Holographic reconstruction of AdS exchanges from crossing symmetry
HJE
Holographic reconstruction of AdS exchanges from crossing symmetry
Luis F. Alday 0 1 2 5 6
Agnese Bissi 0 1 2 3 6
Eric Perlmutter 0 1 2 4 6
Andrew Wiles Building 0 1 2 6
Radcli e Observatory Quarter 0 1 2 6
0 Jadwin Hall , Princeton, NJ 08544 , U.S.A
1 17 Oxford Street, Cambridge, MA 02138 , U.S.A
2 Woodstock Road , Oxford, OX2 6GG , U.K
3 Center for the Fundamental Laws of Nature, Harvard University
4 Department of Physics, Princeton University
5 Mathematical Institute, University of Oxford
6 nite spin , required by crossing. The method
Motivated by AdS/CFT, we address the following outstanding question in large N conformal eld theory: given the appearance of a singletrace operator in the O OPE of a scalar primary O, what is its total contribution to the vacuum fourpoint function hOOOOi as dictated by crossing symmetry? We solve this problem in 4d conformal eld theories at leading order in 1=N . Viewed holographically, this provides a eld theory reconstruction of crossingsymmetric, fourpoint exchange amplitudes in AdS5.
1/N Expansion; Conformal and W Symmetry; Conformal Field Theory

O
2.1
2.2
2.3
2.4
3.1
3.2
1 Introduction
1.1
of [1], largely focused on the question, \Which families of large N conformal eld theories
have weakly coupled, local gravity duals?" The conjecture of [1], which has withstood the
.
ga2p in 4d CFTs [2, 3], and what underlying structures
govern the organization of the CFT data as a whole.
{ 1 {
Still, we are far from a full de nition of \holographicness" from the CFT side, both
in the 1=N and 1= gap expansion. This is true on a basic level. For illustration, consider
the sparsest possible holographic CFTd: the theory of the stress tensor, T , dual to pure
Einstein gravity in AdSd+1. This is, at the least, a consistent subsector of a full edged
holographic CFT with parametrically large
gap at leading nontrivial order in 1=N . The
only light operators are T
and its multitrace composites. What is the lowlying spectrum
of this theory? For d > 2, the answer is not known, even at leading nontrivial order in 1=N .
The same is true for generalized free scalar elds in holographic CFTs, dual to
perturbative scalar elds in AdS, which is the case of interest in this work. Even in the minimal
setting in which O couples only to the stress tensor, we do not know the leading order
OPE data of the doubletrace operators [OO]n;` for general n; ` and
O  in particular,
the anomalous dimensions, n;`, and the leading 1=N correction to the squared OPE
coe cients, an;`
C2
OO[OO]n;`
information about the emergence of the holographic dimension. In the CFT, existence of
Lorentzian bulkpoint singularities [1, 4, 5] and Regge scaling of correlators can be read o
from
n;` at large n; in the bulk, n;` is interpreted as a binding energy of a twoparticle
state, and is intimately related to causality (as we discuss more below). One goal of this
. Both
n;` and an;` are rich quantities that contain essential
paper is to obtain more complete information about n;` and an;`.
A related angle on our work comes from developments in the Lorentzian conformal
bootstrap. In any CFT, crossing symmetry of hOOOOi in the lightcone limit demands the
existence of large spin \doubletwist" primary operators [OO]n;`, with small anomalous
dimensions n;` in the regime `
n [6, 7], see also [8]. In this regime, n;` is a negative,
monotonic, convex function of `. For n = 0, convexity follows from Nachtmann's
theorem [9] and the asymptotic decay
0;`
`
is the lowest nonzero twist in
the O
O OPE. On the other hand, in a CFT with a 1=N expansion, the doubletwist
operators exist for all `, with
n;` suppressed by powers of 1=N instead of 1=`. A natural
question is to understand the behavior of n;` in large N CFT as a function of n and `:
in particular, we would like to understand whether negativity, monotonicity and convexity
persist down to
nite `. The few known results for n;` from topdown computations in
supergravity suggest that this may be the case for all n [10{16]. Moreover, bulk causality
constraints on scattering through shock waves implies that n;` < 0 in the highenergy,
largespin regime n; `
1 [
2, 17, 18
]. And so we ask: in what kinds of large N CFTs do
negativity, monotonicity and convexity of n;` hold for nite n and `?
It may seem surprising that we lack a complete picture of holographic CFT OPE
data at leading order in 1=N , since the AdS amplitudes are largely known. For treelevel
scattering of external scalars in AdS, there are known expressions for arbitrary fourpoint
contact and exchange amplitudes, in both position space (e.g. [10, 19{22]) and Mellin space
(e.g. [21, 23{28]). While all OPE data is, in principle, contained in these known amplitudes,
there is no known systematic way1 to extract it for arbitrary quantum numbers of the elds
1For OPE data at n = 0, there is a formula in Mellin space [
21
], and evaluating it analytically for generic
requires techniques recently developed in [29]. For n > 0 there is no Mellin formula. In position space,
one can apply brute force methods to exchange amplitudes. But as in Mellin space, techniques do not exist
for generic
, where the simplest known form of the amplitude involves an in nite sum of Dfunctions or
a contour integral with in nitely many poles. For
for all n, as we will do in appendix D.
2 Z, however, one can nd results in position space,
{ 2 {
HJEP08(217)4
AdS amplitude due to ' ;s exchange. O is the boundary operator on the external legs, and O ;s
is the operator dual to ' ;s. In this paper, we solve for the righthand side of this equation using
CFT crossing symmetry at large N .
involved: while the decomposition of individual exchange diagrams into CFT conformal
blocks of the same channel is understood [30], it is not understood for crossedchannel
blocks. Alternatively, we do not know the crossing kernel for conformal blocks in arbitrary
spacetime dimension (but for recent progress, see [31{34]).
In light of this, the bootstrap approach to elucidating holography is especially powerful,
and begs the inverse question, posed in [1] but left unsolved: given some spectrum of
singletrace operators in a large N CFT, can we derive the doubletrace OPE data purely from
the CFT side, thus reconstructing the bulk amplitudes without using gravity? In this
paper, we provide an a rmative answer to this question for 4d CFTs. By solving crossing
symmetry for hOOOOi at leading order in 1=N in the presence of a singletrace exchange,
we fully reconstruct the dual crossingsymmetric AdS exchange amplitude. Our results
apply to singletrace operator exchanges in any large N CFT, not only those with local
bulk duals, though they have interesting consequences for the latter.
1.1
Summary of results
We consider the following CFT problem, at leading nontrivial order in 1=N . (See gure 1.)
Consider two singletrace operators: a scalar primary O, of dimension
, and a spins
primary O ;s, of twist . Suppose that hOOO ;si 6= 0. What is the contribution of O ;s to
n;` and an;` for the doubletrace operators [OO]n;`? In what follows we will often use the
{ 3 {
n;` ( ;s) = the contribution to n;` due to O ;s exchange at order 1=N 2
(1)
an;` ( ;s) = the contribution to an;` due to O ;s exchange at order 1=N 2
(1.1)
n;` ( ;s) and a(n1;`) ( ;s)?
This maps to the following AdS dual problem. Suppose there exists a bulk vertex of the
form
' ;s, where
and ' ;s are dual to O and O ;s, respectively. At treelevel in AdS,
this contributes a sum of three exchange diagrams, one from each channel, to the fourpoint
amplitude that computes hOOOOi. What is the total contribution of these diagrams to
Our method here is to solve crossing symmetry at leading order in 1=N , starting from
the large spin perturbation theory recently introduced in [35, 36].3 By utilizing \twist
conformal blocks," which sum up in nite towers of conformal blocks of identical twist, one
can e ciently solve the crossing equations. Working exclusively in 4d CFT, we provide
and demonstrate an algorithm for the complete solution of n;` ( ;s) and an;` ( ;s), for
arbitrary
,
and s. The use of twist conformal blocks allows us to both improve upon
the techniques of [38], and to extend to n > 0. For the present paper we focus mainly on
eveninteger twist , working out several examples explicitly, with extra simpli cations at
(1)
= 2. (See section 2.3.)
The anomalous dimensions organize themselves into a sum of two pieces:
nas;`, is the \asymptotic" piece coming from resummation of large spin
perturbation theory; this is an analytic function of `. The second piece, n;n`, is the \ nite"
piece required to furnish a full solution to crossing, which has support only for `
s. The
OPE coe cients, a(n1;`), also can be written as a sum of two pieces:
and n;` [1]. For truncated solutions to crossing corresponding to AdS contact interactions,
a^(n1;`) = 0, as found experimentally in [1] and proven in [40]. Having derived the nite n
data, we are now able to answer the question  negatively  of whether this relation
holds in the presence singletrace exchanges. In particular, we nd that ^a(n1;`) ( ;s) = 0 only
for
= 2; 3; : : : ; =2 + 1 + s. That it holds at all for these values of
, all the way down
to ` = 0, is fairly remarkable in light of the
nite pieces in eq. (1.2). At n
1, deviations
from the derivative relation appear to be suppressed as ^a(1)
n
prediction, that is consistent with bounds from eikonal gravity calculations [
17, 18
].
