Download - SYK€¦ · SYK: cubic couplings Vladimir Rosenhaus KITP YITP, Strings and Fields 2017 August 7, 2017
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SYK: cubic couplings
Vladimir Rosenhaus KITP
YITP, Strings and Fields 2017 August 7, 2017
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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
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𝒩 = 4, maximally supersymmetric SU(N) Yang-Mills, is a remarkable theory
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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
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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
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The O(N) model is dual to Vasiliev higher spin theory
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𝒩 = 4 and the vector O(N) model are very different theories
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Correspondingly, the bulk duals, string theory and Vasiliev theory, are very different.
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Large N CFTs that are solvable at strong coupling are rare.
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SYK is a new class of solvable, strongly coupled, large N, CFTs.
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SYK is harder than the O(N) model, but easier than 𝒩 = 4.
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It is plausible that the bulk dual of SYK is a truly new theory.
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What is the bulk dual of SYK?
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planar diagrams melon diagrams bubble diagrams
𝒩 = 4 SYK O(N)
Matrix model SYK Vector model
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planar diagrams melon diagrams bubble diagrams
𝒩 = 4 SYK O(N)
Bulk: Large gap Tower massive particles
Tower massless particles
String theory Vasiliev?
Matrix model SYK Vector model
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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
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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?
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a) Gravity
In two-dimensions, there is no Einstein gravity.
The natural analog is dilaton gravity (Jackiw- Teitelboim)
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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)
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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.
A large N theory that is solvable at strong coupling, and that is conformal, is rare.
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,
Thursday 3rd August, 2017
Contents
1. 1
2. 1
1.
On =NX
i=1
�i @2n+1�i (1.1)
�n (1.2)
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.
A large N theory that is solvable at strong coupling, and that is conformal, is rare.
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
1
-
If we had the bulk dual of SYK, it would fully specify the masses of the bulk fields
and all their couplings.
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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.
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The hope is to eventually guess an underlying string theory that gives rise to
these couplings.
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Masses
SYK
Bulk
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Cubic Couplings
SYK
Bulk
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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
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
-
Let us look at the cubic couplings in the classical limit, of large n, m, k.
-
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
Thursday 3rd August, 2017
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.
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.
A large N theory that is solvable at strong coupling, and that is conformal, is rare.
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
1
for is exponentially large
-
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.
A large N theory that is solvable at strong coupling, and that is conformal, is rare.
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
Recall
Thursday 3rd August, 2017
Contents
1. 1
2. 1
1.
On =NX
i=1
�i @2n+1�i (1.1)
�n (1.2)
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.
A large N theory that is solvable at strong coupling, and that is conformal, is rare.
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
1
Thursday 3rd August, 2017
Contents
1. 1
2. 1
1.
On =NX
i=1
�i @2n+1�i (1.1)
�n (1.2)
$ (1.3)
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.
A large N theory that is solvable at strong coupling, and that is conformal, is rare.
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
1
dimension
Thursday 3rd August, 2017
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=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.
A large N theory that is solvable at strong coupling, and that is conformal, is rare.
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
1
Thursday 3rd August, 2017
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.
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.
A large N theory that is solvable at strong coupling, and that is conformal, is rare.
SYK is a new class of solvable large N models. It is harder to solve than O(N), but
easier than N = 4.
1
mass
Thursday 3rd August, 2017
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.
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.
A large N theory that is solvable at strong coupling, and that is conformal, is rare.
SYK is a new class of solvable large N models. It is harder to solve than O(N), but
easier than N = 4.
1
for
-
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.
A large N theory that is solvable at strong coupling, and that is conformal, is rare.
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
Recall
Thursday 3rd August, 2017
Contents
1. 1
2. 1
1.
On =NX
i=1
�i @2n+1�i (1.1)
�n (1.2)
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.
A large N theory that is solvable at strong coupling, and that is conformal, is rare.
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
1
Thursday 3rd August, 2017
Contents
1. 1
2. 1
1.
On =NX
i=1
�i @2n+1�i (1.1)
�n (1.2)
$ (1.3)
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.
A large N theory that is solvable at strong coupling, and that is conformal, is rare.
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
1
dimension
Thursday 3rd August, 2017
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=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.
A large N theory that is solvable at strong coupling, and that is conformal, is rare.
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
1
Thursday 3rd August, 2017
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.
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.
A large N theory that is solvable at strong coupling, and that is conformal, is rare.
SYK is a new class of solvable large N models. It is harder to solve than O(N), but
easier than N = 4.
1
mass
Thursday 3rd August, 2017
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.
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.
A large N theory that is solvable at strong coupling, and that is conformal, is rare.
SYK is a new class of solvable large N models. It is harder to solve than O(N), but
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.
-
Thursday 3rd August, 2017
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)
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.
A large N theory that is solvable at strong coupling, and that is conformal, is rare.
1
Let us be more specific. We find that at large n, m, k,
Thursday 3rd August, 2017
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)
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.
A large N theory that is solvable at strong coupling, and that is conformal, is rare.
1
D. Gross, V.R., in progress
-
Thursday 3rd August, 2017
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)
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.
A large N theory that is solvable at strong coupling, and that is conformal, is rare.
1
Let us be more specific. We find that at large n, m, k,
Thursday 3rd August, 2017
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)
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.
A large N theory that is solvable at strong coupling, and that is conformal, is rare.
1
D. Gross, V.R., in progress
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)
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.
A large N theory that is solvable at strong coupling, and that is conformal, is rare.
1
Saturday 5th August, 2017
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
Jijkl �i�j�k�l (1.3)
�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)
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.
1
Saturday 5th August, 2017
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
Jijkl �i�j�k�l (1.3)
�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)
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.
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.
Saturday 5th August, 2017
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
Jijkl�i�j�k�l (1.3)
�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)
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.
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.
Saturday 5th August, 2017
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
Jijkl�i�j�k�l (1.3)
�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)
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.
1
-
planar diagrams melon diagrams bubble diagrams
𝒩 = 4 SYK O(N)
Bulk: Large gap Tower massive particles
Tower massless particles
String theory Vasiliev?
Matrix model SYK Vector model
cubic couplings ✓
-
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
-
Thursday 3rd August, 2017
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)
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.
A large N theory that is solvable at strong coupling, and that is conformal, is rare.
1
At large n, m, k,
Thursday 3rd August, 2017
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)
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.
A large N theory that is solvable at strong coupling, and that is conformal, is rare.
1
D. Gross, V.R., in progress
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
2.2.1. 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
2.2.1. 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
(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