syk€¦ · syk: cubic couplings vladimir rosenhaus kitp yitp, strings and fields 2017 august 7,...

SYK: cubic couplings

Vladimir Rosenhaus KITP

YITP, Strings and Fields 2017 August 7, 2017

Based on:

D. Gross, V.R. ``The bulk dual of SYK: cubic couplings’’, 1702.08016

D.Gross, V.R. ``A line of CFTs: from generalized free fields to SYK’’, 1706.07015

D.Gross, V.R, in progress

𝒩 = 4, maximally supersymmetric SU(N) Yang-Mills, is a remarkable theory

𝒩 = 4 is conformally invariant, at any value of the ’t Hooft coupling

𝒩 = 4 is integrable, leading to impressive calculations of anomalous dimensions and summation of certain classes of Feynman

diagrams

𝒩 = 4, at large N, is dual to string theory in AdS. At large ’t Hooft coupling, there is a gap,

leading to Einstein gravity and black holes

𝒩 = 4, is a large N theory that is solvable at strong coupling, and conformally invariant.

This is rare.

The free/critical vector O(N) model is another solvable, large N, CFT

The O(N) model is dual to Vasiliev higher spin theory

𝒩 = 4 and the vector O(N) model are very different theories

Correspondingly, the bulk duals, string theory and Vasiliev theory, are very different.

Large N CFTs that are solvable at strong coupling are rare.

SYK is a new class of solvable, strongly coupled, large N, CFTs.

SYK is harder than the O(N) model, but easier than 𝒩 = 4.

It is plausible that the bulk dual of SYK is a truly new theory.

What is the bulk dual of SYK?

planar diagrams melon diagrams bubble diagrams

𝒩 = 4 SYK O(N)

Matrix model SYK Vector model


𝒩 = 4 SYK O(N)

Bulk: Large gap Tower massive particles

Tower massless particles

String theory Vasiliev?


SYK

July 24, 2016

Contents

N � 1Jijkl1 ≤ j < k < l ≤ N

L =�i

1

2

�id

d⌧�i − �

i,j,k,l

Jijkl �i�j�k�l (0.1)

H = �1≤i

The dual of large N 𝒩 = 4, at large ’t Hooft coupling, has:

a) Einstein gravity

b) Stringy modes

What is analog for the dual of SYK?

a) Gravity

In two-dimensions, there is no Einstein gravity.

The natural analog is dilaton gravity (Jackiw- Teitelboim)

The dual of SYK contains dilaton gravity.

This is responsible for the maximal chaos. (In same way that, in higher dimensions, Einstein

gravity leads to maximal chaos)

b) The rest of the bulk fields

SYK has an O(N) symmetry, after disorder averaging.

The primary, O(N), singlets are,

Thursday 3rd August, 2017

Contents

1. 1

2. 1

1.

On =NX

i=1

�i @2n+1�i (1.1)

2.

N=4 is a remarkable theory.

1) It is conformally invariant, at any value of the ’t Hooft coupling

2) It is solvable, but not easily. Has integrability properties, allowing for computation

of anomalous dimensions, and summation of sum classes of Feynman diagrams.

3) It is dual to string theory in AdS. At large N and strong ’t hooft coupling, the bulk

has Einstein gravity, including black holes.

A large N theory that is solvable at strong coupling, and that is conformal, is rare.

Another example is the free, or critical O(N) model. This is very familiar. Much easier

to solve than N = 4. More realistic; describes condensed matter systems. The protype of

large N models. The bulk dual is Vasiliev higher spin theory. An unfamiliar theory.

These two theories are completely di↵erent. And the bulks are completely di↵erent.


SYK is a new class of solvable large N models. It is harder to solve than O(N), but

easier than N = 4.

Table

SYK is related to Tensor models, so maybe I should have put it on the left side, not in

the middle. Tensor, matrix, vector. It is shocking that a tensor model could be simpler than

a matrix model.

What is SYK dual to? Most likely, a new theory. Not some small variant of what we

already know. Because it is an inherently new kind of CFT.

1

Each of these is dual to a bulk scalar,


Contents

1. 1

2. 1

1.

On =NX

i=1

�i @2n+1�i (1.1)

�n (1.2)

2.














easier than N = 4.

Table



a matrix model.


1

If we had the bulk dual of SYK, it would fully specify the masses of the bulk fields

and all their couplings.

