Witten diagrams revisited: the AdS geometry of conformal blocks
JHE
Witten diagrams revisited: the AdS geometry of
Eliot Hijano 0 1 3
Per Kraus 0 1 3
Eric Perlmutter 0 1 2
River Snively 0 1 3
0 Princeton , NJ 08544 , U.S.A
1 Los Angeles , CA 90095 , U.S.A
2 Department of Physics, Princeton University
3 Department of Physics and Astronomy, University of California
We develop a new method for decomposing Witten diagrams into conformal blocks. The steps involved are elementary, requiring no explicit integration, and operate directly in position space. Central to this construction is an appealingly simple answer to the question: what object in AdS computes a conformal block? The answer is a \geodesic Witten diagram", which is essentially an ordinary exchange Witten diagram, except that the cubic vertices are not integrated over all of AdS, but only over bulk geodesics connecting the boundary operators. In particular, we consider the case of fourpoint functions of scalar operators, and show how to easily reproduce existing results for the relevant conformal blocks in arbitrary dimension.
AdSCFT Correspondence; Conformal and W Symmetry

HJEP01(26)4
1 Introduction 2
Conformal blocks, holographic CFTs and Witten diagrams
CFT fourpoint functions and holography
A Witten diagrams primer 2.2.1 2.2.2
Mellin space
Looking ahead
3
The holographic dual of a scalar conformal block
3.3
Comments
3.2.1
3.2.2
3.2.3
3.3.1
3.3.2
3.3.3
Proof by direct computation Proof by conformal Casimir equation The Casimir equation Embedding space
Geodesic Witten diagrams satisfy the Casimir equation
Geodesic versus ordinary Witten diagrams
Simpli cation of propagators and blocks
Relation to Mellin space
2.1
2.2
2.3
2.4
3.1
3.2
4.1
4.2
4.3
4.4
5.1
5.2
5.3
5.4
5.5
5.6
5.7
Known results
Geodesic Witten diagrams with spin` exchange: generalities
Evaluation of geodesic Witten diagram: spin1
Evaluation of geodesic Witten diagram: spin2
General `: proof via conformal Casimir equation
Comparison to double integral expression of Ferrara et al.
Decomposition of spin1 Witten diagram into conformal blocks
5.7.1
5.7.2
AAA
AA and A A
{ i {
Logarithmic singularities and anomalous dimensions
What has been computed?
HJEP01(26)4
The conformal block decomposition of scalar Witten diagrams
An AdS propagator identity
Fourpoint contact diagram
Fourpoint exchange diagram
Further analysis
4.4.1
4.4.2
OPE factorization
Recovering logarithmic singularities
4.5
Taking stock
5
Spinning exchanges and conformal blocks
5.7.3
5.7.4
Summary
6
Discussion and future work
1
Introduction
The conformal block decomposition of correlation functions in conformal eld theory is a
HJEP01(26)4
powerful way of disentangling the universal information dictated by conformal symmetry
from the \dynamical" information that depends on the particular theory under study; see
e.g. [1{7]. The latter is expressed as a list of primary operators and the OPE coe cients
amongst them. The use of conformal blocks in the study of CFT correlation functions
therefore eliminates redundancy, as heavily utilized, for instance, in recent progress made
in the conformal bootstrap program, e.g. [8, 9].
In the AdS/CFT correspondence [10{12], the role of conformal blocks has been
somewhat neglected. The extraction of spectral and OPE data of the dual CFT from a
holographic correlation function, as computed by Witten diagrams [12], was addressed early on
in the development of the subject [13{20], and has been re ned in recent years through the
introduction of Mellin space technology [21{27]. In examining this body of work, however,
one sees that a systematic method of decomposing Witten diagrams into conformal blocks
is missing. A rather natural question appears to have gone unanswered: namely, what
object in AdS computes a conformal block? A geometric bulk description of a conformal block
would greatly aid in the comparison of correlators between AdS and CFT, and presumably
allow for a more e cient implementation of the dual conformal block decomposition, as it
would remove the necessity of actually computing the full Witten diagram explicitly. The
absence of such a simpler method would indicate a surprising failure of our
understanding of AdS/CFT: after all, conformal blocks are determined by conformal symmetry, the
matching of which is literally the most basic element in the holographic dictionary.
In this paper we present an appealingly simple answer to the above question, and
demonstrate its utility via streamlined computations of Witten diagrams. More precisely,
we will answer this question in the case of fourpoint correlation functions of scalar
operators, but we expect a similar story to hold in general. The answer is that conformal blocks
are computed by \geodesic Witten diagrams". The main feature of a geodesic Witten
diagram that distinguishes it from a standard exchange Witten diagram is that in the former,
the bulk vertices are not integrated over all of AdS, but only over geodesics connecting
points on the boundary hosting the external operators. This representation of conformal
blocks in terms of geodesic Witten diagrams is valid in all spacetime dimensions, and holds
for all conformal blocks that arise in fourpoint functions of scalar operators belonging to
arbitrary CFTs (and probably more generally).
To be explicit, consider four scalar operators Oi with respective conformal dimensions
i. The conformal blocks that appear in their correlators correspond to the exchange of
{ 1 {
Z
d
12
Z
34
This computes the conformal partial wave for the exchange of a CFTd primary operator of spin `
and dimension .
primaries carrying dimension
and transforming as symmetric traceless tensors of rank
`; we refer to these as spin` operators. Up to normalization, the conformal partial wave1
in CFTd is given by the following object in AdSd+1:
Gbb y( ); y( 0); ; `
(1.1)
ij denotes the bulk geodesic connecting boundary points xi and xj , with
and 0
denoting the corresponding proper length parameters. Gb@ (y; x) are standard scalar
bulktoboundary propagators connecting a bulk point y to a boundary point x. Gbb y( ); y( 0);
; ` is the bulktobulk propagator for a spin`
eld, whose mass squared in AdS units
is m2 =
(
d)
`, pulled back to the geodesics. The above computes the schannel
partial wave, corresponding to using the OPE on the pairs of operators O1O2 and O3O4.
As noted earlier, the expression (1.1) looks essentially like an exchange Witten diagram
composed of two cubic vertices, except that the vertices are only integrated over geodesics.
See gure 1. Note that although geodesics sometimes appear as an approximation used in
the case of high dimension operators, here there is no approximation: the geodesic Witten
diagram computes the exact conformal block for any operator dimension.
As we will show, geodesic Witten diagrams arise very naturally upon dismantling a
full Witten diagram into constituents, and this leads to an e cient implementation of the
conformal block decomposition. Mellin space techniques also provide powerful methods,
but it is useful to have an approach that can be carried out directly in position space, and
that provides an explicit and intuitive picture for the individual conformal blocks.
For the cases that we consider, the conformal blocks are already known, and so one of
our tasks is to demonstrate that (1.1) reproduces these results. One route is by explicit
computation. Here, the most direct comparison to existing results is to the original work of
Ferrara, Gatto, Grillo, and Parisi [1{3], who provided integral representations for
conformal blocks. In hindsight, these integral expressions can be recognized as geodesic Witten
1Conformal partial waves and conformal blocks are related by simple overall factors as we review below.
{ 2 {
diagrams. Later work by Dolan and Osborn [4{6] provided closedform expressions for
some evend blocks in terms of hypergeometric functions. Dolan and Osborn employed the
very useful fact that conformal partial waves are eigenfunctions of the conformal Casimir
operator. The most e cient way to prove that geodesic Witten diagrams compute
conformal partial waves is to establish that they are the correct eigenfunctions. This turns out
to be quite easy using embedding space techniques, as we will discuss.
Having established that geodesic Witten diagrams compute conformal partial waves,
we turn to showing how to decompose a Witten diagram into geodesic Witten diagrams. We
do not attempt an exhaustive demonstration here, mostly focusing on treelevel contact and
exchange diagrams with four external lines. The procedure turns out to be quite economical
and elegant; in particular, we do not need to carry out the technically complicated step of
integrating bulk vertices over AdS. Indeed, the method requires no integration at all, as
all integrals are transmuted into the de nition of the conformal partial waves. The steps
that are required are all elementary. We carry out this decomposition completely explicitly
for scalar contact and exchange diagrams, verifying that we recover known results. These
include certain hallmark features, such as the presence of logarithmic singularities due
to anomalous dimensions of doubletrace operators. We also treat the vector exchange
diagram, again recovering the correct structure of CFT exchanges.
Let us brie y mention how the analysis goes. The key step is to use a formula expressing
the product of two bulktoboundary propagators sharing a common bulk point as a sum
of bulk solutions sourced on a geodesic connecting the two boundary points. The elds
appearing in the sum turn out to be dual to the doubletrace operators appearing in the
OPE of the corresponding external operators, and the coe cients in the sum are closely
related to the OPE coe cients. See equation (4.1). With this result in hand, all that is
needed are a few elementary properties of AdS propagators to arrive at the conformal block
decomposition. This procedure reveals the generalized free eld nature of the dual CFT.
The results presented here hopefully lay the foundation for further exploration of the
use of geodesic Witten diagrams.
We believe they will prove to be very useful, both
conceptually and computationally, in AdS/CFT and in CFT more generally.
The remainder of this paper is organized as follows. In section 2 we review relevant
aspects of conformal blocks, Witten diagrams, and their relation. Geodesic Witten diagrams
for scalar exchange are introduced in section 3, and we show by direct calculation and via
the conformal Casimir equation that they compute conformal blocks. In section 4 we turn
to the conformal block decomposition of Witten diagrams involving just scalar elds. We
describe in detail how single and double trace operator exchanges arise in this framework.