With our solutions in hand, we may now extract their physical consequences for
holographic CFTs and AdS physics. We focus here on two aspects:
2We use a(n0;)` to denote the mean eld theory squared OPE coe cients, hence the superscript on an;`.
We sometimes drop the ( ;s) su x to reduce clutter, if the risk of confusion is low.
3This is built on the algebraic approach developed in [37, 38]. See [39] for a related approach.
{ 4 {
a) Highenergy limits: at n
1, we are probing high energies in the bulk. It is a
matter of series expansion to study our solutions at large n. We content ourselves with
an expansion to rst subleading order in 1=n. For n=` xed, this is the Regge regime; for
` xed, this is the bulkpoint regime. In each case, this yields the rst CFT derivation
of both the leading and subleading asymptotics. In the Regge limit, our leading order
result matches the bulk computation of n;` as an eikonal scattering phase in [
17, 18
],
and reproduces the full structure of the AdS bulktobulk propagator found there; the
subleading term is a new prediction.4 (See eq. (3.11).) In the bulkpoint regime, our leading
order result is the rst derivation of any kind that applies for nite `; the dependence on `
is extremely simple, n 1;` ( ;s)
(` + 1) 1. (See eq. (3.24).) The subleading term is also
new; upon insertion into the conformal block decomposition of the full correlator hOOOOi,
it gives a prediction for the subleading correction to the bulkpoint singularity, and can be
thought of as encoding the leading \ nite size" correction to the at space Smatrix due
to the nonzero AdS curvature.
b) Negativity, convexity and causality in AdS: by studying our eventwist solutions,
we amass strong evidence that the contribution of O ;s to the leading large N anomalous
dimension obeys the following properties:
Negativity :
Monotonicity :
Convexity :
spin s
from s
We are viewing n;`>s ( ;s) as an analytic function of `, even though ` is integral. Some
representative plots can be found in
gures 3 and 4. We emphasize that these results hold
for
nite `, and for all n, going well beyond the purview of the original, leadingorder
lightcone bootstrap. This may be thought of as a \large N Nachtmann's theorem" 
that is, an extension to arbitrary n and ` of the conclusions of the lightcone bootstrap,
made possible by the presence of the small parameter 1=N . For `
s, various behaviors
are possible based on the sensitivity of
investigations indicate that the stronger negativity property
n;n` to s and to the value of
. Preliminary
n;`6=s ( ;s) < 0 may be true,
but we postpone a fuller investigation of these sporadic phenomena to future work.
Of special interest is the universal contribution due to the stress tensor, which
computes the gravitational contribution to binding energies in AdS [
6, 7, 42, 43
]. The explicit
solutions for n;` T and an;` T can be found in eqs. (2.49){(2.51) and eq. (2.60), respectively.
A holographic CFT with
gap ! 1 has a sparse singletrace spectrum of bounded
2. The total n(1;`) at leading order in 1=N is thus a
nite sum of contributions
2 operators. There may also be a
nite set of terms contributing only to `
2
 dual to contact interactions in AdS  where this upper bound is the condition that the
chaos bound be obeyed without spoiling bulk locality [44]. Therefore, we have shown that
4The leading order result can also be derived by solving the crossing equations directly in the Regge
regime [41]. We thank those authors for discussions.
{ 5 {
2
4
6
…
HJEP08(217)4
n;` from individual singletrace operators O ;s is a negative,
monotonic, convex function of ` for ` > s. In holographic CFTs with
gap ! 1, there is a
nite
number of such contributions, all with spin s
2, thus yielding the above behavior. For ` = 0; 2,
various behaviors are possible, due to nonanalytic contributions.
the total anomalous dimensions n;`>2 are negative, monotonic and convex in holographic
CFTs with weakly coupled, local gravity duals. This is depicted in
gure 2.
A corollary to this is that, still assuming unitarity, n;`>2 > 0 is only possible in a theory
containing higher spin singletrace operators. We also know that such theories must have
in nite towers of higher spin operators, organized into Reggelike trajectories [2, 34, 44{47].
Therefore, the only way
n;`>2 > 0 is possible is if a suitably regularized resummation of
an in nite set of negative contributions yields a nonnegative result.5 Said another way, if
n;`>2 > 0 for at least one pair (n; `) in a given large N CFT, its bulk dual is nonlocal.
The connection between negativity and convexity of n;`, and causality properties of
AdS gravity, was made in [2, 6, 7, 17, 18, 46, 49{52] in the context of gravity coupled to
massive particles. For n
1 and `
1, the twoparticle state in the bulk is approximated
by a pair of particles following null geodesics coming in from in nity. The impact parameter
b in this scattering process is, in AdS units,
e
b
1 +
`
+ n
(1.5)
In the large spin regime `
n, the particle separation becomes much greater than the AdS
scale. In the Regge regime, eb
1 + `=n, which corresponds to highenergy, xed impact
parameter scattering. In both cases,
n;` is proportional to the scattering phase, which is
constrained to be positive by causality [
2, 17, 18
]. As n and ` decrease to nite values, the
overlap between the individual particle wave functions becomes signi cant [
42
], and we can
5This has recently been shown to happen for n = 0 in the 3d O(N ) vector model and its
ChernSimonsmatter cousins [48].
{ 6 {
no longer approximate their trajectories by geodesics. Nevertheless, our results show that
the above picture of gravitational interactions continues to hold in this regime, thanks to
large N : unitarity and crossing symmetry imply that the exchange of ' ;s gives an order
GN contribution to the binding energy that is a negative, monotonic and convex function
of `, all the way down to small spins and low centerofmass energies. These features of
n;` thus give a satisfying holographic demonstration that classical bulk forces mediated
by evenspin
elds, such as the graviton, are attractive down to the AdS length scale and
fall o
monotonically with distance.
The rest of this paper is organized as follows. In section 2 we set up and solve the
crossing problem. We give various explicit examples along the way, including results for
stress tensor exchange. In section 3, we analyze the results and discuss holographic
applications. We conclude with a discussion of future problems in section 4. Appendices A{C
contain technical material needed for section 2, as well as a handful of explicit formulas for
xed twists. Appendix D makes contact with previous literature on AdS amplitudes for
exchanges, by extracting
and comparing (successfully) to our results.
n;` and a(n1;`) from positionspace amplitudes for various
Singletrace exchange in holographic CFTs
Consider a generic CFT in four dimensions with a large N expansion. Assume the spectrum
contains a singletrace scalar operator O of dimension
identical operators O takes the form
. The fourpoint function of
(2.1)
(2.2)
(2.3)
(2.4)
where xij = xi
(1
z)(1
z) = xx211234xx222234 . The correlator admits a decomposition in conformal blocks
xj and we have introduced the conformal crossratios zz = xx221123xx222344 and
where the sum runs over primary operators present in the OPE O
O. Each primary, which
we denote Oi, is labelled by its twist i =
is weighted by the square of the OPE coe cient, ai
i `i and its Lorentz spin `i. Each contribution
C2
OOOi , and the conformal blocks
have been written so as to make their small z; z behaviour explicit. In four dimensions,
g ;`(z; z) =
z`+1F =2+`(z)F 2 (z)
z`+1F =2+`(z)F 2 (z)
2
2
z
z
where
hO(x1)O(x2)O(x3)O(x4)i = G(z; z)
Expanding in conformal blocks, the set of intermediate operators is comprised of the
identity and the doubletrace operators [OO]n;` of dimension
n = 0; 1; 2; : : : and ` = 0; 2; 4; : : :, with corresponding squared OPE coe cients a(n0;`). The
explicit form of these OPE coe cients can be found in eq. (D.3). Next, let us consider
n;` = 2
+ 2n + `, where
1=N 2 corrections to the GFF result,
consistent with crossing symmetry. These corrections arise from two sources. First, the
CFT data corresponding to doubletrace operators gets corrected,
G(z; z) = G
where a(n0;`) is given in eq. (D.3). In addition, new \singletrace" operators may arise in the
OPE O
O.