Since SYK is solvable, we will compute correlation functions in SYK and, using the

AdS/CFT dictionary, deduce what bulk couplings are needed to reproduce these.

The hope is to eventually guess an underlying string theory that gives rise to

these couplings.

Masses

SYK

Bulk

Cubic Couplings

SYK

Bulk

Inserting this into (3.37), and for convenience defining z = ⌧12/⌧31, we obtain the conformal

result (3.16) where the coe�cient I(2)nmk is given by (3.34) where s(2)nmk is the triple sum,

s(2)nmk = �X

p1,p2,p3

✓2n

p1

◆✓2m

p2

◆✓2k

p3

◆✓2n+p2�p1

p2 + 1

◆✓2m+p3�p2

p3 + 1

◆✓2k+p1�p3p1 + 1

◆zp1�p2+2m�2k

(�1�z)p3�p2+2n�2k .(3.40)

This expression is symmetric under all permutations of n,m, k. In addition, this sum must

be independent of z, and must match the result we obtained before, Eq. 3.35. Neither of

these properties is manifest, although one can verify that they are both true. Some properties

of this sum are discussed in Appendix C. In fact, this same sum occurs in the computation

of the three-point function of bilinears in a (particular) generalized free field theory, see

Appendix A.

4. The Bulk Cubic Couplings

In the previous section we found the coe�cients cnmk of the conformal three-point func-

tion of the bilinear operators On. In the limit of large q we wrote explicit equations for cnmk.In this section we use these cnmk to determine the cubic couplings of the bulk fields �n dual

to On.The bulk Lagrangian, to order 1/

pN , is,

Sbulk =

Zd2x

pg

1

2(@�n)

2 +1

2m2n�

2n +

1pN

�nmk �n�m�k

�. (4.1)

One could also consider cubic terms with derivatives, however, as shown in Appendix D, at

this order in 1/N they are equivalent to the non-derivative terms up to a field redefinition. We

use this bulk Lagrangian to compute, via the AdS/CFT dictionary, the three-point function

of the boundary dual. Matching the result with what we found for the SYK three-point

function will determine �nmk.

From the tree level Witten diagram (Fig. 2), the three-point function resulting from this

bulk Lagrangian is [9], 11

hOn(⌧1)Om(⌧2)Ok(⌧3)i = 1pN

�nmkNnNmNk

anmk|⌧12|hn+hm�hk |⌧23|hm+hk�hn |⌧13|hk+hn�hm

, (4.2)

11To simplify comparing with the SYK result, we have normalized the operators to have the two-pointfunction, hOn(⌧1)Om(⌧2)i = �nm|⌧12|�2hn .

21

Cubic Couplings

SYK

Bulk



s(2)nmk = �X

p1,p2,p3

✓2n

p1

◆✓2m

p2

◆✓2k

p3

◆✓2n+p2�p1

p2 + 1

◆✓2m+p3�p2

p3 + 1

◆✓2k+p1�p3p1 + 1

◆zp1�p2+2m�2k

(�1�z)p3�p2+2n�2k .(3.40)






Appendix A.





pN , is,

Sbulk =

Zd2x

pg

1

2(@�n)

2 +1

2m2n�

2n +

1pN

�nmk �n�m�k

�. (4.1)








hOn(⌧1)Om(⌧2)Ok(⌧3)i = 1pN

�nmkNnNmNk


, (4.2)


21

Let us look at the cubic couplings in the classical limit, of large n, m, k.



s(2)nmk = �X

p1,p2,p3

✓2n

p1

◆✓2m

p2

◆✓2k

p3

◆✓2n+p2�p1

p2 + 1

◆✓2m+p3�p2

p3 + 1

◆✓2k+p1�p3p1 + 1

◆zp1�p2+2m�2k

(�1�z)p3�p2+2n�2k .(3.40)






Appendix A.





pN , is,

Sbulk =

Zd2x

pg

1

2(@�n)

2 +1

2m2n�

2n +

1pN

�nmk �n�m�k

�. (4.1)








hOn(⌧1)Om(⌧2)Ok(⌧3)i = 1pN

�nmkNnNmNk


, (4.2)


21


Contents

1. 1

2. 1

1.

On =NX

i=1

�i @2n+1�i (1.1)

�n (1.2)

n,m, k � 1 (1.3)

2.














easier than N = 4.