Section 5 is devoted to generalizing all of this to the case of spinning exchange processes.
We conclude in section 6 with a discussion of some open problems and future prospects.
The ideas developed in this paper originated by thinking about the bulk representation
of Virasoro conformal blocks in AdS3/CFT2, based on recent results in this direction [28{
33]. The extra feature associated with a bulk representation of Virasoro blocks is that the
bulk metric is deformed in a nontrivial way; essentially, the geodesics backreact on the
geometry. In this paper we focus on global conformal blocks (Virasoro blocks are of course
special to CFT2), deferring the Virasoro case to a companion paper [34].
{ 3 {
u =
x212x2324 ;
after using conformal invariance to x three positions at 0; 1; 1.
g(u; v) can be decomposed into conformal blocks, G ;`(u; v), as
where O is a primary operator of dimension
be written compactly as a sum of conformal partial waves, W ;`(xi):
and spin `.2 Accordingly, the correlator can
Let us rst establish some basic facts about fourpoint correlation functions in conformal
eld theories, and their computation in AdSd+1/CFTd. Both subjects are immense, of
course; the reader is referred to [35, 36] and references therein for foundational material.
CFT fourpoint functions and holography
We consider vacuum fourpoint functions of local scalar operators O(x) living in d Euclidean
dimensions. Conformal invariance constrains these to take the form
;
(2.1)
(2.2)
(2.3)
(2.4)
(2.5)
(2.6)
(2.7)
hO1(x1)O2(x2)O3(x3)O4(x4)i =
X C12O CO34W ;`(xi)
O
where
as the insertion of a projector onto the conformal family of O, normalized by the OPE
coe cients:
2In this paper we only consider scalar correlators, in which only symmetric, traceless tensor exchanges
can appear. More generally, ` would stand for the full set of angular momenta under the ddimensional
little group.
hO1(x1)O2(x2)O3(x3)O4(x4)i =
xj . g(u; v) is a function of the two independent
conformal crossratios,
One can also de ne complex coordinates z; z, which obey
2
x13
v =
where
X
n
P ;`
j
P nOihP nOj
and P nO is shorthand for all descendants of O made from n raising operators P . We will
sometimes refer to conformal blocks and conformal partial waves interchangeably, with the
understanding that they di er by the power law prefactor in (2.6).
Conformal blocks admit double power series expansions in u and 1 v, in any spacetime
dimension [4]; for ` = 0, for instance,
G ;0(u; v) = u =2 X
1
m;n=0
d
relations [6, 9]. Especially relevant for our purposes are integral representations of the
conformal blocks [1{3]. For ` = 0,
G ;0(u; v) =
2
1
34
u =2Z 1
d
0
where we have de ned a coe cient
2F1
34
The blocks can also be expressed as in nite sums over poles in
associated with null
states of SO(d; 2), in analogy with Zamolodchikov's recursion relations in d = 2 [37{39];
these provide excellent rational approximations to the blocks that are used in numerical
work. Finally, as we revisit later, in even d the conformal blocks can be written in terms
of hypergeometric functions.
Conformal eld theories with weakly coupled AdS duals obey further necessary
conditions on their spectra.3 In addition to having a large number of degrees of freedom, which
we will label4 N 2, there must be a nite density of states below any xed energy as N ! 1;
e.g. [43{46]. For theories with Einsteinlike gravity duals, this set of parametrically light
operators must consist entirely of primaries of spins `
2 and their descendants.
The \singletrace" operators populating the gap are generalized free elds: given any
set of such primaries Oi, there necessarily exist \multitrace" primaries comprised of
conglomerations of these with some number of derivatives (distributed appropriately to make
a primary). Altogether, the singletrace operators and their multitrace composites
comprise the full set of primary elds dual to nonblack hole states in the bulk. In a fourpoint
3Finding a set of su cient conditions for a CFT to have a weakly coupled holographic dual remains an
unsolved problem. More recent work has related holographic behavior to polynomial boundedness of Mellin
amplitudes [40, 41], and to the onset of chaos in thermal quantum systems [42].
4We are agnostic about the precise exponent: vector models and 6d CFTs are welcome here. More
function of Oi, all multitrace composites necessarily run in the intermediate channel at
some order in 1=N .
Focusing on the doubletrace operators, these are schematically of the form
These have spin` and conformal dimensions
(ij)(n; `) =
i +
j + 2n + ` + (ij)(n; `) ;
where (ij)(n; `) is an anomalous dimension. The expansion of a correlator in the schannel
includes the doubletrace terms5
hO1(x1)O2(x2)O3(x3)O4(x4)i
X P (12)(m; `)W (12)(m;`);`(xi) + X P (34)(n; `)W (34)(n;`);`(xi)
m;`
n;`
Following [26, 43], we have de ned a notation for squared OPE coe cients,
P (ij)(n; `)
C12OCO34 ;
where
O = [OiOj ]n;` :
The 1=N expansion of the OPE data,
1
r=0
1
r=1
P (ij)(n; `) =
X N 2rP (ij)(n; `) ;
r
(ij)(n; `) =
X N 2r (ij)(n; `) ;
r
induces a 1=N expansion of the fourpoint function. Orderbyorder in 1=N , the
generalized free elds and their composites must furnish crossingsymmetric correlators. This is
precisely the physical content captured by the loop expansion of Witten diagrams in AdS,
to which we now turn.
2.2
A Witten diagrams primer
See [36] for background. We work in Euclidean AdSd+1, with RAdS
1. In Poincare
coordinates y = fu; xig, the metric is
ds2 =
du2 + dxidxi
u2
:
The ingredients for computing Witten diagrams are the set of bulk vertices, which are read
o from a Lagrangian, and the AdS propagators for the bulk elds. A scalar eld of mass
m2 =
(
d) in AdSd+1 has bulktobulk propagator
Gbb(y; y0; ) = e
(y;y0)2F1
5Unless otherwise noted, all sums over m; n and ` run from 0 to 1 henceforth.
{ 6 {
(2.12)
(2.13)
(2.14)
(2.15)
(2.16)
(2.17)
(2.18)
(y; y0) = log
u2 + jx
xij2
:
Gbb(y; y0; ) is a normalizable solution of the AdS wave equation with a deltafunction
source,6
(r2
m2)Gbb(y; y0; ) =
The bulktoboundary propagator is
There are also exchangetype diagrams, which involve \virtual" elds propagating between
points in the interior of AdS. The simplest treelevel Witten diagrams are shown in gure 2.
We focus henceforth on treelevel fourpoint functions of scalar elds i dual to scalar
CFT operators Oi. For a nonderivative interaction 1 2 3 4, and up to an overall quartic
coupling that we set to one, the contact diagram equals
A4
Contact(xi) = D 1 2 3 4 (xi) =
(2.24)
We will introduce higher spin propagators in due course.
A holographic CFT npoint function, which we denote An, receives contributions from
all possible npoint Witten diagrams. The loopcounting parameter is GN
O(1=N 2), only treelevel diagrams contribute. The simplest such diagrams are contact
1=N 2. At
diagrams, which integrate over a single npoint vertex. Every local vertex in the bulk
Lagrangian gives rise to a contact diagram: schematically,
L
n
where Gb@ (y; xi) are bulktoboundary propagators for elds with quantum numbers ( i;
`i), and pi count derivatives. We abbreviate
where (y; y0) is the geodesic distance between points y; y0. In Poincare AdS,
D 1 2 3 4 (xi) is the Dfunction, which is de ned by the above integral. For generic
i, this integral cannot be performed for arbitrary xi. There exists a bevy of identities
relating various D 1 2 3 4 (xi) via permutations of the
i, spatial derivatives, and/or
shifts in the
i [18, 47]. Derivative vertices, which appear in the axiodilaton sector of
6We use this normalization for later convenience. Our propagator is 2 d=2
d 2 2 = ( ) times the
common normalization found in, e.g., equation (6.12) of [36].
{ 7 {
contact diagram. On the right is an exchange diagram for a symmetric traceless spin` tensor
eld of dual conformal dimension . Here and throughout this work, orange dots denote vertices
integrated over all of AdS.
type IIB supergravity, for instance, de ne Dfunctions with shifted parameters. When
i = 1 for all i,
2x213x224
2
d
2
d=2 D1111(xi) =
1
z
z
2Li2(z)
2Li2(z) + log(zz) log
:
(2.25)
1
1
z
z
This actually de nes the Dbar function, D1111(z; z). For various sets of
ing (2.25) with e cient use of Dfunction identities leads to polylogarithmic representations
i 2 Z,
combinof contact diagrams.
The other class of treelevel diagrams consists of exchange diagrams. For external
scalars, one can consider exchanges of symmetric, traceless tensor elds of arbitrary spin
`. These are computed, roughly, as
A4
Exch(xi) =
Z Z
y y0
Gbb(y; y0; ; `)
(2.26)
Gbb(y; y0; ; `) is shorthand for the bulktobulk propagator for the spin` eld of dimension
, which is really a bitensor [Gbb(y; y0; )] 1::: `; 1::: ` . We have likewise suppressed all
derivatives acting on the external scalar propagators, whose indices are contracted with
those of [Gbb(y; y0; )] 1::: `; 1::: ` .
Due to the double integral, brute force methods of
simplifying exchange diagrams are quite challenging, even for ` = 0, without employing
some form of asymptotic expansion.