As argued in [1], if no new operators are exchanged at order 1=N 2, then all solutions to
crossing have nite support in the spin. These truncated solutions have been constructed
in [1] and we will denote their contribution to
consider the presence of a new exchanged operator, of twist
n;` as ntr;`. In the present paper we will
and spin s. Schematically,6
O
O
1 + X [OO]n;` +
n;`
1
N O ;s +
Our goal is to solve crossing symmetry, given the presence of O ;s in this OPE. In this case
the situation is quite di erent. The correlator now contains the following term
is the standard hypergeometric function. We take crossing symmetry to act as z $ 1
z
in the crossratios. For the fourpoint correlator it implies
1
z
z
z) :
(2.5)
In this paper we will assume for simplicity that
does not depend on N . At in nite N ,
the correlator is that of mean eld theory, i.e. generalised free elds (GFF): one has a sum
of three disconnected contributions,
G
(0)(z; z) = 1 +
(1
zz
z)(1
z)
+ (zz) :
(2.6)
where a ;s is the leading contribution to the squared OPE coe cient with which the
singletrace operator is exchanged,
6Henceforth we leave implicit the bounds on sums over n and `, with nonnegative integer n and `=2.
G
a ;s(zz) =2g ;s(z; z)
z) :
(2.13)
For noninteger 2
this term can only be obtained from an in nite sum of terms on the
l.h.s.,7 so that crossing symmetry implies solutions with in nite support in the spin [6, 7].
We will use the method developed in [35, 36] to compute the CFT data, and in particular
the anomalous dimensions, to all orders in inverse powers of the spin. This series resums
into an analytic asymptotic answer which we denote
solution to crossing, generically we will need to supplement this asymptotic expression by
n
a piece with nite support in the spin, denoted by n;`. The nal answer takes the form
nas;`. In order to obtain an exact
HJEP08(217)4
We will nd that n;n` is di erent from zero only for `
s.8 Similar considerations apply
to the OPE coe cients a(n1;`). In addition, there always exists the freedom to add a
homogeneous solution to crossing, which contributes a truncated piece ntr;`, as explained above.
2.1.1
A Mellin perspective
The Mellin representation of AdS amplitudes [23, 26, 53] provides a fruitful perspective on
why n;` takes the form eq. (2.14). The Mellin amplitude M (s; t) may be de ned by the
double integral transform
(2.14)
(2.15)
(2.16)
t. (We hope there is no confusion between the spin and the Mellin
variable s.) In this convention, crossing means M (s; t) = M (s; u) = M (t; s). The exchange
of a bulk eld ' ;s, dual to O ;s, contributes to M (s; t) as
M ;s(s; t)
a ;s
Subscripts refer to the spin s, not the Mellin variable: the numerators are Mack polynomials
of degree s, and Rs 1(s; t; u) is a totallysymmetric degree(s
1) polynomial. All channels
are summed over explicitly. To compute n;` we develop the conformal block decomposition
7For instance, a divergent term can be generated by acting with the Casimir operator on (1
z) ,
provided
is noninteger.
8In the language of [35, 36], these two contributions will produce enhanced and nonenhanced terms,
with respect to a single conformal block. In the language of [39], they come from the \Casimirsingular"
and \Casimirregular" terms, respectively.
{ 9 {
in a given channel  say, the schannel for concreteness  and evaluate on the poles at
s = 2
+ 2n.9 There are three kinds of contributions:
1) Crossed channel poles (t and uchannel) contribute to all `. In our calculation of
n;`, these are the pieces we compute by resumming the large spin expansion, nas;`.
It is clear here that they are not crossing symmetric.
2) Direct channel (schannel) poles, evaluated on s = 2
+ 2n, become degrees
polynomials, contributing only to `
s. In our calculation of n;`, these are the nite
n
pieces, n;`. They are also not crossing symmetric.
3) Rs 1(s; t; u) contributes only to `
s
1, and is crossingsymmetric. In our
calculation of n;`, these are the truncated pieces, ntr;`, that may be present for `
s
From the AdS perspective, their presence signals contact terms in the spins
bulktobulk propagator [
21
].
In the following we will work out nas;` and
particular, the exchange of the stress tensor.
2.2
Asymptotic anomalous dimensions
n;n` for several examples. This includes, in
The analyticity properties of the sum (2.2) around z = 0 imply that the piece proportional
to log z arises solely from the anomalous dimension n;`. More precisely,
G
As explained above, in the case of the exchange of a singletrace operator, this sum should
reproduce certain divergences: speci cally, eq. (2.13) implies that
n;`
X an;` 2
(0) n;` (zz) n=2g n;`(z; z)
div
=
(zz)
((1
z)(1
z))
where on the left we keep only the \divergent" part as z ! 1, i.e. the contribution that
cannot be obtained by a
nite number of conformal blocks. This includes all noninteger
powers of (1
z). Equation (2.20) is our crossing equation for n;`.
Following [35, 36], we can e ciently compute n;` to all orders in 1=` using eq. (2.20).10
First we introduce the following family of functions which we denote twist conformal blocks,
9For n = 0 one can use the explicit integral transform of [
21
] to derive 0;`. For higher n, there is no known
explicit analog of this formula (which was one motivation for this work), but our discussion still applies.
10Notice that in principle, solutions to the crossing equations whose anomalous dimensions decay faster
than any power of `, for instance e k`, can be added to n;`. We are not considering this situation in our
paper.
(2.18)
(2.19)
log z
(2.20)
(2.21)
where J 2 = (` + n +
1) is the corresponding conformal spin. Note that it
depends on ` and n, but we are suppressing this dependence in our notation. Assuming
n;` admits the following expansion
n;` = 2 X BJ m2m;n ;
m
equation (2.20) turns into
equation
m;n
XBm;n z z
z +n
X
m
n
X Bm;nHn(m)(z; z)
div
=
(zz)
((1
z)(1
z))
=2 a ;sg ;s(1
z)
log z
(2.23)
Note that we are not free to determine the support of m: crossing symmetry will dictate
this support for us. The explicit expression for the conformal blocks in eq. (2.3) implies
the following factorization property for the divergent part of twist conformal blocks:
Hn(m)(z; z)
div
=
z +n
z
z
F +n 1(z)Hn(m)(z)
Plugging eq. (2.24) into eq. (2.23) and matching powers of z and 1
z, we obtain a nice
structure: the factorization properties of the conformal block on the r.h.s. of (2.23) allows
us to write the anomalous dimension as
n;` =
In other words, Bm;n obeys the factorization property
Bm;n =
2(n)b(m+2s)(n)
+2s(n)b(m 2)(n)
To see this more clearly, let us focus on the rst term contributing to the fourdimensional
conformal blocks (2.3). Then (2.23), together with the factorised form (2.24), leads to the
(2.22)
(2.24)
(2.25)
(2.26)
logz
(2.27)
F +n 1(z)Hn(m)(z) =
a ;s (zz)
((1 z)(1 z))
=2
(1 z)s+1F =2+s(1 z)F 2 (1 z)
2
z z
We see that the dependence on z and z factorises on both sides of the equation. By writing
Bm;n =
2(n)b(m+2s)(n) we obtain two independent equations for
2(n) and b(m+2s)(n).
First, by looking at the z dependence and for each xed n, we obtain
X b(m+2s)(n)Hn(m)(z) =
n
m
z
1
z
(1
z) =2+s+1F =2+s(1
z)a ;s
(2.28)
where the factor
n, given in eq. (A.8), has been included for later convenience. By
matching the powers of 1
z using the explicit form eq. (A.12) of Hn(m)(z), we see that
2
2
m =
+ s + 1;
+ s + 2;
:
(2.29)
X
HJEP08(217)4
By expanding both sides in powers of z, we can nd the coe cients
2(n). In appendix B,
we derive a contour integral representation of
2(n) valid for all , given in eq. (B.3).
Finally, including also the second term in the conformal blocks (2.3) will lead to the
result (2.26). This in turn will lead to (2.25) where the functions f 2(n; J ) are de ned
By using the explicit form of the functions Hn(m)(z) and expanding in powers of 1
can compute arbitrarily many b(m+2s)(n). Having done this, the z dependence of (2.27)
leads to an equation for
2(n). Using
F (1
z) z 1
(2 )
2( ) 2F1( ; ; 1; z) log z + (regular at z = 0)
f 2(n; J ) = 2 X bm
m
( 2)(n)
J 2m ;
These are all of the ingredients needed to solve the problem for general .
The resulting expressions simplify when
is an even integer. First, with our choice
of normalisation,
This leads to a cleaner factorisation for n;` for the case of the exchange of an operator of
twist two. Second, one is able to nd f
2(n; J ) in a closed form. The explicit results for
several even values of , and generic values of
are given in appendix B. In all cases the
dependence in n and J further factorises to yield11
where
Any full edged CFT contains the stress tensor. Furthermore, by conformal Ward
identities it follows that the stress tensor couples to two identical scalar operators with
squared OPE coe cient
where cT / N 2 is the central charge appearing in the stress tensor twopoint function.
