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]