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JHEP01(2018)010

Published for SISSA by Springer

Received: July 17, 2017

Revised: November 20, 2017

Accepted: December 25, 2017

Published: January 2, 2018

Generalized conformal structure, dilaton gravity and

SYK

Marika Taylor

Mathematical Sciences and STAG Research Centre, University of Southampton,

Highﬁeld, Southampton, SO17 1BJ, U.K.

E-mail: m.m.taylor@soton.ac.uk

Abstract: A theory admits generalized conformal structure if the only scale in the quan-

tum theory is set by a dimensionful coupling. SYK is an example of a theory with general-

ized conformal structure and in this paper we investigate the consequences of this structure

for correlation functions and for the holographic realization of SYK. The Ward identities

associated with the generalized conformal structure of SYK are implemented holograph-

ically in gravity/multiple scalar theories, which always have a parent AdS

origin. For

questions involving only the graviton/running scalar sector, one can always describe the

bulk running in terms of a single scalar but multiple running scalars are in general needed

once one includes the bulk ﬁelds corresponding to all SYK operators. We then explore chaos

in holographic theories with generalized conformal structure. The four point function ex-

plored by Maldacena, Shenker and Stanford exhibits exactly the same chaotic behaviour

in any such theory as in holographic realizations of conformal theories i.e. the dimensionful

coupling scale does not aﬀect the chaotic exponential growth.

Keywords: AdS-CFT Correspondence, Gauge-gravity correspondence

ArXiv ePrint: 1706.07812

Open Access,

 The Authors.

Article funded by SCOAP

https://doi.org/10.1007/JHEP01(2018)010

JHEP01(2018)010

aims of this work is to discuss this “nearly” conformal invariance further, from the perspec-

tive of the underlying Ward identities deﬁning both sides of the holographic correspondence.

Symmetries of additional compact directions in the bulk are associated with global sym-

metries of the ﬁeld theory. The SYK model has no such global symmetries, suggesting that

either the holographic dual is non-critical or that the compact directions have no isometries.

Support for the former possibility follows from an analysis of the SYK spectrum, obtained

by extracting the OPE from four point functions. The spectrum contains many low di-

mension operators [19, 20], indicating that the bulk dual may involve low tension strings.

Nevertheless one should be able to bootstrap a bulk description for SYK from knowledge

of the SYK correlation functions, and it is this perspective that will be taken in this paper.

The “nearly” conformal two-dimensional gravity/dilaton theories are particular exam-

ples of holographic theories with generalized conformal structure. Generalized conformal

structure [21–23] is perhaps most easily deﬁned in a ﬁeld theory. A theory with conformal

invariance has a traceless stress energy tensor up to conformal anomalies: hT

i ∼ 0. In a

theory with generalized conformal structure the scale invariance is broken at the quantum

level only by a source for a scalar operator O

i + (d − ∆

)Φ

i ∼ 0, (1.1)

with d the spacetime dimension and ∆

the dimension of the operator. It is therefore the

(dimensionful) coupling Φ

that drives the renormalization group ﬂow; Φ

is the only scale

in the theory. (Later on in this paper we will generalize this concept to allow for a ﬁnite

number of operators driving the ﬂow.)

Generalized conformal structure underlies a number of holographic dualities, the most

prominent of which are the Dp-branes (with p 6= 3) and the fundamental string [24]. In

all such cases, in the regime where the supergravity description is valid the dynamics is

controlled by the dimensionful running of the coupling. The holographic dictionary for

such non-conformal brane backgrounds was developed in detail in [25] and will be reviewed

in section 4. As we discuss later, a particular feature of the holographic dictionary is that

dilaton/gravity models with generalized conformal structure can be understood in terms of

dimensional reductions of AdS gravity on tori [26]. These tori are not in general of integer

dimension — the dimension is related to the running of the dilaton and the thermodynamic

behaviour of the theory.

In a theory with generalized conformal structure, operator correlation functions depend

on the dimensionful coupling Φ

as well as on the operator separation. For example, the

Euclidean two point function for a generic scalar operator O of dimension ∆ behaves as

hO(x)O(0)i =

f(Φ

|x|

d−∆

)

|x|

2∆

(1.2)

where f is a function of the dimensionless coupling Φ

|x|

d−∆

. For a theory in which the

driving operator is irrelevant, the correlator would admit an IR expansion

hO(x)O(0)i =

|x|

2∆



+ f

|x|

∆

−d

+ ···



(1.3)

– 2 –

JHEP01(2018)010

i.e. a Laurent series in Φ

/|x|

∆

−d

with (f

, f

, ···) dimensionless coeﬃcients. In the deep

IR limit |x|

∆

−d

 Φ

this two point function looks conformal; it is the existence of the

series of corrections that distinguishes conformal behaviour from generalized conformal

structure. Precisely this structure is found in the SYK two point function — see for

example [2, 3, 20, 27].