Table



1

for is exponentially large


Contents

1. 1

2. 1

1.

On =NX

i=1

�i @2n+1�i (1.1)

2.














easier than N = 4.

Table



a matrix model.



1

Recall


Contents

1. 1

2. 1

1.

On =NX

i=1

�i @2n+1�i (1.1)

�n (1.2)

2.














easier than N = 4.

Table



a matrix model.


1


Contents

1. 1

2. 1

1.

On =NX

i=1

�i @2n+1�i (1.1)

�n (1.2)

$ (1.3)

2.














easier than N = 4.

Table



1

dimension


Contents

1. 1

2. 1

1.

On =NX

i=1

�i @2n+1�i (1.1)

�n (1.2)

⇡ 2n+ 1, n � 1 (1.3)

2.














easier than N = 4.

Table



1


Contents

1. 1

2. 1

1.

On =NX

i=1

�i @2n+1�i (1.1)

�n (1.2)

⇡ 2n+ 1, n � 1 (1.3)

2.

⇡ 2n (2.1)N=4 is a remarkable theory.













easier than N = 4.

1

mass


Contents

1. 1

2. 1

1.

On =NX

i=1

�i @2n+1�i (1.1)

�n (1.2)

⇡ 2n+ 1, n � 1 (1.3)

2.














easier than N = 4.

1

for


Contents

1. 1

2. 1

1.

On =NX

i=1

�i @2n+1�i (1.1)

2.














easier than N = 4.

Table



a matrix model.



1

Recall


Contents

1. 1

2. 1

1.

On =NX

i=1

�i @2n+1�i (1.1)

�n (1.2)

2.














easier than N = 4.

Table



a matrix model.


1


Contents

1. 1

2. 1

1.

On =NX

i=1

�i @2n+1�i (1.1)

�n (1.2)

$ (1.3)

2.














easier than N = 4.

Table



1

dimension


Contents

1. 1

2. 1

1.

On =NX

i=1

�i @2n+1�i (1.1)

�n (1.2)

⇡ 2n+ 1, n � 1 (1.3)

2.














easier than N = 4.

Table



1


Contents

1. 1

2. 1

1.

On =NX

i=1

�i @2n+1�i (1.1)

�n (1.2)

⇡ 2n+ 1, n � 1 (1.3)

2.














easier than N = 4.

1

mass


Contents

1. 1

2. 1

1.

On =NX

i=1

�i @2n+1�i (1.1)

�n (1.2)

⇡ 2n+ 1, n � 1 (1.3)

2.














easier than N = 4.

1

for Looks like a KK tower of a field

compactified on a circle

The simplest thing would be to consider a single self-interacting scalar in AdS2 × S1.

This, approximately, gives the correct spectrum.

But this would give cubic couplings that are of order 1, not exponentially large for

large n,m,k.


Contents

1. 1

2. 1

1.

On =NX

i=1

�i @2n+1�i (1.1)

�n (1.2)

⇡ 2n+ 1, n � 1 (1.3)

2.

⇡ 2n (2.1)

�nmk ⇡ N !(N � 2n)!(N � 2m)!(N � 2k)! . (2.2)













1

Let us be more specific. We find that at large n, m, k,


Contents

1. 1

2. 1

1.

On =NX

i=1

�i @2n+1�i (1.1)

�n (1.2)

⇡ 2n+ 1, n � 1 (1.3)

2.

N = n+m+ k (2.1)

�nmk ⇡ N !(N � 2n)!(N � 2m)!(N � 2k)! . (2.2)













1

D. Gross, V.R., in progress


Contents

1. 1

2. 1

1.

On =NX

i=1

�i @2n+1�i (1.1)

�n (1.2)

⇡ 2n+ 1, n � 1 (1.3)

2.

⇡ 2n (2.1)

�nmk ⇡ N !(N � 2n)!(N � 2m)!(N � 2k)! . (2.2)













1

Let us be more specific. We find that at large n, m, k,


Contents

1. 1

2. 1

1.

On =NX

i=1

�i @2n+1�i (1.1)

�n (1.2)

⇡ 2n+ 1, n � 1 (1.3)

2.

N = n+m+ k (2.1)

�nmk ⇡ N !(N � 2n)!(N � 2m)!(N � 2k)! . (2.2)













1


This is suggestive of some kind of bit interpretation, but we don’t have a precise

statement yet.

