The SYK model, AdS2 and conformal symmetry

Juan Maldacena

NaFest

September 2016 Instute for Advanced Study Na and I collaborated on 8 papers.

Five were on aspects of two dimensional String theory and their relaon to matrix models. An important paper by Na # 93 1995-1997 Andy Strominger:

The D3 brane near horizon geometry has maximal supersymmetry

16 à 32

This is double the expected number. The extra supersymmetries à conformal symmetry à isometries of the near horizon region

AdS/CFT The SYK model, AdS2 and conformal symmetry

Based on work with Talks by Kitaev Douglas Stanford and Zhenbin Yang Talk Models of holography

Large N quantum system Anomalous Gravity/string dual dimension • Free boundary theories. • Bulk theories with massless S>2 =0 higher spin fields. • Very slighly massive higher • O(N) interacng theories. 1/N spins S>2 ⇠ •

O(1) masses for the higher Easier

Harder • Sachdev Ye Kitaev Model 1 spin fields. S>2 ⇠

• Maximally supersymmetric • Einstein gravity theory. Yang Mills at very strong Higher spin parcles are t’Hoo coupling, g2 N >> 1 very massive. 1 S>2 Solvable large N models A simple solvable model

Rainbow diagrams.

+ + +

+

Eg : 2d QCD, O(N) models, large N Chern Simons theories with fundamental maer in 2 +1 dimensions Giombi, Minwalla, Prakash, Trivedi, Wadia, Yin

Izuka Polchinski Okuda

Summing rainbow diagrams

Rainbow diagrams.

= + + +

=

1 1 = ⌃(!) G(!) G0

⌃(t, t0)=P (t, t0)G(t, t0) Special Case

2 P = J = constant Izuka Polchinski Okuda

Similar to what we get for the following model

N Majorana fermions , = { i j} ij 2 2 H = i Jjk j k J = J /N h iji Xj,k (no sum) Random couplings, gaussian distribuon.

To leading order à treat J as an addional field à same structure as before.

Soluon of the special model

H = i Jjk j k Xj,k

Diagonalize J à semicircle distribuon of energies. 0 ω

Low energies à constant distribuons à like a massless fermion on a circle of size N

Simple emergence of approximate scale invariance.

This model is too simple à no chaos, no black hole like-behavior. The SYK model

N Majorana fermions Sachdev Ye Kitaev i, j = ij Georges, Parcollet { }

H = Ji1i2i3i4 i1 i2 i3 i4 i , ,i 1 ··· 4 X 2 2 3 Random couplings, gaussian distribuon. Ji1i2i3i4 = J /N h i

To leading order à treat Jijkl as an addional field

J = dimensionful coupling. We will be interested in the strong coupling region

1 J, ⌧J N ⌧ ⌧ Spectrum D. Stanford

N dimH =22

Number of random couplings N 4 2N / ⌧

(specific, but random J’s)

Exponenally large number of states contributes to the low energy region we consider Large N limit

Before we had rainbows

+ + +

+ Large N limit

Now:

+ + +

+ 1 1 = ⌃(!) G(!) G0 2 3 ⌃(t, t0)=J G(t, t0)

Generalizaon:

q/2 H = i Ji ,i , ,i i i i 1 2 ··· q 1 2 ··· q 1 1 i1,i2, ,iq = ⌃(!) X··· G(!) G0 2 q 1 ⌃(t, t0)=J G(t, t0)

q=2 à case we had before.

q=4 à SYK

q = Infinity à analycally solvable equaons.

In the IR à Conformal symmetry 1 1 Make a scale invariant ansatz = ⌃(!) G(!) G0 2 3 ⌃(t, t0)=J G(t, t0) 1 1 Gc(⌧, ⌧ 0) is a soluon if = / (⌧ ⌧ )2 q 0

If G is a soluon, and we are given an arbitrary funcon f(τ), we can generate another soluon:

G G (⌧, ⌧ 0)=[f 0(⌧)f 0(⌧ 0)] G (f(⌧),f(⌧ 0)) c ! c,f c Example: Go from zero the temperature to a finite temperature soluon 1 G(⌧, ⌧ 0) / (⌧ ⌧ )2 0 ⇡⌧ f(⌧) = tan

2 ⇡ Gf = ⇡⌧ " sin # General form of the propagator

G(⌧) J =0 1/2 J . 1 1 ⇡⌧ 2 [J sin ]

Deviaon only at very short J 1 distance 0 0 ⌧ Large N effecve acon

Integrate out the fermions and the couplings to obtain an effecve acon for the singlets, the fermion bilinears. 2 N J 4 S = log det(@ ⌃) d⌧d⌧ 0⌃(⌧, ⌧ 0)G(⌧, ⌧ 0)+ G(⌧, ⌧ 0) 2 t 4  Z

Equaons of moon from this acon give the same as the Schwinger Dyson equaons above.