Hence, any complete treatment of a large N CFT at order 1=N 2 must include the stress
tensor. This is the motivation to focus on the case of twisttwo exchange in what follows.
11In the expressions above the dependence on
is kept implicit. See appendix B for the details.
aT =
2
4
9 cT
(2.30)
(2.31)
(2.32)
(2.33)
(2.34)
(2.35)
We now focus on nding the full solution to crossing symmetry  that is, both the
asymptotic and
nite parts of eq. (2.14)  for the exchange of operators with
= 2 and low
values of the spin, with special focus on the stress tensor at s = 2.
The asymptotic part of the anomalous dimension can be computed as described above.
Using the explicit results given in appendix B one arrives at the following expression
where 2+2s(n) is a polynomial of degree 2s. We would like to complete the above
asymptotic solution to an exact solution of crossing symmetry. We will assume the solution has
the following form
n;` = nas;` + n;`
n
where
n;n` has nite support in the spin. In order to nd
n;n` we employ the following
strategy. The structure of the conformal partial wave expansion, together with crossing
symmetry, imply the following analytic structure for the fourpoint correlator around z = 0
and z = 1:
G
(1)(z; z) = 0(z; z) log z log(1
z) + 1(z; z) log z + 2(z; z) log(1
z) + 3(z; z) (2.38)
where the functions i(z; z) do not contain logs in a small z; 1
z expansion.
0(z; z)
receives only contributions from the anomalous dimensions. More precisely,
0
n;`
1 X a(n0;`) nas;`(zz) n=2g n;`(z; z) +
n;`
2
1 X an;` n;n`(zz) n=2g n;`(z; z)A
(0)
1
On the other hand, crossing symmetry implies
(1
z)
z
z
(1
z)
We now set out to solve this equation for n;n`. Plugging in the conformal blocks in eq. (2.3),
we can easily determine the piece proportional to log(1
z) of all sums contributing to
0(z; z) except for the contribution
The reason is that in order to extract the behaviour at z = 1, we rst need to perform the
in nite spin sum, and then expand. This sum is explicitly computed in appendix C. In
order to simplify what follows let us introduce the notation
k (z)
z 2F1( ; ; 2 ; z);
k~ (z)
z 2F1( ; ; 1; 1
z)
(2.42)
(2 )
2( )
(2.36)
(2.37)
HJEP08(217)4
log(1 z)
(2.39)
(2.40)
(2.41)
In terms of these one nds the expression
+ X a(n0;`) nas;`zzk n=2+`(z)k~ n=2 1(z)
X 2 2+2s(m)zk +m 1(z)S~m(z)
1
m=0
where S~n(z) is the piece proportional to log(1
in appendix C this is given by
z) in the sum Sn(z) near z = 1. As shown
S~m(z) =
X
n;`
m;n
n+`;m 1 a(n0;`)zk~ +n 1(z)
where
m;n = (n
m)(2
3). Note that for xed m, only a nite number of
terms contributes to S~m(z), due to the Kronecker delta and the nonnegativity of n and `.
We would like to convert (2.40) into a matrix equation. In order to do this we follow [1]
and introduce the projectors
together with the integral
I
Both contours are taken to run counterclockwise. Plugging (2.43) into the crossing
equation (2.40) and integrating both sides against z 3
z) 3k
1 q=2(1
z) around z = 1, we obtain
k
1 p=2(z) around z = 0 and (1
X an;` n;`
(0) n
n;`
n+`;pIn 1;q
( )
n 1;pIn+`;q
( )
+ X an;`
(0)
n;`
as
n;`
2 2+2s(p + 1)
p+1;n
( )
n+`;pIn 1;q = (p $ q)
This must hold for all nonnegative integer (p; q). This should be viewed as an equation
n
for n;`. The second line arises from
would otherwise be homogeneous in
nas;` and acts as a source term for the equation, which
n
n;`. For a given (p; q) each sum reduces to a
nite
number of terms, due to the Kronecker delta functions. We have solved this equation for
several values of s. In all cases, n;n` is nonzero only for ` = 0; 1;
; s.
Let us give the explicit results for two examples:
Exchange of scalar operator of dimension two.
The asymptotic part of the solution
takes the form (2.36) with 2(n) = 1. The total solution is of the form eq. (2.37), with
n;n` =
(
(
+ n
1)2(n + 1)(2
+ n
1)(2
+ 2n
3)(2
3)
+ 2n
1) a2;0
for ` = 0
(2.48)
and zero for ` > 0. One can explicitly see that both terms contributing to n;0 scale as
n;`
1=n for large n. Requiring this behaviour for large n prohibits the addition of a
truncated solution to crossing ntr;`, but in principle, crossing allows for this ambiguity.
(2.43)
(2.44)
Now crossing symmetry requires the addition of a nite solution with support for ` = 0; 2.
We nd the following for ` = 2,12
n;n2 = 60aT
(n + 1)(n + 2)(n + 3)(
3)(2
+ n)(2
+ n
1)(2
3)(2
+ n
2)(2
+ n
1)
+ 2n + 1)(2
+ 2n + 3)
Together with eq. (2.36), these make up the total contribution to n;` from stress tensor
exchange. Holographically, this computes the contribution to twoparticle binding energies
from their gravitational interactions.
As always, we are free to add truncated solutions to crossing. Note that in this case,
even assuming a speci c large n behaviour, there exists the freedom to add the truncated
solution to crossing with support only for spin zero [1],
ntr;0 =
(n + 1)(
+ n
1)(2
+ n
3)
(2
+ 2n
3)(2
+ 2n
1)
Exchange of stress tensor. In this case the asymptotic part of the solution takes the
form (2.36) with
with any overall coe cient.
We now turn our attention to the computation of order 1=N 2 corrections to the OPE
coe cients, a(n1;`). Together with the anomalous dimensions, these comprise the full solution
for the correlator at this order.
When no singletrace operators are exchanged, the only solutions to crossing are the
truncated solutions ntr;` with
nite support in the spin, and the corrections a(n1;`) are given
in terms of the anomalous dimension by a remarkable formula
(0) tr
:
This is known as the \derivative relation." This relation was found in [1] and proven in [40]
for the case of truncated solutions. Our aim is to understand whether such a relation still
holds for the exchange of singletrace operators, and if it doesn't, what is the correct
expression for a(n1;`). We nd it convenient to split a(n1;`) as follows
12Here and throughout, we use the more physical notation aT
a2;2 in the case of the stress tensor, and
likewise for other quantities with a 2,2 subscript.
for G(z; z) at order 1=N 2 gives
As we will see, the technology introduced above will allow us to compute ^a(n1;`) to all orders
To set up the problem, recall that expanding the conformal partial wave decomposition
G
g2 +2n;`(z; z) :
(2.55)
Plugging (2.54) into this expression and assuming that ^a(n1;`) admits the following expansion
HJEP08(217)4
a^(n1;`) =
1
X
we can write G(1)(z; z) in terms of twist conformal blocks and their derivatives as
G
Bm;nHn(m)(z; z) + dm;nHn(m)(z; z)
:
The crossing equation in the presence of an exchanged operator then implies
X
m;n
Bm;nHn(m)(z; z) + dm;nHn(m)(z; z)
div
=
((1
a ;s(zz)
z)(1
z))
=2 g ;s(1 z; 1 z)
To derive Bm;n, we focused on the term proportional to log z;13 now we focus on the piece
without a log z, which will lead to an equation for dm;n. As before, having computed the
twist conformal blocks for all n the above equation can be solved by expanding both sides
in powers of z and (1
z). The computation is tedious but straightforward. We have
carried out this procedure for the exchange of operators of
= 2 and s = 0; 2, for generic
external . For integer
the results can be written as follows
(1)
a^n;` (2;0) =
(1)
a^n;` T =
3
3
X
k=0
X
k=0
(J 2
(J 2
a2;0(2k + 1)(
k(k + 1))(n +
+ k
1)2
1)(n +
aT (2k + 1)(
1)2P4(k; )
k(k + 1))(n +
+ k
1)(n +
k
k
2)
2)
where P4(k; ) is a fourth order polynomial in k given by
P4(k; ) =
30
2(19 6k(k +1))+12 (2k(k +1) 1)+6k(k2 1)(k +2)
(
1)2 2
For an exchanged operator of
= 2 but generic spin s, the results are of the same form but
with P4(k; ) ! P2s(k; ). We have written the answers in terms of the conformal spin
J 2 = (` + n +
1). Although this is the answer for integer , the sums can
13The l.h.s. produces a log z when the derivative hits the factor z +n in Hn(m)(z; z).
(2.56)
(2.57)
(2.58)
(2.59)
(2.60)
(2.61)
be performed exactly, and the full answer, for arbitrary
, can be expressed in terms of
digamma functions. The resulting expressions, however, are too cumbersome to be shown
here.
