JHEP12(2013)004
Published for SISSA by Springer
Received: June 26, 2013
Accepted: November 22, 2013
Published: December 2, 2013
The analytic bootstrap and AdS superhorizon locality
A. Liam Fitzpatrick,
a
Jared Kaplan,
a,b
David Poland
c
and David Simmons-Duffin
d
a
Stanford Institute for Theoretical Physics, Stanford University,
Stanford, CA 94305, U.S.A.
b
Department of Physics and Astronomy, Johns Hopkins University,
Baltimore, MD 21 218 , U .S.A .
c
Department of Physics , Yale University,
New Haven, CT 06520, U.S.A.
d
School of Natural Sciences, Institute for Advanced Study,
Princeton, NJ 08540, U.S.A.
E-mail:
fitzpatr@stanford.edu, jaredk@pha.jhu.edu,
david.poland@yale.edu, davidsd@gmail.com
Abstract: We take an analytic approach to the CFT bootstrap, studying the 4-pt cor-
relators of d > 2 dimensional CFTs in an Eikonal-type limit, where the confor mal cross
ratios satisfy |u| ≪ |v| < 1. We prove that every CFT with a scalar operator φ must
contain infinite sequences of operators O
τ,ℓ
with twist approaching τ → 2∆
φ
+ 2n for each
integer n as ℓ → ∞. We show how the rate of approach is controlled by the twist and OPE
coefficient of the leading twist operator in the φ ×φ OPE, and we discuss SCFTs and the
3d Ising Mo de l as examples. Additionally, we show that the OPE coefficients of oth er large
spin operators appearing in the OPE are bounded as ℓ → ∞. We interpret these results
as a statement about superhoriz on locality in AdS for gener al CFTs.
Keywords: Conformal and W Symmetry, Nonperturbative Effects, AdS-CFT Correspon-
dence
ArXiv ePrint:
1212.3616
c
SIS S A 2013 doi:
10.1007/JHEP12(2013)004
JHEP12(2013)004
Contents
1 Introduction and review 1
1.1 Lightning bootstrap review 3
2 The boots tr ap and large ℓ operators 4
2.1 An elementary illustration from mean field theory 4
2.2 Existence of twist 2∆
φ
+ 2n + γ(n, ℓ) operators at large ℓ 6
2.2.1 Relation to numerical results and the 3d Ising model 10
2.3 Properties of isolated towers of operators 11
2.3.1 Implications for SCF Ts in 4d 14
2.4 Bounding contributions from operators wi t h general twists 15
2.5 Failure in two dimensions 16
3 AdS interpretation 17
4 Discussion 18
A Properties of conf orm al blocks 20
A.1 Factorization at l ar ge ℓ and small u 20
A.1.1 Factorization in 2 and 4 dimensions 20
A.1.2 Extension to even dimensions via a recursion r e l ati on for F
(d)
(τ, v) 21
A.1.3 Extension to odd dimensions 22
A.1.4 Further approximations for the function k
2ℓ
(1 −z) 22
A.2 Positivity of coefficients and exponential falloff at large τ 23
B ρ(σ) and its crossing equation 24
B.1 Existence of ρ(σ) 24
B.2 Crossing symmetry for ρ(σ) 26
B.3 Bounds on OPE coefficient densities 27
C Generalization to distinct operators φ
1
and φ
2
29
D Comparison with gravity results 30
1 Introduction and review
The last few years have seen a remarkable resurgence of interest in an old approach to Con-
formal Field Theory (CFT), the conformal bootstrap [
1, 2], with a great deal of progress
leading to new results of phenomenological [
3–9] and theoretical [10–15] import. Most of
these new works use numerical methods to constrain the spectrum and OPE coefficients
– 1 –
JHEP12(2013)004
of general CFTs. In a parallel series of developments, there has been significant progress
understanding effec t i ve field theory in AdS and its interpretation in CFT [
10, 13, 16–19].
This has led to a general bottom-up classification of which CFTs have dual [
20–22] de-
scriptions as effective field theories in AdS, providing an understanding of AdS locality
on all length scales greater t han the inverse energy cutoff in the bulk. In fact, these two
developments are closely related, as the semi nal paper [
10] and subsequent work begin by
applying the bootstrap to the 1/N expansion of CFT correlators. This approach has been
fruitful, especially when interpreted in Mellin space [
19, 23–28], but it is an essentially
perturbative approach analogous [
29] to the use of di spersion relations for the study of
perturbative scattering amplitudes.