Inserng the conformal answer here we get the extremal entropy.

q It is non-local. The bilocal terms come from the integral over the couplings. o(1/N )

This effecve acon is correct to leading orders, where we can ignore the replicas,

Similar acons were obtained for usual O(N) style models.

Analog of the ``bulk acon” : two dimensions, but det terms are not local. It is also the wrong space... Zero modes of the acon Recall the conformal symmetry in the IR 1 G(⌧, ⌧ 0) / (⌧ ⌧ )2 0 G G (⌧, ⌧ 0)=[f 0(⌧)f 0(⌧ 0)] G(f(⌧),f(⌧ 0)) ! f

All these soluons have the same acon.

Goldstone bosons à no acon for f à will give a divergence if we do the path integral over f.

Soluon: remember that the symmetry is also slightly broken. Nearly zero modes of the acon

1 G(⌧, ⌧ 0) / (⌧ ⌧ )2 0 G G (⌧, ⌧ 0)=[f 0(⌧)f 0(⌧ 0)] G(f(⌧),f(⌧ 0)) ! f

Deviaons from the conformal soluon were happening at short distances à expect that the effecve acon for f is local in me. We go from bilocal à local.

af + b SL(2) invariance. f ! cf + d

Simplest acon is …. Schwarzian acon

N↵ f 0 1 f 2 S = s dt Sch(f,t) , Sch(f,t)= 00 00 J f 2 f 2 Z ✓ 0 ◆ 0

Numerical coefficient whose determinaon requires knowing the first deviaon of the propagator from the IR conformal soluon. Can be computed numerically.

ghosts ? Thinking of SL(2) as a gauge symmetry à removes ghosts of the higher derivave acon. goldstones = coset = Reparametrizaons/SL(2) Low energy

S[G,⌃] S S [G,⌃] G ⌃e fe f 0G 0⌃e conf D D ! D D D Z Z Z

Measure fixed by SL(2) symmetry Bagrets, Altland, Kamenev Example: Four point funcon

(⌧ ) (⌧ ) (⌧ ) (⌧ ) disconnected h i 1 i 2 j 3 j 4 i/ S[G,⌃] 1 1 G ⌃G(⌧ , ⌧ )G(⌧ , ⌧ )e = G (⌧ , ⌧ )G (⌧ , ⌧ )+ D D 1 2 3 4 c 1 2 c 3 4 N S Z 2

Inverse of the quadrac acon. Since the leading conformal answer has zero modes, this is enhanced. The enhanced terms are given by the Schwarzian acon Enhancement factor

S J ⌧1 ⌧2 ⌧3 ⌧4 4pt fG (⌧ , ⌧ )G (⌧ , ⌧ )e f = F , , , h i/ D c,f 1 2 c,f 3 4 N Z ✓ ◆ 2 G (⌧, ⌧ 0)=[f 0(⌧)f 0(⌧ 0)] [f(⌧) f(⌧ 0)] c,f Four point funcon

(⌧ ) (⌧ ) (⌧ ) (⌧ ) J ⌧ h i 1 i 2 j 3 j 4 i =1+ F i (⌧ ) (⌧ ) (⌧ ) (⌧ ) N h i 1 i 2 ih j 3 j 4 i ✓ ◆

We can use this to compute lorentzian four point funcons by analyc connuaon.

Different analyc connuaons à different orders in Lorentzian signature.

Of parcular interest is to compute the out of me order correlator that is responsible for the growth of commutators. Shenker, Stanford, Kitaev i(0) j(⌧) i(0) j(⌧) J 2⇡⌧ h i 1+i e i(0) i(0) j(⌧) j(⌧) / N Exponenal growth h ih i Saturang chaos bound JM, Shenker, Stanford Full four point funcon

Can be computed by summing some ladder diagrams and using the conformal symmetry, aer removing the Schwarzian contribuon.

1 ⌧ ⌧ 1 4pt JF i + F˜ i + H() h i/N (sin ⇡⌧12 sin ⇡⌧34 )2 " ✓ ◆ ✓ ◆ #

Conformal invariant part à contains informaon about the operator spectrum.

Anomalous dimensions of higher spin fields are of order one. @1+2m h =2 +1+2m + i i ! m m

1 Comparison with previous conformal quantum mechanics

De Alfaro, Fubini, Furlan ˙ 2 2 dt(X + g/X ) Michelson, Strominger, …. Z

Exact SL(2) symmetry acng on the dynamical variables.

No SL(2) invariant ground state.

Under a reparametrizaon the acon changes as S = d⌧X2Sch(f, ⌧) Z Different paern of symmetry realizaon.