These results exhibit some interesting features which we now discuss. For the general
case of exchange of an operator of even twist , we have checked that
a^(n1;`) = 0 for
= 2; 3; : : : ; =2 + 1 + s :
This is evident in equations (2.59){(2.61), upon noting that P4(0; ) = 0 for
and P4(1; 4) = 0.14 Furthermore, note that for the exchange of an operator of
arbitrary s, the fallo
with large n goes like n 4. For generic twist we expect
a^(1)
n 1;`
This agrees with all the cases we have explicitly checked. For instance, the case
= 4; s = 0
with
= 4 may be found in equation (D.8). We have checked several other cases with
higher twist and spin.
We end this section by making the following important remark. We have found the
expression for a^(n1;`) to all orders in 1=`. The nal expression for the correction to the OPE
coe cient takes the form
where a^(n1;`) is an analytic function of the spin. In principle crossing could demand the
addition of extra terms with nite support in the spin. However, in all the cases we have checked
this is not the case. In particular, this expression, with ^a(n1;`) given above, is valid for all
values of the spin, provided the full anomalous dimension n;` is used inside the derivative.
3
Applications to AdS physics
As explained in the introduction, the results of the preceding section provide a complete
CFT reconstruction of the full crossingsymmetric exchange amplitudes in AdS. Our results
are thus guaranteed to reproduce all features of these amplitudes. We now use them to
clarify and derive some new properties of treelevel AdS physics: namely, the leading and
subleading behavior of anomalous dimensions n;` in the Regge and bulkpoint regimes, and
the behavior of n;` for nite n and `. The former are intimately related to the emergence
of bulk locality; the latter, to bulk causality.
Before proceeding, we note that while the method of this work applies to any twist,
we have focused on nding explicit results for
2 2Z+. While the formulas of this section,
namely the leading and subleading behavior of n;` in the Regge and bulkpoint limits, are
14In the case of N = 4 superYangMills, the supergravity contribution to a(n1;)`, where O is the
= 2, 12
BPS operator in the 200 of SU(4)R, satis es the derivative relation, as observed in [16, 27]. Even if it seemed
accidental, we now see why this happens: at large 't Hooft coupling, every operator in the O200
O200
OPE has even twist.
derived with help from the even twist results, they appear to hold for generic . It would
be satisfying to check this directly at generic .
For convenience, we recall our notations for the various parameters on which
depends:
and
: conformal dimension of the external scalar O
(n; `) : quantum numbers of [OO]n;`
( ; s) : twist and spin of the exchanged operators O ;s
n;` ( ;s) = the contribution to n;` due to O ;s exchange
n;`
(3.1)
(3.2)
(3.3)
(3.4)
(3.5)
n 2 .
gap,
(3.6)
The function f , de ned in eq. (2.32), has the following largespin asymptotics:
f2X (n; J
1)
J 2(X+1) 2(
2
( )
1
X)
+ O(J 2(X+2))
The second term of eq. (2.25) dominates, and we recover the results of the lightcone
bootstrap for arbitrary n [6, 7, 54{56],
`
1
1 ; n xed
n;` 1 ( ;s)
1
`
+2s(n) 2( )
2(
2
)
where
was de ned in eq. (B.3). Note that eq. (B.3) gives an alternative, more compact,
expression than the sums in [54].
3.2
High energy limits
We now consider the behavior of n;` for n ! 1. This regime probes highly energetic
twoparticle states in the bulk. We compute in turn for `=n
xed and for ` xed; these probe
the Regge and bulkpoint regimes, respectively. We will explicitly compute the leading and
subleading terms of n;`, leaving higher orders as an exercise.
Let us rst brie y comment on the OPE coe cients. In eq. (2.63), we proposed that for
twist exchange, deviations from the derivative relation (2.53) scale as a^(1)
This is a new prediction that should hold independent of the scaling of `. One check is that
it is consistent with previous computations in the Regge limit, which bounded the fallo
to be faster than n (d 2) (cf. footnote 6 of [18]). It also appears possible to understand
this using crossing symmetry directly in the Regge limit [41]. It would be interesting to
prove (2.63) directly, if true. Note that for heavy singletrace operators with
n
this fallo is highly suppressed.
The following calculations rely on the large n asymptotics of
and f .
only depends
To warmup, we consider the large spin limit, relevant for lightcone physics:
on n, and behaves as
2X (n
1)
n2X 2
(
1)2X 1
42X 1(X
(X
1
2
)
)
)
by looking at the lowest several values of X 2 N. Together with eq. (3.6), this implies that
the second term in eq. (2.25) dominates at large n, through the rst several subleading
orders, for any ( ; s): in the limit (3.7),
Then through rst subleading order, and omitting the ( ; s) subscript for visual clarity,
=
+2s(n)f 2(n; J ) n; `!1
xed
Putting things together, one nds
xed
xed
`
n
2
( )
n2X+2 ( +2)( +1)2X 2 (
1
X)
1+
(
2 ( +( +2)X +1)+ X +2X
)
(3.7)
(3.8)
(3.9)
(3.10)
(3.12)
(3.13)
(3.14)
3
2
+
where we've de ned
with
(2
3) (2s +
2) +
and COOO ;s is the OPE coe cient.
We have written the result this way to facilitate comparison with the leading order
result of [18], as computed using eikonal techniques in gravity. We nd perfect agreement.
If
and ' ;s are dual to O and O ;s, respectively, then
LAdS
;s' ;s 2
;
moving \dimensions"
The Regge limit is
where
In this limit, the result of [18] in d = 4 is
n;` ;
2
h
h
h
2
2
1
h; h ! 1 ;
n;` Regge
`
n
=
xed
;s (4hh)s 1
?(h=h)
?(h=h) =
h
2
with the holographic relation between the cubic coupling
;s and the OPE coe cients
COOO ;s given above (as derived in [
21
]). The CX;Y coe cients set the normalization of
the boundary twopoint function of a dimensionX, spinY operator, as computed from
extrapolation of the bulktobulk propagators. Following [18], de ne the left and
righth =
+ n + ` +
h =
+ n +
n;`
2
(3.15)
(3.16)
(3.17)
(3.18)
(3.20)
(3.21)
(3.22)
This can be seen to match the leading order term of eq. (3.11). In the above expression,
?(h=h) is the bulktobulk propagator on H3 for a eld of dual conformal dimension
1
propagating a geodesic distance log(h=h), which emerges naturally in the eikonal scattering
calculation. Thus, our result (3.11) reproduces the AdS bulktobulk propagator.
The subleading piece of eq. (3.11) is new, and makes a prediction for a bulk calculation.
It is worth explicitly writing the subleading correction due to graviton, i.e. stress tensor,
exchange:
n
2
g
2
T 2 ( + 2)
1 +
2 2(2
3) + (6
n ( + 2)
11)
2
)
(3.19)
Note that the sign of the subleading term can be either positive or negative in a unitary
theory.
3.2.2
Bulk point limit
We consider
In this regime, we nd that
n ! 1 ; ` xed
n!1
2n2X+1(` + 1) 2 (
1
1
2
( )
1
X)
2n
(4X + 2)
+ (4X + 1)`
2(X + 1)
)
Together with eq. (3.6), this implies that the second term in eq. (2.25) dominates at large
n, through the rst several subleading orders, for any ( ; s): in the limit eq. (3.20),
2(n)f +2s(n; J )
+2s(n)f 2(n; J )
n 2s 5
n2s 1
Then through rst subleading order,
Putting things together, one nds
n;` b:p:
=
n!1
n;`>s b:p:
)
exactly for the \asymptotic" part, nas;`.
This holds for arbitrary ` > s. For `
s, we need to incorporate the nite pieces of the
n
full solution to crossing, n;`, which can modify this result. Equivalently, eq. (3.24) holds
While the leading order nscaling was known for general ( ; s), its full `dependence for
arbitrary ( ; s) had never been computed, either from AdS or CFT. We see that it takes a
very simple form. Indeed, if we take the
1 limit of the Regge result eq. (3.11), we nd
n;` Regge;
1
;s
2`
+ : : :
The nite ` correction to this, which gives the bulkpoint result eq. (3.24), is just ` ! ` + 1.
Except for the twistdependence of
;s, the leading order result is independent of the
twist. In a holographic CFT with
gap ! 1, the total leadingorder bulkpoint singularity
is determined by the sum of contributions from all s = 2 exchanges:
n;`>2 b:p:
n
3
2 2(` + 1)
X
O ;2
2
;2 + O(n2) ;
gap ! 1
The subleading term of eq. (3.24) is new. It may be thought of as capturing the
leading correction to the bulkpoint singularity of a holographic CFT fourpoint function.