We have so far discussed the tower of fields in the bulk, their masses and interactions.

Let us return now to the gravitational sector. We said that this is dilaton gravity.

Actually, in our discussion of SYK we have been somewhat imprecise. SYK is not truly a

CFT1 in the infrared.

SYK is ``nearly’’ a CFT1 in the infrared. The fermion four-point function breaks conformal

invariance.

We say ``nearly’’ a CFT1, because only one of the conformal blocks in the four-point function, due to the lowest dimension

operator, breaks conformal invariance.

In fact, this was to be expected.

An old argument, due to Polchinski, is that one can not have a dynamical one-

dimensional CFT. A one-dimensional CFT would have constant entropy - independent of the temperature - which is inconsistent

with dynamics.

So, the infrared of SYK is nearly a CFT1

In fact, dilaton gravity in AdS2 is not AdS2, but ``nearly’’ AdS2. This is because of the

dilaton.

So SYK is an example of nearly AdS2 / nearly CFT1 duality

Nevertheless, it would be nice to have a variant of SYK that is a CFT, and for which

standard AdS/CFT applies.

Moreover, it would be nice to have not just a CFT at strong coupling, but a CFT for any

coupling. Like in 𝒩 = 4.

SYK

cSYK

Saturday 5th August, 2017

Contents

1. 1

2. 1

1.

� = 1/q , q = 4 (1.1)

�n (1.2)

⇡ 2n+ 1, n � 1 (1.3)

2.

N = n+m+ k (2.1)

�nmk ⇡N !

(N � 2n)!(N � 2m)!(N � 2k)! . (2.2)













1


Contents

1. 1

2. 1

1.

� = 1/q , q = 4 (1.1)

S =X

i

1

2

Zd⌧ �i@⌧�i �

Zd⌧

X

i,j,k,l


S =X

i

1

4⇡

Zd⌧1d⌧2 �i(⌧1)

sgn(⌧1 � ⌧2)|⌧1 � ⌧2|2(1��)

�i(⌧2)�Z

d⌧X

i,j,k,l


�n (1.4)

⇡ 2n+ 1, n � 1 (1.5)

2.

N = n+m+ k (2.1)

�nmk ⇡N !

(N � 2n)!(N � 2m)!(N � 2k)! . (2.2)








1


Contents

1. 1

2. 1

1.

� = 1/q , q = 4 (1.1)

S =X

i

1

2

Zd⌧ �i@⌧�i �

Zd⌧

X

i,j,k,l


S =X

i

1

4⇡

Zd⌧1d⌧2 �i(⌧1)

sgn(⌧1 � ⌧2)|⌧1 � ⌧2|2(1��)

�i(⌧2)�Z

d⌧X

i,j,k,l


�n (1.4)

⇡ 2n+ 1, n � 1 (1.5)

2.

N = n+m+ k (2.1)

�nmk ⇡N !

(N � 2n)!(N � 2m)!(N � 2k)! . (2.2)








1

cSYK has an action that is bilocal, rather than local.

To give a Hilbert space interpretation, one can embed this into a local action, by introducing a tower of auxiliary fields and integrating them out.

In this sense, the theory is not self-contained. Perhaps this is why we managed to achieve a CFT1 with dynamics

More generally, any bilocal action can be interpreted in this way.

General and Finite Temperature

To obtain the appropriate action at finite temperature, we could repeat the procedure,

considering an AdS-Schwarzschild background rather than the Poincare patch. 1 This would

be more involved, and is in fact unnecessary, since a clear generalization of (2.9) is,

I =

Zddx

Z 1

0

d�

1

2

�(@'

�

)2 + �2L�2'2�

�+ j(�)'

�

�0

�, (2.12)

for some j(�) that can depend on L and other scales one may choose to introduce. Integrating

out '�

leads to the bilocal action (2.10) with the kernel,

K(x, x0) =

Z 1

0

d� j(�)2 hx| 1⇤� �2

L

2

|x0i . (2.13)

In particular, it is now easy to find the necessary coupling j(�) to achieve the finite

temperature generalized free field action for a boson in 0 + 1 dimension. In other words, we

would like to find a j(�) such that the kernel (2.13) is the inverse of the finite temperature

two-point function, G�(!n), for a conformal scalar field of dimension �,

K(!n

) = �2⇡ tan ⇡�(2�� 1)