The key fact about these treelevel Witten diagrams relevant for a dual CFT
interpretation is as follows. For contact diagrams (2.24), their decomposition into conformal
blocks contains the in nite towers of doubletrace operators in (2.14), and only these. This
is true in any channel. For exchange diagrams (2.26), the schannel decomposition
includes a singletrace contribution from the operator dual to the exchanged bulk
eld, in
addition to in nite towers of doubletrace exchanges (2.14). In the t and uchannels, only
doubletrace exchanges are present. The precise set of doubletrace operators that appears
is determined by the spin associated to the bulk vertices.
{ 8 {
Higherloop Witten diagrams are formed similarly, although the degree of di culty
increases rapidly with the loop order. No systematic method has been developed to compute
these.
Logarithmic singularities and anomalous dimensions
When the external operator dimensions are nongeneric, logarithms can appear in treelevel
Witten diagrams [14, 15, 18]. These signify the presence of perturbatively small anomalous
dimensions, of order 1=N 2, for intermediate states appearing in the CFT correlator. Let
us review some basic facts about this.
In general, if any operator of free dimension
0 develops an anomalous dimension , so
0 + , a small expansion of its contribution to correlators
yields an in nite series of logs:
G 0+ ;`(u; v)
In the holographic context, the doubletrace composites [OiOj ]n;` have anomalous
dimensions at O(1=N 2). Combining (2.13), (2.14) and (2.16) leads to doubletrace contributions
to holographic fourpoint functions of the form
(2.27)
(2.28)
A4(xi) 1=N2
X
m;`
+ X
n;`
1
P (12)(m; `) +
1
P (34)(n; `) +
1 P (12)(m; `) 1(12)(m; `)@m
1 P (34)(n; `) 1(34)(n; `)@n
W 1+ 2+2m+`;`(xi)
W 3+ 4+2n+`;`(xi)
where @mW 1+ 2+2m+`;` / log u and likewise for the (34) terms. These logarithmic
singularities should therefore be visible in treelevel Witten diagrams. In topdown examples
of AdS/CFT, the supergravity elds are dual to protected operators, so the (ij)(n; `) are
the only (perturbative) anomalous dimensions that appear, and hence are responsible for
all logs.
For generic operator dimensions, the doubletrace operators do not appear in both the
O1O2 and O3O4 OPEs at O(N 0), so P (ij)(n; `) = 0. On the other hand, when the
0
i are
related by the integrality condition
1 +
2
3
4 2 2Z, one has P (ij)(n; `) 6= 0 [14].
0
2.2.2
What has been computed?
In a foundational series of papers [13, 15{18, 48{51], methods of direct computation were
developed for scalar fourpoint functions, in particular for scalar, vector and graviton
exchanges. Much of the focus was on the axiodilaton sector of type IIB supergravity on
AdS5
S5 in the context of duality with N = 4 superYang Mills (SYM), but the methods
were gradually generalized to arbitrary operator and spacetime dimensions.
This e ort largely culminated in [17, 51] and [18]. [17] collected the results from all
channels contributing to axiodilaton correlators in N = 4 SYM, yielding the full correlator
at O(1=N 2). In [51], a more e cient method of computation was developed for exchange
diagrams. It was shown that scalar, vector and graviton exchange diagrams can generically
{ 9 {
be written as in nite sums over contact diagrams for external elds of variable dimensions.
These truncate to nite sums if certain relations among the dimensions are obeyed.7 These
calculations were translated in [18] into CFT data, where it was established that
logarithmic singularities appear precisely at the order determined by the analysis of the previous
subsection. This laid the foundation for the modern perspective on generalized free elds.
Further analysis of implications of fourpoint Witten diagrammatics for holographic CFTs
(e.g. crossing symmetry, nonrenormalization), and for N
= 4 SYM in particular, was
performed in [4, 47, 52{62]. A momentum spacebased approach can be found in [63, 64].
More recent work has computed Witten diagrams for higher spin exchanges [65, 66].
These works develop the split representation of massive spin` symmetric traceless tensor
elds, for arbitrary integer `. There is a considerable jump in technical di culty, but the
results are all consistent with AdS/CFT.
2.3
Mellin space
An elegant alternative approach to computing correlators, especially holographic ones, has
been developed in Mellin space [21, 67]. The analytic structure of Mellin amplitudes neatly
encodes the CFT data and follows a close analogy with the momentum space representation
of at space scattering amplitudes. We will not make further use of Mellin space in this
paper, but it should be included in any discussion on Witten diagrams; we only brie y
review its main properties with respect to holographic fourpoint functions, and further
aspects and details may be found in e.g. [22{27, 68{70].
Given a fourpoint function as in (2.1), its Mellin representation may be de ned by
the integral transform
Z i1
i1
The integration runs parallel to the imaginary axis and to one side of all poles of the
integrand. The Mellin amplitude is M (s; t). Assuming it formally exists, M (s; t) can be
de ned for any correlator, holographic [21] or otherwise [71, 72]. M (s; t) is believed to be
meromorphic in any compact CFT. Written as a sum over poles in t, each pole sits at a xed
twist
=
`, capturing the exchange of twist operators in the intermediate channel.
Given rhe exchange of a primary O of twist
O, its descendants of twist
=
O + 2m
contribute a pole
M (s; t)
C12OCO34 t
Q`;m(s)
O
2m
(2.29)
(2.30)
where m = 0; 1; 2; : : :. Q`;m(s) is a certain degree` (Mack) polynomial that can be found
in [
25
]. Note that an in nite number of descendants contributes at a given m. npoint
Mellin amplitudes may be likewise de ned in terms of n(n
3)=2 parameters, and are
known to factorize onto lowerpoint amplitudes [27].
7For instance, an schannel scalar exchange is written as a nite linear combination of Dfunctions if
1 + 2
is a positive even integer [51].
Specifying now to holographic correlators at treelevel,8 the convention of including
explicit Gamma functions in (2.29) has particular appeal: their poles encode the
doubletrace exchanges of [O1O2]m;` and [O3O4]n;`. Poles in M (s; t) only capture the
singletrace exchanges, if any, associated with a Witten diagram. In particular, all local AdS
interactions give rise to contact diagrams whose Mellin amplitudes are mere polynomials
in the Mellin variables. In this language, the counting of solutions to crossing symmetry in
sparse large N CFTs performed in [43] becomes manifestly identical on both sides of the
duality. Exchange Witten diagrams have meromorphic Mellin amplitudes that capture the
lone singletrace exchange: they take the form9
M (s; t) = C12OCO34
1
X
m=0 t
Q`;m(s)
O
2m
+ Pol(s; t) :
(2.31)
Pol(s; t) stands for a possible polynomial in s; t. The polynomial boundedness is a signature
of local AdS dynamics [
25, 40
]. Anomalous dimensions appear when poles of the integrand
collide to make double poles.
A considerable amount of work has led to a quantitative understanding of the above
picture. These include formulas for extraction of the oneloop OPE coe cients P (ij)(n; `)
1
and anomalous dimensions 1
(ij)(n; `) from a given Mellin amplitude (section 2.3 of [26]);
and the graviton exchange amplitude between pairwise identical operators in arbitrary
spacetime dimension (section 6 of [65]).
2.4
Looking ahead
Having reviewed much of what has been accomplished, let us highlight some of what
has not.
issues here.
First, we note that no approach to computing holographic correlators has
systematically deconstructed loop diagrams, nor have arbitrary external spins been e ciently
incorporated. Save for some concrete proposals in section 6, we will not address these
While Mellin space is home to a fruitful approach to studying holographic CFTs in
particular, it comes with a fair amount of technical complication. Nor does it answer the
natural question of how to represent a single conformal block in the bulk. One is, in any
case, left to wonder whether a truly e cient approach exists in position space.
Examining the position space computations reviewed in subsection 2.2, one is led to
wonder: where are the conformal blocks? In particular, the extraction of dual CFT spectral
data and OPE coe cients in the many works cited earlier utilized a double OPE expansion.
8This is the setting that is known to be especially amenable to a Mellin treatment. Like other approaches
to Witten diagrams, the Mellin program has not been systematically extended to loop level (except for
certain classes of diagrams; see section 6). Because highertrace operators appear at higher orders in 1=N ,
some of the elegance of the treelevel story is likely to disappear. The addition of arbitrary external spin
in a manner which retains the original simplicity has also not been done, although see [22].
9For certain nongeneric operator dimensions, the sum over poles actually truncates [21, 23]. The precise
mechanism for this is not fully understood from a CFT perspective. We thank Liam Fitzpatrick and Joao
Penedones for discussions on this topic.
More recent computations of exchange diagrams [65, 66] using the split representation
do make the conformal block decomposition manifest, in a contour integral form [73]:
integration runs over the imaginary axis in the space of complexi ed conformal dimensions,
and the residues of poles in the integrand contain the OPE data. This is closely related
to the shadow formalism. However, this approach is technically quite involved, does not
apply to contact diagrams, and does not answer the question of what bulk object computes
a single conformal block.
diagrams.
Let us turn to this latter question now, as a segue to our computations of Witten
3
The holographic dual of a scalar conformal block
What is the holographic dual of a conformal block? This is to say, what is the geometric
representation of a conformal block in AdS? In this section we answer this question for
the case of scalar exchanges between scalar operators, for generic operator and spacetime
dimensions. In section 5, we will tackle higher spin exchanges. At this stage, these operators
need not belong to a holographic CFT, since the form of a conformal block is xed solely by
symmetry. What follows may seem an inspired guess, but as we show in the next section,
it emerges very naturally as an ingredient in the computation of Witten diagrams.