Is OK to describe brane probes in AdS2, but does not seem to capture gravitaonal features properly. Reparametrizaon symmetry

• SYK and AdS2 both have an emergent, spontaneously broken and explicitly broken reparametrizaon symmetry. • The spontaneous breaking, and the explicit breaking, both preserves an SL(2) gauge-like symmetry. Quesons

• The discussion was mostly through the Euclidean path integral.

• How should we think about this approximate symmetry in a Hilbert space context ?

• Is there a Virasoro algebra ?

• Is there any “central charge” to be computed ? Reparametrizaons in CFT2

• In a CFT2 we also have holomorphic reparametrizaons z f(z), z¯ f¯(¯z) 2 • Spontaneously broken by the vacuum to just SL(2)! ! • Goldstones: modes created by the stress tensor operator. They have a non-zero acon consistent with conformal symmetry. • Symmetry is not explicitly broken • Symmetry algebra is deformed by the central charge (viewed as an operator with an expectaon value = c) à Virasoro algebra. • Only in the case that all other Δ>>1 it dominates. Near extermal black holes ( Not a field theory.)

Extremal black hole M Q M 2 J Low energies, near horizon

Conformal quantum

AdS2 region mechanics? Conformal symmetry in quantum mechanics in a finite Hilbert space • No go:

A. Strominger…

Density of states, scale invariance:

1 ⇢(E) , or (E)? / E

Either divergent in IR or no dynamics. Gravity in two dimensions

• No go:

Naïve two dimensional gravity :

pg(R 2⇤)+S M Z Einstein term topological à no contribuon to equaons of moon. Equaons of moon à set stress tensor to zero.

No dynamics !

OK for extremal entropy. See Murthy’s talk Nearly AdS2

Keep the leading effects that perturb away from AdS 2 Teitelboim Jackiw Almheiri Polchinski 2 2 d xpg(R + 2) + 0 d xpgR Z Z

Ground state entropy

Comes from the area of the addional dimensions, if we are geng this from 4 d gravity for a near extremal black hole. pg(R + 2) Z

Equaon of moon for φ à metric is AdS2

Equaon of moon for the metric à phi is almost completely fixed

ds2 = d⇢2 +sinh2 ⇢d⌧ 2

= cosh ⇢ Value at the horizon h Posion of the horizon.

In the full theory: when φ is sufficiently large à change to a new UV theory Asymptoc boundary condions:

2 1 2 ds = du Fixed proper length |Bdy ✏2 1 = (u) |Bdy ✏ r 1 ds2 = du2 |Bdy ✏2

Infinite number of soluons.

ds2 =d⇢2 +sinh2 ⇢d⌧ 2 ⇢(⌧) 2 1 2 2 d⌧ =(⇢0 +sinh ⇢) ✏2 du ✓ ◆ 1 ds2 = du2 |Bdy ✏2 1 = (u) |Bdy ✏ r

One one soluon

Acon à related to Schwarzian (u) S = d2xpg(R + 2) 2 r duK ✏2 ! Z Z 1 S = du (u)Sch(t, u) ✏2 r Z

t = Usual AdS2 me coordinate t(u) u = Boundary system (quantum mechanical) me coordinate Some qualitave relaons 1 G (t ,t ) Background AdS2 metric. c 1 2 / t t 2 | 1 2| Both SL(2) invariant

Non-zero mode perturbaons of G Fields propagang on AdS2

Gravitaonal interacons , Nearly zero modes à t = f(u) , u is via dilaton gravity à reduce to physical me , t = me set by the correlators, the same Schwarzian acon. internal me. t = AdS2 coordinate me, u = boundary proper me.

S[G, ⌃] S = Sdil.grav. + Smatter

Gc(tL,tR) Wormhole or WdW patch of AdS2 Properes fixed by the Schwarzian

• Common to NAdS2 and in NCFT1 (SYK, for example). N • S Temperature dependence of the free energy / J

• Part of the four point funcon that comes from the explicit conformal symmetry breaking. This part leads to a chaos-like behavior with maximal growth in the commutator. 1 2⇡t/ growth of commutators (J)e Kitaev ⇠ N • Wormhole becomes traversable as we add a double trace interacon linking the two sides,

Gao, Jafferis, Wall dtg(t)OL(t)OR(t) Z All these properes Happy Birtday Nat !

Extra slides Bulk coordinates ?

t = t1 + t2 Conformal casimir acng on t1 , t2 à same as Wave operator on z = t1 t2 dt2 + dz2 ds2 = ± z2 Finite temperature. ⌧ + ⌧ ⌧ ⌧ d2 d⌧ 2 ⌧ = L R , = L R ds2 = 2 2 ! ± cosh2 ⌧

Words of cauon:

Does not work for Euclidean space. It is not how it works in gravity. The two point funcons are geodesics in space, not the space directly. # 93