1

G�(!n), G�(!n) =

✓2⇡

�

◆2��1 ⇡cos ⇡��(2�)

�⇣�+ �!n2⇡

⌘

�⇣1��+ �!n2⇡

⌘ ,

(2.14)

where !n

= 2⇡n/� are the Matsubara frequencies. One can see that G�(!n) satisfies the

property,1

G�(!n)=

(2�0 � 1)2⇡ tan ⇡�0

G�0(!n) , (2.15)

where we have defined �0 = 1��. From (2.13) we have that,

K(!n

) = �Z 1

0

d�j(�)2

!2n

+ �2

L

2

. (2.16)

Therefore, we can achieve the desired (2.14) with the choice of coupling,

j(�)2 =2

L

�

L⇢�0(�/L) =

2

L

�

L

1

⇡�(2�0)

✓2⇡

�

◆2�0�1sinh

✓��

2L

◆�

✓�0 � i��2⇡L

◆�

✓�0 +

i��

2⇡L

◆,

1 Note that in AdS2 one get between di↵erent backgrounds through a change of coordinates. However, ifone chooses to maintain Poincare patch coordinates, then the necessary source will no longer be at constantz = ✏, but rather along some trajectory z(t) that one can find through a change of coordinates to AdS-Schwarzschild coordinates, with the radial coordinate there set to ✏.

6

General and Finite Temperature

To obtain the appropriate action at finite temperature, we could repeat the procedure,

considering an AdS-Schwarzschild background rather than the Poincare patch. 1 This would

be more involved, and is in fact unnecessary, since a clear generalization of (2.9) is,

I =

Zddx

Z 1

0

d�

1

2

�(@'

�

)2 + �2L�2'2�

�+ j(�)'

�

�0

�, (2.12)

for some j(�) that can depend on L and other scales one may choose to introduce. Integrating

out '�

leads to the bilocal action (2.10) with the kernel,

K(x, x0) =

Z 1

0

d� j(�)2 hx| 1⇤� �2

L

2

|x0i . (2.13)

In particular, it is now easy to find the necessary coupling j(�) to achieve the finite

temperature generalized free field action for a boson in 0 + 1 dimension. In other words, we

would like to find a j(�) such that the kernel (2.13) is the inverse of the finite temperature

two-point function, G�(!n), for a conformal scalar field of dimension �,

K(!n

) = �2⇡ tan ⇡�(2�� 1)

1

G�(!n), G�(!n) =

✓2⇡

�

◆2��1 ⇡cos ⇡��(2�)

�⇣�+ �!n2⇡

⌘

�⇣1��+ �!n2⇡

⌘ ,

(2.14)

where !n

= 2⇡n/� are the Matsubara frequencies. One can see that G�(!n) satisfies the

property,1

G�(!n)=

(2�0 � 1)2⇡ tan ⇡�0

G�0(!n) , (2.15)

where we have defined �0 = 1��. From (2.13) we have that,

K(!n

) = �Z 1

0

d�j(�)2

!2n

+ �2

L

2

. (2.16)

Therefore, we can achieve the desired (2.14) with the choice of coupling,

j(�)2 =2

L

�

L⇢�0(�/L) =

2

L

�

L

1

⇡�(2�0)

✓2⇡

�

◆2�0�1sinh

✓��

2L

◆�

✓�0 � i��2⇡L

◆�

✓�0 +

i��

2⇡L

◆,

1 Note that in AdS2 one get between di↵erent backgrounds through a change of coordinates. However, ifone chooses to maintain Poincare patch coordinates, then the necessary source will no longer be at constantz = ✏, but rather along some trajectory z(t) that one can find through a change of coordinates to AdS-Schwarzschild coordinates, with the radial coordinate there set to ✏.

6

of the normalizable modes). In the limit that z0 ! 0,

Gbulk

(z, x | z0, x0) ! 12�0 � dz

0�0K�0(z, x | x0) , as z0 ! 0 , (2.7)

so (2.5) implies (2.3), and in particular that ↵ = 2�0 � d. The action (2.4) is the kindof bilocal action we are seeking; it is the action for a generalized free field �0 of dimension

� = d��0. With standard quantization, as we have discussed, one can achieve �0 � d/2,where the lower bound is set by the BF bound on the mass. With alternate quantization [21]

this can be extended to �0 � d/2� 1.