Let us state the main result. We want to compute the scalar conformal partial wave
W ;0(xi), de ned in (2.6), corresponding to exchange of an operator O of dimension
between two pairs of external operators O1; O2 and O3; O4. Let us think of the external
operators as sitting on the boundary of AdSd+1 at positions x1;2;3;4, respectively. Denote
the geodesic running between two boundary points xi and xj as
ij . Now consider the
scalar geodesic Witten diagram, which we denote W ;0(xi), rst introduced in section 1
and drawn in gure 1:
Gbb y( ); y( 0);
W ;0(xi)
Z
Z
where
(3.1)
(3.2)
(3.3)
HJEP01(26)4
Z
12
Z 1
1
d ;
Z
34
Z 1
1
d 0
W ;0(xi) =
12
34W ;0(xi) :
denote integration over proper time coordinates
and 0 along the respective geodesics.
Then W ;0(xi) is related to the conformal partial wave W ;0(xi) by
The proportionality constant
34 is de ned in equation (2.11) and
12 is de ned
analogously.
The object W ;0(xi) looks quite like the expression for a scalar exchange Witten
diagram for the bulk
eld dual to O. Indeed, the form is identical, except that the bulk
vertices are not integrated over all of AdS, but rather over the geodesics connecting the
pairs of boundary points. This explains our nomenclature. Looking ahead to conformal
partial waves for exchanged operators with spin, it is useful to think of the bulktobulk
propagator in (3.1) as pulled back to the two geodesics.
Equation (3.3) is a rigorous equality. We now prove it in two ways.
Consider the piece of (3.1) that depends on the geodesic 12, which we denote '12:
:
HJEP01(26)4
In terms of '12 y( 0) , the formula for W ;0 becomes
It is useful to think of '12(y), for general y, as a cubic vertex along 12 between a
bulk
eld at y and two boundary
elds anchored at x1 and x2. We may then solve for
'12(y) as a normalizable solution of the KleinGordon equation with a source concentrated
on 12. The symmetries of the problem turn out to specify this function uniquely, up to a
multiplicative constant. We then pull this back to 34, which reduces W ;0(xi) to a single
onedimensional integral along 34. This integral can then be compared to the wellknown
integral representation for G ;0(u; v) in (2.10), which establishes (3.3).
To make life simpler, we will use conformal symmetry to compute the geodesic Witten
diagram with operators at the following positions:
1
solving the wave equation rst in global AdS, then moving to Poincare coordinates and
comparing with CFT. We work with the global AdS metric
We implement the above strategy by
Z
12
Z
The relation between mass and conformal dimension is
and so the wave equation for '12(y) away from 12 is
1
ds2 =
cos2 (d 2 + dt2 + sin2 d 2d 1) :
m2 =
(
d)
r
2
(
d) '12(y) = 0 ;
or
2
t
(
d) '12(y) = 0 :
(3.4)
(3.5)
(3.6)
(3.7)
(3.8)
(3.9)
(3.10)
At 12, there is a source. To compute (3.6), we take t1 !
1, t2 ! 1, in which limit the
geodesic becomes a line at
= 0, the center of AdS. This simpli es matters because this
source is rotationally symmetric. Its timedependence is found by evaluating the product
of bulktoboundary propagators on a xed spatial slice,
12t:
Therefore, we are looking for a rotationallysymmetric, normalizable solution to the
following radial equation:
tion (3.4), is
'12( ; t) =
12
e 1t1
2t2 2F1
relation between coordinates is
+
2
12 ;
2
12 ;
d
2
2
Now we need to transform this to Poincare coordinates, and pull it back to 34. The
2
12
(
d) '12 = 0 :
(3.12)
The full solution, including the timedependence and with the normalization xed by
equae 2t = u2 + jxj2;
cos2
=
u
2
u2 + jxj2
:
Although the eld '12 is a scalar, it transforms nontrivially under the map from global to
Poincare AdS because the bulktoboundary propagators in its de nition (3.4) transform as
After stripping o the power of x1 needed to de ne the operator at in nity, the eld in
In terms of the proper length parameter 0,
z( 0) =
z( 0) =
u( 0) = jz3
z4j :
z3 + z4 +
z3 + z4 +
2
2
2 cosh 0
z3
z3
2
2
z4 tanh 0;
z4 tanh 0;
(3.11)
12t:
(3.13)
(3.14)
(3.15)
(3.17)
(3.18)
Poincare coordinates is
'12(u; xi) =
12
2F1
where on the right hand side
and t are to be viewed as functions of the Poincare
coordinates u; xi via (3.14).
Now we want to evaluate our Poincare coordinates on the geodesic 34. With our
choice of positions in (3.6), 34 is a geodesic in an AdS3 slice through AdSd+1. Geodesics
in Poincare AdS3 are semicircles: for general boundary points z3; z4,
2u2 + (z
z4)(z
z3) + (z
z4)(z
z3) = 0 :
+
2
12 ;
2
12 ;
d
2
2
Therefore, the pullback of '12 to 34 is
'12 u( 0); xi( 0) =
12(2 cosh 0)
12
2
2F1
+
2
12 ;
into the propagator (2.21), we get
z Gb@ y( 0); 1 in (3.5). Plugging (3.18)
We may now assemble all pieces of (3.5){(3.6): plugging (3.20) and (3.21) into (3.5)
gives the geodesic Witten diagram. Trading z; z for u; v we nd, in the
parameterization (3.6),
W ;0(u; v) =
De ning a new integration variable,
we have
W ;0(u; v) =
2
d
0
2
Comparing (3.24) to the integral representation (2.10), one recovers (3.3) evaluated at the
appropriate values of x1; : : : ; x4. Validity of the relation (3.3) for general operator positions
is then assured by the identical conformal transformation properties of the two sides.
3.2
In the previous section we evaluated a geodesic Witten diagram W ;0 and matched the
result to a known integral expression for the corresponding conformal partial wave W ;0,
thereby showing that W ;0 = W ;0 up to a multiplicative constant. In this section we
give a direct argument for the equivalence, starting from the de nition of conformal partial
waves.
Section 3.2.1 reviews the de nition of conformal partial waves W ;` as eigenfunctions
of the conformal Casimir operator. We prove this de nition to be satis ed by geodesic
Witten diagrams W ;0 in section 3.2.3, after a small detour (section 3.2.2) to de ne the
basic embedding space language used in the proof.
The arguments of the present section are generalized in section 5.5 to show that W ;` =
W ;` (again, up to a factor) for arbitrary exchanged spin `.
The generators of the ddimensional conformal group SO(d + 1; 1) can be taken to be the
Lorentz generators LAB of d + 2 dimensional Minkowski space (with LAB antisymmetric in
A and B as usual). The quadratic combination L2
12 LABLAB is a Casimir of the algebra,
i.e. it commutes with all the generators LAB. As a result, L2 takes a constant value on
any irreducible representation of the conformal group, which means all states jP n
conformal family of a primary state jOi are eigenstates of L2 with the same eigenvalue.
Oi in the
The eigenvalue depends on the dimension
and spin ` of jOi, and can be shown to be [4]
The SO(d + 1; 1) generators are represented on conformal elds by
C2( ; `) =
(
d)
`(` + d
2) :
where L1AB is a di erential operator built out of the position x1 of O1 and derivatives
with respect to that position. The form of the L1AB depends on the conformal quantum
numbers of O1. Equation (3.26) together with conformal invariance of the vacuum imply
the following identity, which holds for any state j i:
(L1AB + L2AB)2h0jO1(x1)O2(x2)j i = h0jO1(x1)O2(x2)L2j i
:
Consistent with the notation for L2, we have de ned
1
2
(L1AB + L2AB)
2
(L1AB + L2AB)(L1 AB + L2 AB) :
As discussed in section 2.1, one obtains a conformal partial wave W ;` by inserting into
a fourpoint function the projection operator P ;` onto the conformal family of a primary
O with quantum numbers
; `:
Xh0jO1(x1)O2(x2)jP nOihP nOjO3(x3)O4(x4)j0i :
(3.25)
(3.26)
(3.27)
(3.28)
(3.29)
One can take this secondorder di erential equation, plus the corresponding one with 1; 2 $
3; 4, supplemented with appropriate boundary conditions, as one's de nition of W ;` [74].
Regarding boundary conditions, it is su cient to require that W ;` have the correct leading
behavior in the x2 ! x1 and x4 ! x3 limits. The correct behavior in both limits is dictated
by the fact that the contribution to W ;` of the primary O dominates that of its descendants
since those enter the OPE with higher powers of x12 and x34.
We will prove that geodesic Witten diagrams W ;0 are indeed proportional to
conformal partial waves W ;0 by showing that W ;0 satis es the Casimir equation (3.30) and
has the correct behavior in the x2 ! x1 and x4 ! x3 limits. The proof is very transparent
in the embedding space formalism, which we proceed now to introduce.
The embedding space formalism has been reviewed in e.g. [35, 65, 74]. The idea is to embed
the ddimensional CFT and the d + 1 dimensional AdS on which lives the geodesic Witten
diagram both into d + 2 dimensional Minkowski space. We give this embedding space the
metric
ds2 =
(dY 1)2 + (dY 0)2 + X(dY i)2:
The CFT will live on the projective null cone of embedding space, which is the
Lorentzinvariant ddimensional space de ned as the set of nonzero null vectors X with scalar
multiples identi ed: X
aX.
We will use null vectors X to represent points in the
projective null cone with the understanding that X and aX signify the same point. The
plane Rd can be mapped into the projective null cone via
X+(x) = ajxj2;
X (x) = a ;
Xi(x) = axi
where we have introduced light cone coordinates X
choice of the parameter a de nes the same map.