This may be computed explicitly using the techniques of [1, 4, 5]. Alternatively, because
the at space Smatrix may be obtained from the sum over doubletrace exchanges in the
large n regime [
42
], the insertion of the subleading correction into the conformal block
captures the leading \ nite size" correction due to the curvature of AdS. Note that its sign
can be either positive or negative in a unitary theory.15
3.3
Negativity, convexity and causality
To demonstrate the negativity, monotonicity and convexity properties in eq. (1.4), we take
an experimental approach by taking various slices through the ( ; n; `; ; s) parameter
space. The cases shown here, as well as many other similar checks and plots, provide
convincing evidence that in general, the anomalous dimensions are negative, monotonic
and convex functions of `, for all n and ` > s. The overall picture for holographic CFTs is
given in gure 2. For `
s, the nonanalytic n;n` contributions to n;` can potentially spoil
these properties.16 We table these for now but return to them shortly. We also take as
15For s > 0 exchange where
2 by unitarity, the sign is positive for su ciently large `. On the other
hand, for s = 0 exchange with
< 3=2, the sign remains negative for su
ciently large `.
16These nonanalyticities are related to the limits of applicability of the OPE inversion formula in [34].
(3.23)
(3.24)
(3.25)
(3.26)
10
15
20
25
ℓ
HJEP08(217)4
as we move downwards through the rainbow. The left plot is at n = 1. The right plot is at integer
100
104, where each thick line is comprised of ve individual lines. In all cases, the result is
negative, monotonic and convex.
implied the freedom to add the homogeneous solutions to crossing; in a holographic CFT
that obeys the chaos bound and has a higher spin gap, these can only contribute to n;` 2.
For
= 2 exchange, we can easily prove this. Recall that in this case,
n;`>s (2;s) =
1)2
(` + 1)(2
+ 2n + `
This is manifestly negative, monotonic and convex for all n, assuming unitarity.
For
> 2, both terms in eq. (2.25) contribute, and negativity, monotonicity and
convexity are not obvious.
Nevertheless, these properties still hold. For example, for
massless scalar exchange between
= 3 scalars, one nds
n;`>0 (4;0) =
(` + 1)(` + n + 2)(` + n + 3)(` + 2n + 4) a4;0
24(n + 1)(n + 2)
which is manifestly negative, monotonic and convex. Likewise at
= 4,
n;`>0 (4;0) = 24
1
1
1
1
4
`+n+3
`+n+4
`+n+5
Similar expressions are easily obtained for other
2 Z using the results of appendix B,
which are valid for arbitrary
. In gures 3 and 4, we plot results for
= 4 exchanges as a
function of ` for xed values of n and
. All results are negative, monotonic and convex.
To study the complete solution to crossing, we must include the
nite nonanalytic
n
contributions, n;`. For concreteness, we focus on stress tensor exchange. From the results
eq. (2.36) and eqs. (2.49){(2.51) for generic
, one can check that they are indeed manifestly
negative even for ` = 0; 2, and it is a matter of algebra to check that for all n, n;` increase
monotonically as a function of `, starting from ` = 0. (The reader may nd it useful to
(3.27)
(3.28)
400
600
800
1000
1200
2 × 106
= 4; s = 2 exchange, nas;` (4;2). In both
as a function of
in the right plot.
cases, the result is negative and convex as a function of `. Notice that in this case, n;` (4;2) increases
see the full result specialized to
= 4; 5, given in eq. (D.12) and eq. (D.16).) Thus, we
conclude that for generic
, negativity, monotonicity and convexity hold all the way down
to ` = 0, for all n:
T
exchange; generic
:
n;` T < 0 ;
n;` T
> 0 ;
n;` T
< 0 ; 8 n; `
In a moment we will discuss an exception at n = 0 and small .
The implications of these results for AdS physics, and their relation to bulk causality,
were discussed in the Introduction. Note that even for higher spin exchanges, s > 2, the
causality violation of [2] is not manifest as a \wrong sign" of the anomalous dimension
 indeed, nas;` ( ;s)
re ects the twopronged nature of causality: signals must propagate forward in time, and
inside the lightcone. If either property is not obeyed, a theory is not causal.
0  but rather in its behavior at large n; `, as in eq. (3.11). This
3.3.1
Holographic causality for low spins
For general exchanges, what happens to negativity and convexity upon reinstating
This only a ects n;` s, the opposite regime of the lightcone bootstrap. We now show that
To demonstrate, we can determine the range of
for which the stress tensor
contribution becomes positive:
n;` T > 0. To make the point as clear as possible, take n = 0.
Then from eq. (2.36) and eqs. (2.49){(2.50),
Here we see an odd fact:
0;2 T = 10aT
( 4 3 + 9
+ 7)
12
(4 (
+ 2) + 3)
0;2 T
0 for 1
(3.30)
(3.31)
(3.32)
where the only real solution is
1
12
2
q
3
271 + 9p822 + 3
q
2168
72p822
4
1:41
(3.33)
Perhaps surprisingly, the stress tensor contribution is positive for su ciently small, but still
unitary,
. In an AdS compacti cation that contains a free scalar coupled to gravity 
and perhaps a 4 potential, but no other cubic couplings  this is the only contribution
to 0;2, which is positive despite being due to gravity alone. This shows how one must
be careful in applying arguments relating the sign of n;` to the sign of the gravitational
force at very small n; `. It also gives new credence to the perspective, explored in [43],
that even in a sensiblelooking theory of weakly coupled gravity, anomalous dimensions
can be positive for ` = 0; 2. It remains an open question whether further UV consistency
constraints forbid this.
However, there are fewer possible positive contributions to n;` than rst meet the eye.
In [57] it was argued that for
= 2, the following three properties hold:
0;` (4;s) = 0
0;`6=s ( ;s)
0
0;s ( ;s) > 0 possible only for
< 4
With the results herein, we can address whether these extend to arbitrary n and
4 ! 2 . Here we make only preliminary remarks, deferring an indepth investigation to
the future. First, the generalization of eq. (3.34a) can be checked using our formulas for
f , which indeed has double zeroes
F(2mX)(n; J )(
m)2 + O(
m)3 for` m = 1; 2; : : : ; X + 1
(3.35)
for some functions F(2mX)(n; J ) that are independent of . These zeroes are visible in the
large n limits of eq. (3.8) and eq. (3.21). This implies nas;` (2 ;s) = 0. For several cases, we
n
have also checked that the same holds for n;` (2 ;s). On this basis, we conclude that, at
least for
2 2Z+,
n;` (2 ;s) = 0
Next, we test the generalizations of eqs. (3.34b), (3.34c) for stress tensor exchange.
Eq. (3.34b) indeed holds, as shown in
gure 5. As for eq. (3.34c), it does appear to
extend to arbitrary
, but still requires n = 0. For instance, stress tensor exchange
contributes negatively for n > 0 for all unitary
, in particular, including the range eq. (3.32),
see eqs. (2.49){(2.50). Extending this analysis to other cases, in order to formulate the
strongest possible statement of negativity of anomalous dimensions that is consistent with
all known data, is an intriguing question for future work.
We close with a comment on our result eq. (3.32). In [49], the bound 0;2 < 0 was
proven using CFT causality at leading order in 1=N , but only in the absence of exchange
of the stress tensor or other operators of twist
2 .17 (See [51] for a related proof.) It is
17We thank Tom Hartman for clari cation.
(3.36)
10 Δ
4
6
8
4
6
8
10 Δ
γn,0
1000
2000
3000
4000
5000
6000
γn,0
5000
10 000
15 000
(3.37)
(3.38)
HJEP08(217)4
(a)
(b)
we move through the rainbow. All results are negative. In the right plot, we show only 1
n
5
to make clear that these lines sit below the xaxis.
plausible that, for asyetunknown reasons, 0;2 < 0 must in fact hold in general holographic
CFTs with
gap ! 1. Then eq. (3.32) would imply a nogo theorem for e ective actions
in AdS5: a theory of the form
LAdS = R + 2
1
2
m2 2 +
1
2
would be inconsistent with alternate quantization of the scalar
4
4)
3:64
(mLAdS)2 <
3 ;
We are not aware of any topdown compacti cations containing a scalar sector of the form
eq. (3.37) with mass in the range eq. (3.38). Such a nogo result is only speculation, but it
would be interesting to understand whether 0;2 > 0 is truly possible in a consistent large
N CFT.
4
The main technical result of this work is the construction of full solutions to fourpoint
crossing symmetry in the presence of singletrace operator exchanges, at leading order in
1=N . Together with the work of [1], this completes the program of computing the building
blocks of planar correlators in large N CFTs using crossing symmetry, thus reconstructing
arbitrary treelevel fourpoint AdS amplitudes. It is remarkable how much crossing
symmetry knows about operator products in holographic CFTs and their gravitational images
in AdS.