2.1. A Tower of Auxiliary Fields

For generalizing (2.5) it is useful to rewrite it as an inherently d dimensional theory. We

decompose the field in terms of its radial eigenfunctions, which are Bessel functions,

�(z, t, x) =

Z 1

0

d� zd/2 J⌫

(�z)p�'

�

(t, x) , (2.8)

where �0 = d/2 + ⌫, with ⌫ =q

m2 + d2/4. Inserting this expansion of �(z, t, x) into the

action (2.5) and restoring the AdS scale L, we get,

I =

Zddx

Z 1

0

d�

1

2

�(@'

�

)2 + �2L�2'2�

�+ L�⌫�1 ↵�⌫+

12 '

�

�0

�, (2.9)

where we have defined ↵ = 21�⌫/�(⌫). So we have a tower of fields '�

, of mass �2L�2, all of

which are linearly coupled to �0. Let us integrate out '�. This gives an e↵ective action,

Ieff

=1

2

Zddxddx0 �0(x)K(x, x

0)�0(x0) , (2.10)

where,

K(x, x0) = ↵2Z 1

0

d��2⌫+1 hx| 1⇤� �2 |x

0i , (2.11)

and so reproduces (2.4), wherein the non-locality arises since we have integrated out massless

fields.

5

Integrating out,

cSYK is a CFT. The bulk dual is a nongravitaional theory in a

rigid AdS background.

There is a line of fixed points. The action is conformally invariant for any coupling J,

which is now dimensionless

More specifically, the acton is classically conformally invariant. That is clear.


Contents

1. 1

2. 1

1.

� = 1/q , q = 4 (1.1)

S =X

i

1

2

Zd⌧ �i@⌧�i �

Zd⌧

X

i,j,k,l

Jijkl�i�j�k�l (1.2)

S =X

i

1

4⇡

Zd⌧1d⌧2 �i(⌧1)

sgn(⌧1 � ⌧2)|⌧1 � ⌧2|2(1��)

�i(⌧2)�Z

d⌧X

i,j,k,l


�n (1.4)

⇡ 2n+ 1, n � 1 (1.5)

2.

N = n+m+ k (2.1)

�nmk ⇡N !

(N � 2n)!(N � 2m)!(N � 2k)! . (2.2)








1

We have coupled the fermion to its shadow

However, one would have expected that this would not be true at the quantum level.

This would manifest itself through UV divergences.

SYK was super-renormalizable, and manifestly UV finite. cSYK is not.

In fact, at least to leading order in 1/N, cSYK is UV finite. This is surprising.

At weak coupling, cSYK is a generalized free field theory. At strong coupling, it looks like the strong coupling (infrared) of SYK.

It will be interesting to see how the bulk dual changes as we vary the coupling.


Contents

1. 1

2. 1

1.

� = 1/q , q = 4 (1.1)

S =X

i

1

2

Zd⌧ �i@⌧�i �

Zd⌧

X

i,j,k,l


S =X

i

1

4⇡

Zd⌧1d⌧2 �i(⌧1)

sgn(⌧1 � ⌧2)|⌧1 � ⌧2|2(1��)

�i(⌧2)�Z

d⌧X

i,j,k,l


�n (1.4)

⇡ 2n+ 1, n � 1 (1.5)

2.

N = n+m+ k (2.1)

�nmk ⇡N !

(N � 2n)!(N � 2m)!(N � 2k)! . (2.2)








1


𝒩 = 4 SYK O(N)

Bulk: Large gap Tower massive particles

Tower massless particles

String theory Vasiliev?


cubic couplings ✓

Cubic Couplings

SYK

Bulk



s(2)nmk = �X

p1,p2,p3

✓2n

p1

◆✓2m

p2

◆✓2k

p3

◆✓2n+p2�p1

p2 + 1

◆✓2m+p3�p2

p3 + 1

◆✓2k+p1�p3p1 + 1

◆zp1�p2+2m�2k

(�1�z)p3�p2+2n�2k .(3.40)






Appendix A.





pN , is,

Sbulk =

Zd2x

pg

1

2(@�n)

2 +1

2m2n�

2n +

1pN

�nmk �n�m�k

�. (4.1)








hOn(⌧1)Om(⌧2)Ok(⌧3)i = 1pN

�nmkNnNmNk


, (4.2)


21


Contents

1. 1

2. 1

1.