Conformal transformations on the plane are implemented by Lorentz transformations
in embedding space. As a speci c example, we may consider a boost in the 0 direction with
rapidity . This leaves the Xi coordinates unchanged, and transforms X
according to
= X 1
X0. Of course, any nonzero
d
i=1
(3.31)
(3.32)
(3.33)
Applying the identity (3.27) to the equation above and recalling that each state jP n
an eigenstate of L2 with the same eigenvalue C2( ; `), we arrive at the Casimir equation
(L1AB + L2AB)2W ;`(xi) = C2( ; `)W ;`(xi) :
(3.30)
! e X+;
X
! e
X :
A point X(x) = (jxj2; 1; xi) gets mapped to (e jx2j; e
; xi) which is projectively equivalent
to X(e xi). Thus boosts in the 0 direction of embedding space induce dilatations in the
plane.
it as
Any eld O^ on the null cone de nes a eld O on the plane via restriction: O(x)
O^ X(x) . Since O^ is a scalar eld in embedding space, the SO(d + 1; 1) generators act on
[LAB; O^(X)] = (XA@B
(3.34)
and only if O^ satis es the homogeneity condition
The induced transformation law for O is the correct one for a primary of dimension
if
O^(aX) = a
O^(X) :
Thus in the embedding space formalism a primary scalar eld O(x) of dimension
represented by a eld O^(X) satisfying (3.35). Below, we drop the hats on embedding space
is
elds. It should be clear from a eld's argument whether it lives on the null cone (as O(X))
or on the plane (as O(x)). Capital letters will always denote points in embedding space.
Meanwhile, AdSd+1 admits an embedding into d + 2 dimensional Minkowski space, as
the hyperboloid Y 2 =
1. Poincare coordinates (u; xi) can be de ned on AdS via
The induced metric for these coordinates is the standard one, (2.17).
The AdS hyperboloid sits inside the null cone and asymptotes toward it. As one takes
u ! 0, the image of a point (u; xi) in AdS approaches (Y +; Y ; Y i) = u 1
(jx2j; 1; xi) which
is projectively equivalent to X(xi). In this way, the image on the projective null cone of
the point xi 2 R
d marks the limit u ! 0 of the embedding space image of a bulk point
(u; xi).
so are generated by
Isometries of AdS are implemented by embedding space Lorentz transformations, and
as long as Y is on the AdS slice. This fact, which is not surprising given that L2 is a
secondorder di erential operator invariant under all the isometries of AdS, can be checked
directly from (3.37).
3.2.3
Geodesic Witten diagrams satisfy the Casimir equation
The geodesic Witten diagram W ;0(xi) lifts to a function W ;0(Xi) on the null cone of
embedding space via a lift of each of the four bulktoboundary propagators with the
appropriate homogeneity condition
i Gb@ (y; Xi) ; i = 1; 2; 3; 4 :
The geodesics in AdS connecting the boundary points X1 to X2 and X3 to X4 lift to curves
in embedding space which can be parameterized by
Y Y =
1. That is, for Y belonging to the AdS slice, L2f (Y ) depends only on the values
of f on the slice. In fact, applied to scalar functions on AdS the operator L2 is simply the
negative of the Laplacian of AdS:
L2f (Y ) =
r2Y f (Y )
(3.35)
(3.36)
(3.37)
(3.38)
(3.39)
(3.40)
we see that terms for which m2m = m2n give rise to derivatives of conformal blocks, and
hence to logarithms. This is equivalent to the condition
d 2 Z, both of these are equivalent to
integrality condition stated above. Identical structure is visible in (4.15): logarithms will
appear when any of m2m, m2n, m2 coincide.
As an explicit example, let us consider D
(xi). Then (4.11) can be split into
m 6= n and m = n terms, the latter of which yield logarithms:
D
(xi) =
1
X 2an
n=0
X
m6=n
am
m2n
m2m
(an )
W2 +2n;0(xi) +
HJEP01(26)4
This takes the form of the ` = 0 terms in (2.28), with
P1(n; 0) = 2 (22 +2n)
a
n
X
m6=n
am
m2n
m2m
+
(an )
2
and
As an aside, we note the conjecture of [43], proven in [26], that
1
2 P0(n; 0) 1(n; 0) =
(an )
2
2
:
P1(n; `) =
1
(4.24)
(4.25)
(4.26)
(4.27)
We have checked in several examples that this is obeyed by (4.25){(4.26). It would be
interesting to prove it using generalized hypergeometric identities.
4.5
Taking stock
We close this section with some perspective. Whereas traditional methods of computing
Witten diagrams are technically involved and require explicit bulk integration [16] and/or
solution of di erential equations [51], the present method skips these steps with a minimum
of technical complexity. It is remarkable that for neither the contact nor exchange diagrams
have we performed any integration: the integrals have instead been absorbed into sums
over, and de nitions of, conformal partial waves.
For the contact diagram/Dfunction, we have presented an e cient algorithm for its
decomposition into spin0 conformal blocks in position space. Speci c cases of such
decompositions have appeared in previous works [43, 45], although no systematic treatment had
been given. Moreover, perhaps the main virtue of our approach is that exchange diagrams
are no more di cult to evaluate than contact diagrams.
Dfunctions also appear elsewhere in CFT, including in weak coupling perturbation
theory. For example, the fourpoint function of the 20' operator in planar N = 4 SYM at
weak coupling is given, at order , by [76]
hO200 (x1)O200 (x2)O200 (x3)O200 (x4)i
/ D1111(z; z)
(4.28)
where D1111(z; z) was de ned in (2.25). The ubiquity of Dfunctions at weak coupling may
be related to constraints of crossing symmetry in the neighborhood of free xed points [71].
The OPE of two scalar primary operators yields not just other scalar primaries but also
primaries transforming in symmetric traceless tensor representations of the Lorentz group.
We refer to such a rank` tensor as a spin` operator. Thus, for the full conformal block
decomposition of a correlator of scalar primaries we need to include blocks describing spin`
exchange. The expression for such blocks as geodesic Witten diagrams turns out to be the
natural extension of the scalar exchange case. The exchanged operator is now described by
a massive spin` eld in the bulk, which couples via its pullback to the geodesics connecting
the external operator insertion points. This was drawn in
gure 1.
In this section we do the following. We give a fairly complete account of the spin1
case, showing how to decompose a Witten diagram involving the exchange of a massive
vector eld, and establishing that the geodesic diagrams reproduce known results for spin1
conformal blocks. We also give an explicit treatment of the spin2 geodesic diagram, again
checking that we reproduce known results for the spin2 conformal blocks. More generally,
we use the conformal Casimir equation to prove that our construction yields the correct
blocks for arbitrary `.
Conformal blocks with external scalars and internal spin` operators were studied in the
early work of Ferrara et al. [1]. They obtained expressions for these blocks as double
integrals. It is easy to verify that their form for the scalar exchange block precisely coincides
with our geodesic Witten diagram expression (3.1). We thus recognize the double integrals
as integrals over pairs of geodesics. Based on this, we expect agreement for general `,
although we have not so far succeeded in showing this due to the somewhat complicated
form for the general spin` bulktobulk propagator [65, 66]. Some more discussion is in
section 5.6. We will instead use other arguments to establish the validity of our results.
Dolan and Osborn [5] studied these blocks using the conformal Casimir equation.
Closedform expressions in terms of hypergeometric functions were obtained in dimensions
d = 2; 4; 6. For example, in d = 2 we have
G ;`(z; z) = jzj
and in d = 4 we have
G ;`(z; z) = jzj
`
z`2F1
2F1
z
1
z
z`+12F1
2F1
12 + `
2
2
12
+
+
34 + `
2
2
34
+ `; z
`; z
+ (z $ z)
`
;
;
;
`
2
12
2
`
12 + `
+
34 + `
+ `; z
2
+
34
2
1;
`
2; z
(z $ z)
;
;
1;
(5.1)
(5.2)
The d = 6 result is also available, taking the same general form, but it is more complicated.
Note that the d = 2 result is actually a sum of two irreducible blocks, chosen so as to be
even under parity. The irreducible d = 2 blocks factorize holomorphically, since the global
conformal algebra splits up as sl(2; R)
sl(2; R). An intriguing fact is that the d = 4 block
is expressed as a sum of two terms, each of which \almost" factorizes holomorphically.
Results in arbitrary dimension are available in series form.
Since the results of Dolan and Osborn are obtained as solutions of the conformal
Casimir equation, and we will show that our geodesic Witten diagrams are solutions of the
same equation with the same boundary conditions, this will constitute exact agreement.
Note, though, that the geodesic approach produces the solution in an integral
representation. It is not obvious by inspection that these results agree with those in [5], but we will
verify this in various cases to assuage any doubts that our general arguments are valid. As
noted above, in principle a more direct comparison is to the formulas of Ferrara et al. [1].
Geodesic Witten diagrams with spin` exchange: generalities
Consider a CFTd primary operator which carries scaling dimension
and transforms in
the rank` symmetric traceless tensor representation of the (Euclidean) Lorentz group. The
AdSd+1 bulk dual to such an operator is a symmetric traceless tensor eld h 1::: ` obeying
the eld equations
Our proposal is that the conformal partial wave W ;`(xi) is given by the same expression
as in (3.1) except that now the bulktobulk propagator is that of the spin` eld pulled
back to the geodesics. The latter de nes the spin` version of the geodesic Witten diagram,
W ;`(xi): its precise de nition is
W ;`(xi)
Z
Z
(5.3)
(5.4)
:
(5.5)
Gbb y( ); y( 0); ; `
and Gbb y( ); y( 0); ; ` is the pulledback spin` propagator,
Gbb y( ); y( 0); ; `
[Gbb(y; y0; )] 1::: `; 1::: ` d
dy 1
: : :
dy ` dy0 1
d
d 0
: : :
dy0 `
d 0 y=y( ); y0=y( 0)
To explicitly evaluate this we will use the same technique as in section 3.1. Namely,
the integration over one geodesic can be expressed as a normalizable spin` solution of the
equations (5.3) with a geodesic source. Inserting this result, we obtain an expression for
the geodesic Witten diagram as an integral over the remaining geodesic. If we call the
above normalizable solution h 1::: ` , then the analog of (3.5) is
Z
34
h 1::: ` y( 0)
dy0 1
d 0
: : :
dy0 `
d 0
(5.6)
As in section 3.1, we will speci cally compute
= jzj
1
3
now written in terms of (z; z) instead of (u; v) to facilitate easier comparison with (5.1)
and (5.2). We recall that this reduces 12 to a straight line at the origin of global AdS.