In light of recent progress, one naturally wonders if an analytic approach to the boot-
strap coul d yield interesting new exact results. In fact, in [14] bounds on operator product
expansion (OPE) coefficients for large dimension operators have already been obtained.
We will obtain a different sort of bound on both OPE coefficients and operator dimension s
in the limit of large angular momentum, basically providing a non-perturbative bootstrap
proof of some results that Alday and Maldacena [30] have also discussed.
1
Specifically, we will study a general scalar primary operator φ of dimension ∆
φ
in a
CFT in d > 2 dimensions. We will prove that for each non-negative integer n there must
exist an infinite tower of operators O
τ,ℓ
with twist τ → 2∆
φ
+ 2n appearing in the OPE
of φ with itself. This m ean s that at large ℓ, and we can define an ‘anomalous dimension’
γ(n, ℓ) which vanis he s as ℓ → ∞. If there ex i st s one such operator at each n and ℓ, we will
argue that at large ℓ the anomalous dimensi ons should roughly approach
γ(n, ℓ) ≈
γ
n
ℓ
τ
m
, (1.1)
where τ
m
is the twist of the minimal twist operator appearing in the OPE of φ with itself.
Related predictions can be made about the OPE coeffici e nts. Finally, we wi l l show that the
OPE coefficients of other operators appearing in the OPE of φ with its e l f at lar ge ℓ must
be bounded, so that they fall off even faster as ℓ → ∞. Simi l ar results also hold for the
OPE of pairs of operators φ
1
and φ
2
, although for simplicity we will leave the discussion
of this generali z ati on to appendix
C.
Our arguments fail for CFTs in two dimensions, and in fact we will see that the
c = 1/2 minimal model provides an explicit counter-example. Two dimensional CFTs are
distinguished because there is no gap between the twist of the identity operator and the
twist of other operators, such as conserved currents and the energy-mome ntum tensor.
Our results can be interpreted as a pro of that all CFTs in d > 2 dimensions have cor-
relators that are dual to local AdS physics on superhorizon scales. That is, CF T processes
that are dual to bulk interactions will effectively shut off as the bulk impact parameter is
taken to b e much greater than t he AdS length . This can also be v i e wed as a strong form
of the clu st er decomposition principle in the bulk. Since the early days of AdS/CFT it
1
The authors of [
30] explicitly discuss minimal twist double-trace operators in a large N gauge theory;
however their elegant argument can be applied in a mo r e general context, beyond perturbation theory and
for general twists. We thank J. Maldacena for discussions of this point.
– 2 –
JHEP12(2013)004
has been argued that this notion of “coarse locality” [10] could be due to a decoupling of
modes of very different wavelengths, but it has been challenging to make this qualitative
holographic RG intuition precise. The bootstrap offers a precise and general method for
addressing coarse l ocality.
For the rem ai nde r of this section we will give a quick review of the CFT bootstrap.
Then in section
2 we del ve into the argument, first giving an illustrative example from
mean field th eor y (a Gaussian CFT, with all correlators fixed by 2-pt functions, e.g. a free
field theory in AdS). We give the comp l et e argument in sections
2.2 and 2.4, with some
more specific results and examples that follow from further assumptions in secti on
2.3. We
provide more de t ai l on how two dimensi onal CFTs escape our con cl us i ons in section
2.5.
In section
3 we connect our results to superhorizon locality in AdS, and we conclude with
a brief discus si on in section
4. In appe ndi x A we collect some r e sul t s on relevant approxi-
mations of the conformal blocks in four and general dimensions. In appendix
B we give a
more formal and rigorous version of the argument in section
2. In appendix C we explain
how our results generalize to terms occurring in the OPE of disti nct operators φ
1
and φ
2
.
In appendix
D we connect our results wi th perturbative gravity computations in AdS.
Note added. After this work was complete d we learned of the r e l at ed work of Komargod-
ski and Zhi boedov [
31]; they obtain very similar results using somewhat different methods.