On =NX

i=1

�i @2n+1�i (1.1)

�n (1.2)

⇡ 2n+ 1, n � 1 (1.3)

2.

⇡ 2n (2.1)

�nmk ⇡ N !(N � 2n)!(N � 2m)!(N � 2k)! . (2.2)













1

At large n, m, k,


Contents

1. 1

2. 1

1.

On =NX

i=1

�i @2n+1�i (1.1)

�n (1.2)

⇡ 2n+ 1, n � 1 (1.3)

2.

N = n+m+ k (2.1)

�nmk ⇡ N !(N � 2n)!(N � 2m)!(N � 2k)! . (2.2)













1


This is suggestive of some kind of bit interpretation, but we don’t have a precise

statement yet.

SYK

ChaosNon-Fermi liquid

String Theory?

Tensor models

dilaton gravity

Generalizations

H =X

Ji1,...,iq�i1 · · ·�iq (32)

H =X

a

0

@X

ijkl

Jaijkl�ai�

aj�

ak�

al +

X

ijkl

J̃aijkl�ai�

aj�

a+1k �

a+1l

1

A (33)

AdS2/CFT1 (34)

3

higher dimensions

Tensors and quantum gravity

Tensor SYK

• variables

• symmetry

Contents

1. Tensor SYK 1

2. Order 1/N 2

2.1. SYK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.2. Tensor SYK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2.1. Four-point function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.3. Free Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3. Mean Field 7

1. Tensor SYK

The disorder free SYK is given by,

H =X

�0ijk�1klm�

2mjp�

3pli (1.1) {H}

The invariance O(n)6 is

�0ijk ! O03ia0O02jb0O

01kc0 �

0a0b0c0 (1.2)

�1klm ! O01ka1O13lb1O

12mc1 �

1a1b1c1 (1.3)

�2mjp ! O12ma2O02jb2O

23pc2 �

2a2b2c2 (1.4)

�3pli ! O23pa3O12lb3O

03ic3 �

3a3b3c3 (1.5)

We can write the Hamiltonian as a matrix model. This will be a matrix model with 4n

matrices which are n by n. That gives a total of 4n3 degrees of freedom. There are three

ways to do this.

1. Let (�j)ij = �ijk. Then

H =X

tr(�0j�1l�

2j�

3l ) (1.6) {H1}

2. Let (�0i )jk = �0ijk, (�

1m)kl = �

1klm, (�

2m)jp = �

2mjp, (�

3i )pl = �

3pli. Then,

H =X

tr(�0i�1m�

3,ti �

2,tm ) (1.7) {H2}

3. Let (�0k)ij = �0ijk, (�

1k)lm = �

1klm, (�

2p)mj = �

2mjp, (�

3p)li = �

3pli. Then,

H =X

tr(�0k�2,tp �

1,tk �

3p) (1.8) {H3}

1

Contents

1. Tensor SYK 1

2. Order 1/N 2

2.1. SYK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.2. Tensor SYK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3


2.3. Free Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3. Mean Field 7

1. Tensor SYK


H =X

�0ijk�1klm�

2mjp�

3pli (1.1) {H}



01kc0 �

0a0b0c0 (1.2)


12mc1 �

1a1b1c1 (1.3)


23pc2 �

2a2b2c2 (1.4)


03ic3 �

3a3b3c3 (1.5)



ways to do this.


H =X

tr(�0j�1l�

2j�

3l ) (1.6) {H1}

2. Let (�0i )jk = �0ijk, (�

1m)kl = �

1klm, (�

2m)jp = �

2mjp, (�

3i )pl = �

3pli. Then,

H =X

tr(�0i�1m�

3,ti �

2,tm ) (1.7) {H2}

3. Let (�0k)ij = �0ijk, (�

1k)lm = �

1klm, (�

2p)mj = �

2mjp, (�

3p)li = �

3pli. Then,

H =X

tr(�0k�2,tp �

1,tk �

3p) (1.8) {H3}

1

Contents

1. Tensor SYK 1

2. Order 1/N 2

2.1. SYK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.2. Tensor SYK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3


2.3. Free Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3. Mean Field 7

1. Tensor SYK


H =X

�0ijk�1klm�

2mjp�

3pli (1.1) {H}



01kc0 �

0a0b0c0 (1.2)


12mc1 �

1a1b1c1 (1.3)


23pc2 �

2a2b2c2 (1.4)


03ic3 �

3a3b3c3 (1.5)



ways to do this.