The form of 34 is given in (3.18), from which the pullback is computed using
cos2
e2t
e2i
34
34
34
=
=
=
1
;
Carrying out this procedure for all dimensions d at once presents no particular
complications. However, it does not seem easy to
nd the solution h 1::: ` for all ` at once. For
this reason, below we just consider the two simplest cases of ` = 1; 2, which su ce for
illustrating the general procedure.
Evaluation of geodesic Witten diagram: spin1
In the global AdSd+1 metric
we seek a normalizable solution of
1
ds2 =
cos2 (d 2 + dt2 + sin2 d 2d 1
)
r2A
d)
= 0 ;
r A
= 0
A dx = At( ; t)dt + A ( ; t)d :
12 tand 1 At = 0
which is spherically symmetric and has time dependence e
12t. A suitable ansatz is
Assuming the time dependence e
12t, the divergence free condition implies
and the components of the wave equation are
cosd 1
sind 1
cosd 1
sind 1
212 cos2
212 cos2
d 1
sin2
1)(
d+1) At
2 12 cos sin A = 0
1)(
d+1) A
+2 12 cos sin At = 0 :
(5.7)
(5.8)
(5.9)
(5.10)
(5.11)
(5.12)
(5.13)
(5.14)
where we have inserted a factor of
12 in A to ensure a smooth
particular, setting
12 = 0 we have A = 0 and
12 ! 0 limit. In
At = (cos )
12F1
+ 1
2
;
2
1
;
2
It is now straightforward to plug into (5.6) to obtain an integral expression for the conformal
block. Because the general formula is rather lengthy we will only write it out explicitly
12 = 0. In this case we nd (not paying attention to overall normalization
W ;1(z; z) = jzj
Z 1
0
3
4 1(1
+ 34 1
2
(1
2F1
+ 1
2
;
j
1
zj2)
)
2
1
;
2
34 1
1
2
2
1
zj2)
j
1
jzj2 (1
(1
j
1
+1
2
)
zj2)
:
Setting d = 2; 4, it is straightforward to verify that the series expansion of this integral
reproduces the known d = 2; 4 results in (5.1), (5.2) for
12 = 0. We have also veri ed
agreement for
12 6= 0.
5.4
Evaluation of geodesic Witten diagram: spin2
In this section we set
12 = 0 to simplify formulas a bit. We need to solve
2
r h
d)
2]h
= 0 ;
= 0 ;
h = 0 :
(5.18)
The normalizable solution is
A =
At =
12 sin (cos ) 2F1
1
12 + 1
2
;
2
;
2
e
12t
r h
1
d
1
h = f
+ ftt + f :
f
=
f
ftt :
should be static and spherically symmetric, which implies the general ansatz
h dx dx = f ( )g d 2 + ftt( )gttdt2 +
f ( ) tan2 d 2d 1
:
We rst impose the divergence free and tracelessness conditions. We have
We use this to eliminate f ,
Moving to the divergence, only the component r h
is not automatically zero. We nd
r h
= f 0 +
d + 1
cos sin
f
cos
sin
cos
sin
f
+
ftt = 0
(5.15)
(5.16)
(5.17)
(5.19)
(5.20)
(5.21)
(5.22)
which we solve as
ftt =
tan f 0 +
1
We then work out the
component of the eld equation,
2
r h
d)
Setting this to zero, the normalizable solution is
d + 1
cos2
f :
2 cot
f 0
2)
cos2
= (cos ) +22F1
2
;
+ 2
2
;
2
:
This completely speci es the solution, and we now have all we need to plug into (5.6). We
refrain from writing out the somewhat lengthy formulas. The series expansion of the result
matches up with (5.1) and (5.2) as expected.
General `: proof via conformal Casimir equation
As in the case of scalar exchange, the most e cient way to verify that a geodesic Witten
diagram yields a conformal partial wave is to check that it is an eigenfunction of the
conformal Casimir operator with the correct eigenvalue and asymptotics.
We start from the general expression (5.4). A rankn tensor on AdS is related to a
tensor on the embedding space via
The bulktoboundary propagator lifted to the embedding space is
Y ( ) =
p
e X1 + e
X2
:
i :
(5.23)
f :
(5.24)
(5.25)
(5.26)
(5.27)
(5.28)
(5.29)
(5.30)
In particular, this holds for the bulktobulk propagator of the spin` eld, and so we can
write
Gbb(y; y0; ; `) = [Gbb(Y; Y 0; )]M1:::M`;N1:::N` d
: : :
d
d 0
dY M1
dY M` dY 0N1
: : :
dY 0N`
d 0
:
Now, [Gbb(Y; Y 0; )]M1:::M`;N1:::N` only depends on Y and Y 0. Since Y M dY M = 12 dd (Y Y ) =
d
0, when pulled back to the geodesics the only contributing structure is
T 1::: n =
TM1:::Mn :
[Gbb(Y; Y 0; )]M1:::M`;N1:::N` = f (Y Y 0)Y M01 : : : Y M0` YN1 : : : YN` :
We also recall a few other useful facts. Lifted to the embedding space, the geodesic
connecting boundary points X1 and X2 is
We follow the same strategy as in the case of scalar exchange. We start by isolating
the part of the diagram that contains all the dependence on X1;2,
FM1:::M` (X1; X2; Y 0; ) =
Z
12
M1:::M`;N1:::N` d
dY M1
: : :
dY M`
:
Here, Y ( ) lives on 12, but Y 0 is left arbitrary. This is the spin` generalization of '12(y)
de ned in (3.4), lifted to embedding space. We now argue that this is annihilated by the
SO(d + 1; 1) generators L1AB + L2AB + LYAB0. This generator is the sum of three generators
in the scalar representation, plus a \spin" term acting on the free indices N1 : : : N`. This
operator annihilates any expression of the form g(X1 X2; X1 Y 0; X2 Y 0)XN1 : : : XN` , where
each X stands for either X1 or X2. To show this, we just note the SO(d + 1; 1) invariance of
the dot products, along with the fact that XN is the normal vector to the X2 = 0 surface
and so is also SO(d + 1; 1) invariant. From (5.28){(5.30) we see that FN1:::N` (X1; X2; Y 0; )
is of this form, and so is annihilated by L1AB + L2AB + LYAB0. We can therefore write
(L1AB + L2AB)2FN1:::N` (X1; X2; Y 0; ) = (LYAB0)2FN1:::N` (X1; X2; Y 0; )
= C2( ; `)FN1:::N` (X1; X2; Y 0; )
(5.32)
(5.31)
where we used that (LYAB0)2 is acting on the spin` bulktobulk propagator, which is
an eigenfunction of the conformal Casimir operator13 with eigenvalue (3.27). The
relation (5.32) holds for all Y 0, and hence holds upon integrating Y 0 over 34 with any weight.
Hence we arrive at the conclusion
(L1AB + L2AB)2W ;`(xi) = C2( ; `)W ;`(xi)
(5.33)
which is the same eigenvalue equation obeyed by the spin` conformal partial wave,
W ;`(xi). The short distance behavior as dictated by the OPE is easily seen to match
in the two cases, establishing that we have the same eigenfunction.
We conclude that the spin` geodesic Witten diagram is, up to normalization, equal to the spin` conformal partial wave.
5.6
Comparison to double integral expression of Ferrara et al.
It is illuminating to compare our expression (5.4) to equation (50) in [2], which gives the
general result (in d = 4) for the scalar conformal partial wave with spin` exchange, written
as a double integral. We will rewrite the result in [2] in a form permitting easy comparison
to our formulas. First, it will be useful to rewrite the scalar bulktobulk propagator (2.18)
by applying a quadratic transformation to the hypergeometric function,
Gbb(y; y0; ) =
2F1
2
;
+ 1
2
;
+ 1
; 2 :
d
2
(5.34)
13Note that the conformal Casimir is equal to the spin` Laplacian up to a constant shift: (LYAB0)2 =
r`2 + `(` + d
1) [77].
Next, recall that in embedding space the geodesics are given by (3.40), from which we
compute the quantity
with one point on each geodesic
1 =
Y ( ) Y ( 0) =
d 0
Comparing to [2], we have
+ and
e
1 =
the form
2
2
1 =
Z
Z
12
34
x12x34
.
x12x34
With these de nitions in hand, it is not hard to show that the result of [2] takes
Here Gbb y( ); y( 0);
is the scalar bulktobulk propagator (5.34), and C`0(x) is a
Gegenbauer polynomial. This obviously looks very similar to our expression (5.4), and indeed
agrees with it for ` = 0. The two results must be equal (up to normalization) since they
are both expressions for the same conformal partial wave. If we assume that equality
holds for the integrand, then we nd the interesting result that the pullback of the
spin` propagator, as written in (5.5), is equal to C`0(2
1)Gbb y( ); y( 0);
. The general
spin` propagator is very complicated (see [65, 66]), but apparently has a simple
relation to the scalar propagator when pulled back to geodesics. It would be interesting to
verify this.