It would be worthwhile to further explore some of the conclusions in this paper, to check
them more thoroughly for arbitrary internal twist, and to nd direct proofs. Extension to
other spacetime dimensions is clearly possible using the same technology, particularly in
It would also be interesting to formulate a precise connection between possible
contwo dimensions, where the result will take a form essentially identical to one of the terms
in eq. (2.25).
Having now understood treelevel exchange in AdS using crossing symmetry in CFT, we
can consider holographically computing oneloop AdS amplitudes using crossing symmetry
at order 1=N 4, using the method of [29]. That work did not include the e ects of
singletrace exchange, precisely because knowing the data derived in this paper is a prerequisite to
such a calculation: the OPE data at order 1=N 2 acts as a source in the crossing equations
at order 1=N 4. In particular, we can now consider the computation of fourpoint, oneloop
bulk amplitudes involving virtual gravitons, or the scalar box diagram.
straints on the doubletrace data and Reggeization of the singletrace spectrum. Perhaps
this could lead to a sharper, su cient set of criteria for a holographic CFT to have a local
bulk dual. Similarly, we would like to understand the dependence on
gap of the negativity,
monotonicity and convexity properties of n;`. When is n;` > 0 possible, as a function of
gap? At large n and `, there is a causality constraint from classical gravity on the sign of
n;`; is there a generalization of this constraint in classical higher spin theories? One can
formulate this question in AdS as a scattering experiment in a \higher spin shock wave"
background, where the metric and all higher spin gauge elds are activated and couple to
an incoming particle.
In this paper we have considered the exchange of a nite number of singletrace
operators. There are situations, for instance supergravity, in which there are in nite towers
of singletrace operators exchange, but the nal answer has a surprisingly simple form. It
would be interesting to understand these cases.
Acknowledgments
We wish to thank O. Aharony, S. Giombi, D. Li, D. Meltzer, D. Poland, and
D.SimmonsDu n for helpful discussions. LFA acknowledges Harvard University for hospitality where
part of this work has been done. The work of LFA was supported by ERC STG grant
306260. LFA is a Wolfson Royal Society Research Merit Award holder. AB acknowledges
the University of Oxford and the Weizmann Institute for hospitality where part of this work
has been done. AB is partially supported by Templeton Award 52476 of A. Strominger and
by Simons Investigator Award from the Simons Foundation of X. Yin. EP is supported by
the Department of Energy under Grant No. DEFG0291ER40671.
A
Twist conformal blocks
In this appendix we construct the twist conformal blocks H(0)(z; z) together with the
sequence of functions H(m)(z; z) in four dimensions and for the speci c case of deformations
of generalised free elds. In four dimensions the conformal blocks are given by
g ;s(z; z) =
zs+1F =2+s(z)F 2 (z)
zs+1F =2+s(z)F 2 (z)
2
(A.1)
2
z
z
For us it will be important that they are eigenfunctions of a quadratic Casimir operator18
C z =2z =2g ;s(z; z) = J
2 z =2z =2g ;s(z; z)
where J 2 = (s + =2)(s + =2
1) and, in four dimensions,
C = D + D +
2zz
z
1
Hn(m)(z; z) =
X an;`
`
(0) z n=2z n=2
J 2m
g n;`(z; z)
z2@. The sequence of functions Hn(m)(z; z) is de ned by
HJEP08(217)4
(1
zz
X Hn(0)(z; z)
Hn(0)(z) =
n (1 + n(1
Expanding both sides in powers of z we can nd Hn(0)(z) casebycase. They take the nal
form
with
n =
p 2
4 n
1)2
n
2
n 3
2
1
;
n =
(4
+ ( n
4(
6) n + 4)
1)2
where recall n = 2
+ 2n and J is the corresponding conformal spin J 2 = (` + n +
)(` +
n +
1). Here
is the dimension of the external operator. We will be concerned with
the singular contribution as z ! 1. From the explicit expression for the conformal blocks,
it follows that the factorised form
1
z
Hn(0)(z; z) =
z n=2F n 2 (z)Hn(0)(z)
captures all power law divergent terms  only the sum over spins can generate a power
law singularity at z = 1, so the rst term in eq. (A.1) does not participate. The functions
Hn(0)(z) can be found by decomposing the divergent part of the fourpoint function,
Hn(m)(z; z) =
1
z
z
2
z n=2F n 2 (z)Hn(m)(z)
18This operator is the usual quadratic Casimir shifted by a constant.
(A.2)
(A.3)
(A.4)
(A.5)
(A.6)
(A.7)
(A.8)
(A.9)
(A.10)
1
z
+ 2n
3
+ 2n + 1
2
n
Let us now turn our attention to the divergent piece of the sequence of functions H(m)(z; z)
for m > 0. They are de ned by the following recurrence relation
CH(m+1)(z; z) = H(m)(z; z)
For the same reasons above, the sequence of functions has the same factorisation properties
conditions
desired order.
B
Explicit results
The relation (A.11) then leads to recursion relations for the functions h(km), which can
be solved iteratively. The dependence on the twist, or n, enters through the boundary
h(00) = n; h(10) = n; h(k0) = 0; for k > 1
With enough patience and/or computers, the functions Hn(m)(z) can be constructed to any
Hn(m)(z) =
1
m
h(m) 1 + h(1m)(1
0
z) + h(2m)(1
z)2 +
Having constructed the sequence of functions Hn(m)(z; z) in the previous appendix, we can
expand both sides of (2.23) in powers of (1
z) and z and solve for the coe cients Bm;n.
As explained in the text, the dependence on (1
z) and z factorises and the nal answer
for n;` takes the form eq. (2.25), which we reproduce here:
n;` =
The recurrence relation then leads to
D4dHn(m+1)(z) = Hn(m)(z);
D4d = zDz 1
In this paper we will consider the case in which
is generic and m is an integer. For each
n the solution has the following structure
(A.11)
(A.12)
(A.13)
(B.1)
(B.2)
2(n) as
(B.3)
The functions
2(n) and f 2(n; J ) arise from two decomposition problems, one in the
variable z and the other in the variable 1
z. The coe cients
2(n) satisfy
1
X
n=0
2(n)zk +n 1(z) =
1
n (1
z)
1 2
k~ 2 (1
z)
where k +n 1(z) and k~ 2 (1
and the integral (2.46), w2e can write down the following closed expression for
z) were de ned in eq. (2.42). Using the projector (2.45)
2(n) =
1 I( )
n 2 1 ;n 1
2(n) is a polynomial of degree
4. For the rst few examples
a contour integral:
For even it turns out
we obtain 0(n) = 0, and
2(n) = 1;
4(n) =
6 (
6(n) = 30
(
1)2 2
1)2
6)n
;
6n4 +12(2
3)n3 +6 5 2 14 +11 n2 +6 2 3 7 2 +10
6 n
+1
and so on. The functions f 2(n; J ) admit the following decomposition
where
f 2(n; J ) = 2 X1 bm 2(n)
m=1
J 2m
1
X bm
m=1
1
( 2)(n)Hn(m)(z) =
z) =2F =2 1(1
z)
This can be solved to arbitrarily high order. We have solved this equation for several cases.
The simplest solution corresponds to
= 2. In this case
2(
1)2
1) + (
1)2 n
J 2 + (
= 4; 6;
p
J 2 + (
The expressions for f 2(n; J ) for
a closed form. They all have the structure
are more complicated but can be found in
2( 1 + 4J 2) + P =2 2(J 2) (`)
where RY (x) is a rational function whose numerator is a degreeY polynomial in x, PY (J 2)
is a degreeY polynomial in J 2, and we have introduced
(n + ` +
2)
(n + ` +
+ 1) + (n + ` + 2
2) (B.8)
where (x) = ddx log (x) is the digamma function. For the rst cases
= 4; 6 we obtain
P0(J 2) =
P1(J 2) =
2
12
2
(
2
3
3
+ 2 2
+ 2 2
2)2
(
2)2(
2
4
1)2
2J 2 + 3
(B.4)
(B.5)
(B.6)
(B.7)
(B.9)
while
R2(x) =
R4(x) =
8(
24(3
1)2( 7
3) (4x2
+ 4x(
+ (
4)x) + 12)
1) ( 2
+ 2x + 3)
)( (8
+ 4x(
(2x
+ 2x(2
3) (4x2
+ 2x(
1) ( 2
+ x + 1)
+ 2x + 3)
3) + 1)
23) + 16)
With enough patience one can nd f 2(n; J ) for arbitrarily high even .