The generalized conformal structure underlying two dimensional dilaton/gravity mod-

els and the relation to AdS

gravity was discussed previously in [28], where the holographic

dictionary between the bulk metric, gauge ﬁeld and dilaton and the dual Hamiltonian H,

global current J and O

was derived. Further discussion of the relation between SYK and

AdS

can be found in the recent work [29].

In this paper we point out an implication of the holographic dictionary: the two point

functions of these operators vanish (up to contact terms). This result follows directly from

the Ward identities and can also be understood by reducing the two point function of the

stress energy tensor of a 2d CFT on a circle. The Hamiltonian, global current and O

can

however acquire (time independent) expectation values, as they indeed do in the thermal

state. Both properties indicate that these particular operators cannot be used to probe

chaotic behaviour in the theory via out of time correlators: according to [17] one needs

to probe chaos using operators that have non-vanishing correlators and vanishing thermal

expectation values. Holographically, this implies that one will need to include additional

scalar ﬁelds with appropriate couplings to the running scalar to explore chaos.

In the SYK model (before Gaussian averaging) the ﬂow is not driven by a single

operator but by multiple four fermion operators ψ

, each with a coupling λ

ijkl

This prompts the question of how to realise generalised conformal structure with multiple

scalars holographically. In section 5 we construct two dimensional models with multiple

running scalars and show that these can again always be interpreted in terms of AdS gravity

theories compactiﬁed on (non-integer) dimension tori.

The uplifted dimension (2σ + 1) is determined by the thermodynamics of the theory

e.g. the entropy scales with temperature as T

2σ−1

. Hence (even for multiple dilatons)

linear scaling of the entropy with temperature picks out a parent AdS

structure in the

bulk realization of SYK.

The SYK model after Gaussian averaging has equal sources J

for bilocal operators

i.e. the trace Ward identity is

H =

dtO(t) =

(t) ≡

)

) (1.4)

where the operators

denote collectively ψ

i.e. the sum over A is equivalent to

summing the indices (i, j, k, l) over N .

Here the driving term can either be thought of as a single operator O, dual to a single

bulk scalar φ, or as a set of operators O

(with identical sources) dual to a set of scalars

. The required Ward identities can be realised holographically using

I = −NJ

√

(R + 2) = −N J

√

(R + 2) . (1.5)

– 3 –

JHEP01(2018)010

As we show in section 5, both possibilities for the action result in a bulk geometry that ad-

mits an uplift to AdS

. The two possibilities cannot be distinguished if one is interested in

questions involving only the metric/scalar sector. If however one includes additional bulk

ﬁelds, corresponding to other operators in the SYK theory, then one needs to take into ac-

count that these bulk ﬁelds will in general have diﬀerent couplings to each of the scalars φ

These couplings are determined by the OPEs of the operators

with the operator

under consideration. We argue in section 7 that the requirement of preserving the gener-

alized conformal structure constrains the interactions between generic bulk ﬁelds and the

metric/running scalars in the holographic action. These constraints are such that there is

always a parent AdS

description of the theory.

Next we proceed to discuss the characterisation of chaos via out of time four point

functions in a holographic realisation of a non-conformal theory with generalized conformal

structure. We consider holographic non-conformal theories in general dimensions, thus

including not just putative duals to SYK but also non-conformal Dp-branes, fundamental

strings etc.

The bulk action is then of the generic form [25]

I = −N

d+1

√

γφ



R + β(∂φ)

+ C



(1.6)

where (γ, β, C) are related in such a way that the equations admit AdS

d+1

solutions with

the scalar running as α log(ρ) and (a, b) are constants. This action can be uplifted to an

AdS action in (2σ + 1) dimensions where 2σ = d −αγ:

I = −N

2σ+1

√

G (R(G) + 2σ(2σ + 1)) . (1.7)

Here L

is interpreted in terms of the volume of the (2σ −d)-dimensional torus on which the

theory is compactiﬁed and L can be interpreted as the length scale set by the dimensionful

coupling i.e. the length scale associated with the coupling Φ

appearing in (1.1).

The parameter N is related to the number of degrees of freedom in the dual theory.

In top down realizations, all terms originating from supergravity will have the same N

scaling, with loop and α

corrections subleading in N . The constants (a, b) relevant for

non-conformal Dp-branes and fundamental strings can be found in [25]. For the conformal

cases (β = γ = 0) of M5-branes, D3-branes and M2-branes, the constant a is 3, 2 and 3/2

respectively.

As we discussed above, probing chaos requires scalar operators that do not acquire

thermal expectation values i.e. we need to include additional ﬁelds to the graviton and

running scalars. Let V and W be such generic scalar operators that do not acquire thermal

expectation values, with corresponding bulk scalar realisations ϕ

and ϕ

respectively.

We can then characterize chaos in terms of the normalized out of time four point function

f(t) =

Tr (yV (0)yW (t)yV (0)yW (t))

Tr (y

V (0)y

V (0)) Tr (y

W (t)y

W (t))

(1.8)

where y moves an operator a quarter of the way around the thermal circle.

– 4 –

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