5.7
Decomposition of spin1 Witten diagram into conformal blocks
In the case of scalar exchange diagrams, we previously showed how to decompose a Witten
diagram into a sum of geodesic Witten diagrams, the latter being identi ed with conformal
partial waves of both single and doubletrace exchanges. We now wish to extend this to
the case of higher spin exchange; we focus here on the case of spin1 exchange for simplicity.
A picture of the nal result is given in gure 6.
As discussed in section 2, given two scalar operators in a generalized free eld theory,
we can form scalar double trace primaries with schematic form [O1O2]m;0
dimension
(12)(m; 0) =
1 +
2 + 2m + O(1=N 2), and vector primaries [O1O2]m;1
(12)(m; 1) =
1 +
2 + 1 + 2m + O(1=N 2). The analysis
of [43], and later [26, 65, 66] demonstrated that these conformal blocks, and their cousins
[O3O4]n;0 and [O3O4]n;1, should appear in the decomposition of the vector exchange Witten
diagram, together with the exchange of a singletrace vector operator. The computations
below will con rm this expectation.
The basic approach is the same as in the scalar case, although the details are more
complicated.
Before diving in, let us note the main new features. In the scalar case
a basic step was to write, in (4.1), the product of two bulktoboundary propagators
partial waves. The term in the upper right captures the singletrace exchange of the dual vector
operator. The second line captures the CFT exchanges of the ` = 0 doubletrace operators [O1O2]m;0
and [O3O4]n;0. Likewise, the
operators [O1O2]m;1 and [O3O4]n;1.
nal line captures the CFT exchanges of the ` = 1 doubletrace
Gb@ (y; x1)Gb@ (y; x2) as a sum over solutions '12(y) of the scalar wave equation sourced on
the 12 geodesic. Here, we will similarly need a decomposition of Gb@ (y; x1)r Gb@ (y; x2),
where r is a covariant derivative with respect to bulk coordinates y. It turns out that this
can be expressed as a sum over massive spin1 solutions and derivatives of massive scalar
solutions. This translates into the statement that the spin1 exchange Witten diagram
decomposes as a sum of spin1 and spin0 conformal blocks, as noted above.
Our rst task is to establish the expansion
cmAm; (y) + bmr 'm(y)
X
m
where Am; (y) and 'm(y) denote the solutions to the massive spin1 and spin0 equations
sourced on 12, found earlier in sections 5.3 and 4.1, respectively.14 m labels the masses of
the bulk elds, to be determined shortly. We will not attempt to compute the coe cients
cm and bm, which is straightforward but involved, contenting ourselves to determining
the spectrum of conformal dimensions appearing in the expansion, and showing how the
expansion coe cients can be obtained if desired.
Following the scalar case, we work in global AdS and send t1 !
1, t2 !
Dropping normalizations, as we shall do throughout this section, we have
12t:
12t
(`) denote the dimension of the corresponding spin, we have, from (5.15)
m
Now to the computation. We consider a theory of massive scalars coupled to a massive
vector eld via couplings ir
j A . The Witten diagram with vector exchange is then
Z Z
y y0
Gbb (y; y0; )
12 ; (0)
m
(m1) + 12 +1
2
d
2
2
; cos2
(1)
m
2
(cos ) m e
(0)
12t
12 +1
; (1)
m
d 2
2
e
12t
Letting
and (3.13),
'm = 2F1
(m0) + 12 ;
2
m
2
Am; =
12 sin (cos ) m 2F1
(1)
Am;t =
1
The various terms have the following powers (cos2 )k in an expansion in powers of cos2 ,
Am; : k =
r 'm : k =
Am;t : k =
rt'm : k =
+ q
1 +
(1)
m
2
(0)
m
1 +
2
2
2
(1)
m
(0)
m
2
2
2
+ q
+ q
1
2
1
+ q
+ q
14'm is just '1m2, whose superscript we suppress for clarity, and likewise for 'n and '3n4.
(5.38)
(5.39)
(5.40)
(5.41)
(5.42)
where q = 0; 1; 2; : : :. Comparing, we see that we have the right number of free coe cients
for (5.39) to hold, provided we have the following spectrum of dimensions appearing
(m0) =
(m1) =
1 +
1 +
2 + 2m
2 + 1 + 2m
with m = 0; 1; 2; : : :. The formulas above can be used to work out the explicit coe cients
cm and bm. We noted at the beginning of this subsection that this spectrum of dimensions
coincides with the expected spectrum of doubletrace scalar and vector operators appearing
in the OPE, at leading order in large N .
We may now rewrite (5.38) as15
We expand this out in an obvious fashion as
cmAm; (y) + bmr 'm(y) Gbb (y; y0; ) cnAn; (y0) + bnr 'n(y0) :
A4Vec(xi) = AAA(xi) + AA (xi) + A A(xi) + A
(xi) :
The next step is to relate each term to geodesic Witten diagrams, which we now do in
turn.
The solution Am; (y) can be expressed as
X cmcn
m;n
Z Z
y y0
Am; (y)Gbb (y; y0; )An; (y0) :
Z
12
2
Z
12
dy ( )
Gbb y( ); y; (1) :
m
(5.47)
which is easily veri ed for a straight line geodesic at the center of global AdS, and hence
dy ( )
d
is true in general. Using this we obtain (dropping the normalization, as usual)
AAA =
X cmcn
m;n
dy ( )
d
Gbb y( ); y; (1) Gbb; (y; y0; )Gbb y0; y( 0); (n1)
m
dy0 ( 0)
:
(5.48)
15Following the precedent of section 4, all quantities with an m subscript refer to the doubletrace
operators appearing in the O1O2 OPE, and those with an n subscript refer to the doubletrace operators
appearing in the O3O4 OPE.
(5.43)
(5.44)
(5.45)
(5.46)
The bulktobulk propagator for the vector eld obeys
m2)Gbb (y; y0; ) =
where
y0) denotes a linear combination of g
this, and the fact that the propagator is divergence free at noncoincident points, we can
y0) and r r
(y
y0). Using
verify the composition law
Z
y0
Gbb (y; y0; )Gbb; (y0; y00; 0) =
1
(m0)2 Gbb (y; y00; )
Gbb (y; y00; 0) :
We use this relation twice within (5.48) to obtain a sum of three terms, each with a single
vector bulktobulk propagator. Note also that these propagators appear pulled back to
the geodesics. Each term is thus a geodesic Witten diagram with an exchanged vector,
that is, a spin1 conformal partial wave. The spectrum of spin1 operators that appears is
1 +
2 + 1 + 2m ;
3 +
4 + 1 + 2n ;
m; n = 0; 1; 2; : : :
(5.51)
So the contribution of AAA is a sum of spin1 conformal blocks with internal dimensions
corresponding to the original exchanged eld, along with the expected spin1 double trace
operators built out of the external scalars.
Next we integrate by parts in y, use r Gbb (y; y0; ) / r
y0), and integrate by parts
again, to get
A A =
X bmcn
m;n
Z
y0
r 'm(y0)An(y0) :
Now we write An(y0) as an integral over 34 as in (5.47) and then again remove the
bulktobulk propagator by integrating by parts. This yields
A A =
X bmcn
m;n
Z
34
r 'm y( 0) :
(5.54)
dy ( 0)
d 0
Writing 'm as an integral sourced on 12 we obtain
A A = X bmcn
m;n
Z
Z
5.7.2
AA and A A
We start with
A A =
X cmbn
m;n
Z Z
y y0
r 'm(y)Gbb (y; y0; )An; (y0) :
Integrating by parts and using
Gb@ y( 0); x3
Gb@ y( 0); x4 we see that A A decomposes into a sum of spin0 exchange geodesic Witten
diagrams. That is, A A contributes a sum of spin0 blocks with conformal dimensions
1 +
m = 0; 1; 2; : : :
(5.49)
(5.50)
(5.52)
(5.53)
(5.55)
(5.56)
By the same token AA yields a sum of spin0 blocks with conformal dimensions
3 +
4 + 2n ;
n = 0; 1; 2; : : :
Integration by parts reduces this to
A
X bmbn
Z Z
y y0
r 'm(y)Gbb (y; y0; )r 'n(y0) :
A
X bmbn
Z
y
r 'm(y)r 'n(y) :
Now rewrite the scalar solutions as integrals over the respective geodesic sources,
X bmbn
Z Z
Z
y0 12
34
r
Gbb y( ); y0; (0)
m
r 0 Gbb y0; y( 0); (n0) :
The composition law analogous to (5.50) is easily worked out to be
r
0
Gbb y( ); y0; (0)
m
r 0 Gbb y0; y( 0); (n0) = cmnGbb y( ); y( 0); (0)
m
A
Z
y0
+ dmnGbb y( ); y( 0); (n0)
with some coe cients cmn and dmn that we do not bother to display here. Inserting this
in (5.60) we see that A
decomposes into a sum of scalar blocks with conformal dimensions
1 +
3 +
4 + 2n ;
m; n = 0; 1; 2; : : :
(5.62)
5.7.4
We have shown that the Witten diagram involving the exchange of a spin1 eld of
dimension
decomposes into a sum of spin1 and spin0 conformal blocks. The full spectrum of
conformal blocks appearing in the decomposition is
scalar:
vector:
1 +
2 + 2n ;
3 +
1 +
4 + 2n
2 + 1 + 2n ;
3 +
4 + 1 + 2n
(5.63)
where n = 0; 1; 2; : : :. This matches the spectrum expected from 1=N counting, including
single and doubletrace operator contributions. With some patience, the formulas above
can be used to extract the coe cient of each conformal block, but we have not carried this
out in full detail here.