C
Resumming the asymptotic contribution
In the body of the paper we have encountered the following sum
Sn(z) =
X a(n0;`) nas;`zk =2+`(z)
(C.1)
`
where nas;` is the asymptotic solution corresponding to the exchange of a
spin s,
2(
1)2
n (` + 1)(` + 2
+ 2n
2)
where in this appendix we take n
2+2s(n). Due to the precise form of nas;` we have
(D4d
+ n
+ n
2))Sn(z) =
2 nRn(z)
where Rn(z) is a simpler sum
Rn(z) =
X a(n0;`)zk =2+`(z)
`
There is an elegant way to compute Rn(z). First, we note that it also arises in the conformal
block decomposition of the treelevel result. More presicely
n;`
X a(n0;`)zzk n=2+`(z)k 2 (z)
2
X zk 2n 1(z)Rn(z) = (z
z) G(0)(z; z)
1
(C.5)
Using the projectors (2.45) we can obtain a closed form expression for Rn(z)
Rm(z) =
X
n;`=0
n+`;m 1a(n0;`)zk 2n 1(z) +
z (c1m + c2mz)
(1
z)
+ (c3m + c4mz)z
(C.6)
where the coe cients cim are given by
2 i z3
k
z) G(0)(z; z)
1 =
(c3m + c4mz)z
(C.7)
z (c1m + c2mz)
(1
z)
Due to the Kronecker delta function in eq. (C.6) and the fact that n
0 and `
0, only a
nite number of terms contributes to Rm(z) for a given m; this is completely unobvious in
the form eq. (C.4). Moreover, the integral eq. (C.7) is very easy to evaluate for any value
of m. In order to obtain the sum Sn(z) we started with we note
D4dzk (z) = (
1)zk (z)
so that in the basis of functions zk (z) the operator D4d acts diagonally. Hence, from
eq. (C.3) and eq. (C.6), this implies that the solution we are looking for contains the piece
Sm(z) =
2 m
X
n;`=0
m;n
n+`;m 1 a(n0;`)zk 2n 1(z) +
(C.2)
(C.3)
(C.4)
(C.8)
(C.9)
where
m;n = (
+ n
1)(
+ n
2) (
+ m
1)(
+ m
2) = (n
m)(2
Inverting the extra terms in (C.6) we obtain the nal result
Sm(z) =
0
n+`;m 1 a(n0;`)zk 2n 1(z) + cm (1
m;n
z
z)
1
1 + dmz A
(C.11)
For any m this can be expanded around z = 1. The piece proportional to log(1 z) exactly
agrees with the one quoted in the body of the paper.
D
Comparison with literature
In this appendix we perform the conformal partial wave decomposition of explicitly known
examples of crossingsymmetric fourpoint function contributions from scalar and
spintwo exchange. In particular, we decompose the AdS amplitudes for scalar and graviton
exchange, respectively. These amplitudes have been computed using explicit position space
Witten diagram computations, and using Mellin space, but had not been decomposed. All
of the OPE data arising from these conformal partial wave decompositions are consistent
with our calculations of section 2.
The fourpoint function up to order 1=N 2 is
and admits the following conformal partial wave decomposition
X a(n0;`)(zz) +ng2 +2n;`(z; z)
N 2 (zz) 2 a ;sg ;s(z; z) + : : :
X(zz) +n
n;`
a(n1;`) +
1 (0)
2 an;` n;` log(zz) +
g2 +2n;`(z; z)
where the last line represents the contribution of the exchanged operator of twist
and
spin s. At leading order in N , the OPE coe cients are
where we have introduced
cm =
dm =
p 2 2
2m+5 (m +
( )2 (m + 1)
1) (m + 2
3)
m +
3
2
p ( 1)m+12 2
2m+5 (m +
1) (m + 2
3)
( )2 (m + 1)
m +
3
2
(C.12)
(D.1)
(D.2)
(D.3)
(D.4)
(D.5)
HJEP08(217)4
with
a(n0;`) =
2(` + 1)(` + 2(
+ n
1))
1)2
Cn;
1C`+n+1;
1
Cn; =
2(n +
) (n + 2
n! 2( ) (2n + 2
1)
1)
For the scalar exchange, the fourpoint function can be written in terms of Dfunctions
(see e.g. appendix D of [14] for their de nition) as
T (z; z) =
(zz) (
1)4 g2
c
16 8
=2
X
p=1
p; ;
p; (z; z)
where
r(p) =
8 ( )4
2
p
2 +
2 +
( p + 2
2)
p + 2 +
1
and we de ne gc = gN , with g being the bulk cubic coupling.
We would like to consider the fully symmetrized amplitude which is given by
+ 1
2
z
G
(1)(z; z) = T (z; z) + T
z
z
+ (zz) T
z z
HJEP08(217)4
(D.6)
(D.7)
(D.8)
(D.9)
(D.10)
We performed the conformal partial wave decomposition eq. (D.2) for several combinations
of
and , and nd agreement with the predictions of section 2: namely, we obtain that the
anomalous dimensions and the OPE coe cients are of the form eq. (2.37) and eq. (2.54),
respectively, with the correct functions.
First we compute for various scalar exchanges, s = 0. For
= 2; 4; 5; 6,
g2
the n;` and a(n1;`) agree with equations (2.48), (2.54) and (2.59), provided that a2;0 = 128c 8 .
an;` =
1 @(a(n0;`) n;`)
2
27gc2a(n0;`)
128 8(n+1)(n+2)(n+3)(n+4)(`+n+2)(`+n+3)(`+n+4)(`+n+5)
where, consistently with eqs. (2.36) and (2.48), the anomalous dimensions are
n;0 =
c
7n5 +103n4 +584n3 +1584n2 +2031n+965
9gc2 256 8(n+2)(n+3)(n+4)(n+5)(2n+5)(2n+7)
2 9`2 2n2 +10n+9 +9` 4n3 +34n2 +88n+63 +18 n4 +12n3 +51n2 +89n+51
128 8(`+1)(`+n+2)(`+n+3)(`+n+4)(`+n+5)(`+2n+6)
Using the data in appendices A{C, one can derive the full solution to crossing, which agrees
with the above data provided that a4;0 = 2034g8c2 8 . Note that this is consistent with the large
n fallo
(2.63).
Turning now to the exchange of
= 2; s = 2 operators, corresponding to AdS graviton
exchange, this amplitude has been computed for generic
using position space Witten
diagrams in [
20
], and in Mellin space in [
21, 23
].
We explicitly perform the conformal
partial wave decomposition for
= 4; 5, and in both cases the results are consistent
with our solution of crossing. We will nd it more practical to deal with its space time
counterpart in computing OPE data.
= 4, the amplitude is given by eq. (D.7) with
T (z;z) = (zz)4
1
3
4D1414(z;z) 20D2424(z;z)
+12(zz +(1 z)(1 z))D2525(z;z)+23D3434(z;z)+9D4444(z;z)
(D.11)
+5(4(zz +(1 z)(1 z))D3535(z;z)+3( z +z( 1+2z))D4545(z;z)
where we have used the fact that the Newton constant GN = 2N2 . We obtain the anomalous
dimensions
n;0 =
n;2 =
n;`>2 =
(`+1)(`+2n+6)
(D.12)
(D.13)
(D.14)
which are equivalent to eqs. (2.49){(2.51) provided that aT = 1960 , which is in agreement
with the Ward identities (2.35) with cT = 40N 2. While it is not obvious, one can check
using eq. (2.51) that n;0 contains a contribution from the truncated solution (2.52) with
coe cient 2596 . The OPE coe cients in this cases are given by a(n1;`) = 12 @n(a(n0;`) n;`), this is
consistent with our ndings in the body of the paper.
= 5, the amplitude is given by eq. (D.7) with
60D1515(z;z) 420D2525(z;z) 615D3535(z;z)
443D4545(z;z)+672D5555(z;z)+3(zz +(1 z)(1 z)) 15(4D2626(z;z) (D.15)
168D5656(z;z)
The anomalous dimensions are
n;0 =
n;2 =
n;`>2 =
297n8 +8736n7 +109635n6 +765000n5 +3236708n4 +8469584n3 +13308320n2
11372520n+3976000
381n8 +12936n7 +187491n6 +1509546n5 +7348219n4 +21995444n3 +39146909n2
37297234n+14187600
4 3n4 +42n3 +198n2 +357n+200
3(`+1)(`+2n+8)
(D.16)
(D.17)
(D.18)
The OPE coe cients can also be computed and read
20a(n0;`)
3
1
3
(n + 2)(n + 5)(` + n + 3)(` + n + 6)
2
(n + 1)(n + 6)(` + n + 2)(` + n + 7)
(n + 3)(n + 4)(` + n + 4)(` + n + 5)
These results agree with the expression eq. (2.60), with aT = 158 , in agreement with
eq. (2.35). Again, n;0 contains a contribution from the truncated solution (2.52) with
coe cient 73030 . That a^(n1;`) 6= 0 for this case is consistent with the observed behavior eq. (2.62).
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