H =X

tr(�0j�1l�

2j�

3l ) (1.6) {H1}

2. Let (�0i )jk = �0ijk, (�

1m)kl = �

1klm, (�

2m)jp = �

2mjp, (�

3i )pl = �

3pli. Then,

H =X

tr(�0i�1m�

3,ti �

2,tm ) (1.7) {H2}

3. Let (�0k)ij = �0ijk, (�

1k)lm = �

1klm, (�

2p)mj = �

2mjp, (�

3p)li = �

3pli. Then,

H =X

tr(�0k�2,tp �

1,tk �

3p) (1.8) {H3}

1

(a) (b)

i i

j

k l

l k4pt

(c)

Figure 2 {Fig2}

0 2

1

3(a)

Figure 3: The color coding is as follows: 01: red, 02: blue, 03: green, 12: black, 13: orange,23: yellow. {Fig3}

2.2. Tensor SYK

We now consider the tensor SYK. The vertex is shown in Fig. 3. A melon diagram is

shown in Fig. 4. There are two ways to count the order of the diagram. The first is by

counting directly. The second is to consider the three di↵erent matrixations, (1.6, 1.7, 1.8).

Graphically these correspond to deleting the green and black lines, or the blue and orange

lines, or the red and yellow lines. We construct these three matrixations and count the power

of 1/n for each one (the genus). Then taking the sum of the genuses,

! =X

J

gJ , (2.1)

the power of a diagram is1

n!. (2.2) {PC}

3

Bonzom, Gurau, Riello, Rivasseau Gurau

Witten Klebanov- Tarnopolsky

Colored Tensor Models

• Only melon diagrams at leading order!

• Therefore, same large N properties as SYK(a)

Figure 4 {Fig4}

(a) (b) (c)

Figure 5 {Fig5}

For the melon diagrams, the three matrixations are shown in Fig. 5, and we see that ! = 0.

Now consider the diagram in Fig. 1 (a). In tensor notation it is Fig. 6 and we see that it

is of order j4n5 = 1/n. If we look at the three matrixations, Fig. 7, then the first two scale

as 1/n2 (genus zero), while the third (c) goes as 1/n4 (genus 1). Thus ! = 1, and using (2.2)

we recover the scaling 1/n. As for SYK, one can generate more 1/n diagrams by melonizing

Fig. 6.

Consider now Fig. 6, but with five loops instead of three. Then we find that the analog

of Fig. 6 (a) and Fig. 6 (b) are still genus zero, while the analog of Fig. 6 (c) becomes genus

2 (with scaling j6n2 ⇠ 1/n6).Next, we note that the tensor analogs of Fig. 1(b) and Fig. 1(c) are 1/n2 suppressed.

We show this for Fig. 1(b) in Fig. 8; the diagram is of order j4n2 ⇠ 1/n2. In terms of thematrixations, from Fig. 9, we have that Fig. 9 (a) scales as j4n2 ⇠ 1/n4 (so genus 1), whileFig. 9 (b) and (c) scale as j4n3 ⇠ 1/n3 (so genus 1/2). Thus, ! = 2.

Thus, the only diagrams at order 1/n are Fig. 6 and its melonizations.

4

(a) (b)

i i

j

k l

l k4pt

(c)

Figure 2 {Fig2}

0 2

1

3(a)

Figure 3: The color coding is as follows: 01: red, 02: blue, 03: green, 12: black, 13: orange,23: yellow. {Fig3}

2.2. Tensor SYK

We now consider the tensor SYK. The vertex is shown in Fig. 3. A melon diagram is

shown in Fig. 4. There are two ways to count the order of the diagram. The first is by

counting directly. The second is to consider the three di↵erent matrixations, (1.6, 1.7, 1.8).

Graphically these correspond to deleting the green and black lines, or the blue and orange

lines, or the red and yellow lines. We construct these three matrixations and count the power

of 1/n for each one (the genus). Then taking the sum of the genuses,

! =X

J

gJ , (2.1)

the power of a diagram is1

n!. (2.2) {PC}

3

syk€¦ · syk: cubic couplings vladimir rosenhaus kitp yitp, strings and fields 2017 august 7,...

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