While we have not explored this in any detail, it seems likely that the above method
can be directly generalized to the case of arbitrary spin` exchange. The split (5.45) will
still be natural, and a higher spin version of (5.50) should hold.
(5.57)
(5.58)
(5.59)
(5.60)
(5.61)
HJEP01(26)4
diagrams, and hence decomposed into conformal blocks using our methods.
6
Discussion and future work
In this paper, we have shed new light on the underlying structure of treelevel
scattering amplitudes in AdS. Fourpoint scalar amplitudes naturally organize themselves into
geodesic Witten diagrams; recognizing these as CFT conformal partial waves signals the
end of the computation, and reveals a transparency between bulk and boundary with little
technical e ort required. We are optimistic that this reformulation extends, in some
manner, to computations of generic holographic correlation functions in AdS/CFT. To that
end, we close with some concrete observations and proposals, as well as a handful of future
directions.
Adding loops. It is clearly of interest to try to generalize our techniques to loop level.
We rst note that there is a special class of loop diagrams that we can compute already
using these methods: namely, those that can be written as an in nite sum of treelevel
exchange diagrams [21]. For the same reason, this is the only class of loop diagrams whose
Mellin amplitudes are known [21]. These diagrams only involve bulktobulk propagators
that all start and end at the same points; see gure 7 for examples. Careful study of the
resulting sums would be useful.
More generally, though, we do not yet know how to decompose generic diagrams into
geodesic objects. This would seem to require a \geodesic identity" analogous to (4.1) that
applies to a pair of bulktobulk propagators, rather than bulktoboundary propagators.
It would be very interesting to nd these, if they exist. Such identities would also help to decompose an exchange Witten diagram in the crossed channel.
Decomposition of Witten diagrams in the crossed channel.
The present work
studies the decomposition of an schannel exchange Witten diagram into schannel partial
waves. It is clearly of interest to understand, using the language of geodesic Witten
diagrams, how the same diagram decomposes into tchannel partial waves, which corresponds
to using the OPE on the pairs of operators O1O3 and O2O4. As explored in [45], the
statement is that the basic scalar exchange Witten diagram decomposes into tchannel partial
waves involving only scalar double trace operators [O1O3]n;0 and [O2O4]n;0. As we mention
vepoint treelevel Witten diagram. However, it is not equal to the vepoint conformal
partial wave, as discussed in the text.
above, this decomposition seems to require new geodesic identities involving bulktobulk
propagators.
Derivative interactions.
We have primarily focused on decomposing Witten diagrams
with nonderivative interactions. For example, the contact diagram of gure 4 is based
on the interaction
1 2 3 4
. We would like to be able to e ciently decompose Witten
diagrams with derivative interactions too, like 1r 1
r 2 : : : r k 2 3r 1 r 2 : : : r k 4; a
precise version of the identity in equation (5.39) would be su cient to treat the k = 1 case,
but we would like to generalize that to add more Lorentz indices. We anticipate that this
identity exists and involves propagators of massive elds of spin l
k.
Adding legs. Consider for example a vepoint correlator of scalar operators hO1(x1) : : :
O5(x5)i. We can de ne associated conformal partial waves by inserting projection
opera
W ;`; 0;`0 (xi) = hO1(x1)O2(x2)P ;`O5(x5)P 0;`0 O3(x3)O4(x4)i :
(6.1)
Using the OPE on O1O2 and O3O4, reduces this to threepoint functions. The question
is, can we represent W ;`; 0;`0 (xi) as a geodesic Witten diagram?
Suppose we try to dismantle a tree level vepoint Witten diagram. For de niteness,
we take `0 = ` = 0. All tree level vepoint diagrams will lead to the same structures
upon using our geodesic identities: namely, they can be written as sums over geodesictype
diagrams, each as in
gure 8, which we label Wc a;0; b;0(xi). This is easiest to see starting
from a 5 contact diagram, and using (4.1) on the pairs of propagators (12) and (34). In
that case,
a 2 f mg and
b 2 f ng. As an equation, gure 8 reads
Wc a;0; b;0(xi) =
Gbb y( ); y5; a Gb@ (y5; x5)Gbb y5; y( 0); b
(6.2)
Z
Z
12
Z
y5
34
{ 42 {
Note that the vertex at y5, indicated by the orange dot in the gure, must be integrated
over all of AdS. Could these diagrams be computing W a;0; b;0(xi) as de ned above? The
answer is no, as a simple argument shows. Suppose we set
5 = 0 in (6.2), which requires
a =
b
. From (6.1) it is clear that we must recover the fourpoint conformal partial
wave with the exchanged primary ( ; 0). So we should ask whether (6.2) reduces to the
expression for the fourpoint geodesic Witten diagram, W ;0. Using Gb@ (y5; x5)j 5=0 / 1,
the integral over y5 becomes
Z
y5
Gbb y( ); y5;
Gbb y5; y( 0);
:
Therefore, the
5 = 0 limit of (6.2) does not give back the fourpoint partial wave, but
rather its derivative with respect to m2 , which is a di erent object.
We conclude that although we can decompose a vepoint Witten diagram into a sum
of diagrams of the type in
gure 8, this is not the conformal block decomposition. This
raises two questions: what is the meaning of this decomposition in CFT terms, and (our
original question) what diagram computes the vepoint partial wave?
External operators with spin.
Another obvious direction in which to generalize is to
consider correlation functions of operators carrying spin. As far as the conformal blocks
go, partial information is available. In particular, [7] obtained expressions for such blocks
as di erential operators acting on blocks with external scalars, but this approach is limited
to the case in which the exchanged operator is a symmetric traceless tensor, since only
such operators appear in the OPE of two scalar operators. The same approach was taken
in [78]. Explicit examples of mixed symmetry exchange blocks were given in [79].
Our formulation in terms of geodesic Witten diagrams suggests an obvious proposal
for the AdS computation of an arbitrary conformal partial wave: take our usual
expression (1.1), now with the bulktoboundary and bulktobulk propagators corresponding to
the elds dual to the respective operators. Of course, there are many indices here which
have to be contracted, and there will be inequivalent ways of doing so. But this is to
be expected, as in the general case there are multiple conformal blocks for a given set of
operators, corresponding to the multiplicity of ways in which one spinning primary can
appear in the OPE of two other spinning primaries. It will be interesting to see whether
this proposal turns out to be valid. As motivation, we note that it would be quite useful for
bootstrap purposes to know all the conformal blocks that arise in the fourpoint function
A related pursuit would be to decompose all fourpoint scalar contact diagrams,
including any number of derivatives at the vertices. This would involve a generalization
of (5.39) to include more derivatives.
Virasoro blocks and AdS3/CFT2.
Our calculations give a new perspective on how to
construct the dual of a generic Virasoro conformal block: starting with the geodesic Witten
diagram, we dress it with gravitons. Because Virasoro blocks depend on c, a computation
in semiclassical AdS gravity would utilize a perturbative 1=c
GN expansion. In [34], we
put the geodesic approach to use in constructing the holographic dual of the heavylight
Virasoro blocks of [30], where one geodesic essentially backreacts on AdS to generate a
conical defect or black hole geometry. It would be worthwhile to pursue a 1=c expansion
around the geodesic Witten diagrams more generally.
A closely related question is how to decompose an AdS3 Witten diagram into Virasoro,
rather than global, blocks. For a treelevel diagram involving light external operators like
those considered here, there is no di erence, because the large c Virasoro block with light
external operators reduces to the global block [38]. It will be interesting to see whether
loop diagrams in AdS3 are easier to analyze using Virasoro symmetry.
Assorted comments.
The geodesic approach to conformal blocks should be useful in
deriving various CFT results, not only mixed symmetry exchange conformal blocks. For
example, the conformal blocks in the limits of large , ` or d [80{86], and subleading
corrections to these, should be derivable using properties of AdS propagators. One can
also ask whether there are similar structures present in bulk spacetimes besides AdS. For
instance, an analog of the geodesic Witten diagram in a thermal spacetime would suggest a
useful ingredient for parameterizing holographic thermal correlators. Perhaps the existence
of a dS/CFT correspondence suggests similar structures in de Sitter space as well.
It is natural to wonder whether there are analogous techniques to those presented
here that are relevant for holographic correlators of nonlocal operators like Wilson loops
or surface operators, perhaps involving bulk minimal surfaces.
Let us close by noting a basic fact of our construction: even though a conformal
block is not a semiclassical object per se, we have given it a representation in terms of
classical elds propagating in a smooth spacetime geometry. In a bulk theory of quantum
gravity putatively dual to a nite N CFT, we do not yet know how to compute amplitudes.
Whatever the prescription, there is, evidently, a way to write the answer using geodesic
Witten diagrams. It would be interesting to understand how this structure emerges.
Acknowledgments
We thank Eric D'Hoker, Liam Fitzpatrick, Tom Hartman, Daniel Ja eris, Juan Maldacena,
Joao Penedones and Sasha Zhiboedov for helpful discussions. E.P. wishes to thank the
KITP and Strings 2015 for hospitality during this project. P.K. is supported in part by
NSF grant PHY1313986. This research was supported in part by the National Science
Foundation under Grant No. NSF PHY1125915. E.P. is supported by the Department of
Energy under Grant No. DEFG0291ER40671.
Open Access.
This article is distributed under the terms of the Creative Commons
Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
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