1.1 Lightning bootstrap review
In CFTs, the bootstrap equation follows from the constraints of conformal invariance and
crossing symmetry applied to the operator product expansion, which says that a product
of local operators is equivalent to a sum
φ(x)φ(0) =
X
O
c
O
f
O
(x, ∂)O(0). (1.2)
Conformal invariance relates the OPE coefficients of all operators in the same irreducible
conformal multiplet, and this allows one to reduce the sum above to a sum over diff er e nt
irreducible multiplets, or “conformal blocks”. When this expansion is performed inside of
a four-poi nt function, the contribution of each block is just a constant “conformal b l ock
coefficient” P
O
∝ c
2
O
for the entire multiplet times a func t i on of the x
i
’s whose functional
form depe nds only on the spin ℓ
O
and dimension ∆
O
of the lowest-weight (i.e. “pri mar y ”)
operator of t he multiplet:
hφ(x
1
)φ(x
2
)φ(x
3
)φ(x
4
)i =
1
(x
2
12
x
2
34
)
∆
φ
X
O
P
O
g
τ
O
,ℓ
O
(u, v), (1.3)
where x
ij
= x
i
− x
j
, the twist of O is τ
O
≡ ∆
O
− ℓ
O
, and
u =
x
2
12
x
2
34
x
2
24
x
2
13
, v =
x
2
14
x
2
23
x
2
24
x
2
13
, (1.4)
are the conformally invariant cross-ratios. The functions g
τ
O
,ℓ
O
(u, v) are also usually re-
ferred to as conformal blocks or conformal partial waves [
32–35], and they are crucial
elementary ingredients in the bootstr ap program.
– 3 –
JHEP12(2013)004
In the above, we took the OPE of φ(x
1
)φ(x
2
) and φ(x
3
)φ(x
4
) in si de the four-point
function, but one can also take the OPE between different pairs of operators, and the
result should be the same. For example, swapping 2 ↔ 3 gives the bootstrap equation
1
(x
2
12
x
2
34
)
∆
φ
X
O
P
O
g
τ
O
,ℓ
O
(u, v) =
1
(x
2
14
x
2
23
)
∆
φ
X
O
P
O
g
τ
O
,ℓ
O
(v, u). (1.5)
(Meanwhile, swapping 1 ↔ 2 or 3 ↔ 4 gives the constraint that only even spin operators
can appear in the OPE. Other permutations give no new constraints.) Much of the power
of th i s constraint follows from the fact that by unitarity, the conformal block coefficients
P
O
must all be non-negative in each of these channels, because the P
O
can be taken to be
the squares of real OPE coefficients.
2 The bootstrap and large ℓ operators
Although some of the arguments below are technical, the idea behind them is very sim-
ple. By way of analogy, consider the s-channel partial wave d ec omposition of a tree-level
scattering amplitude with poles in both the s and t channels. The center of mass energy is
simply
√
s, so the s-channel poles will appear explicitly in the partial wave decomposition.
However, the t-channel poles will not be manifest. They wi l l arise from the infinite sum
over angular momenta, because the large angular momentum region encodes long-distance
effects. Crossing symmetry will impose constraints between the s-wave and t-wave de-
compositions, relating the large ℓ behavior in one channel with the pole structure of the
other channel.
We will be studying an anal ogous phenomenon in the conformal block (sometimes
called confor m al partial wave) decompositions of CFT correlation functions. The metaphor
between scattering amplitudes and CFT correlation functions is very direct when the CFT
correlators are expressed in Mellin space, but in what follows we will stick to position space.
In position space CFT correlators, the poles of t he scattering amplitude are analogous to
specific p ower-laws in conformal cross-ratios, with the smallest power-laws corresponding
to the leading poles.
2.1 An elementary illustration from mean field th eory
Let us begin by considering what naively appears to be a paradox. Consider the 4-point
correlation function in a CFT with only Gaussian or ‘mean field theory’ (MFT) type
correlators. These mean field theories are the dual of free field theories in AdS. We will
study the 4-pt correlator of a dim ens i on ∆
φ
scalar operator φ in such a theory. By definition,
in mean field theory the 4-pt correlator is given as a sum over the 2-pt function contractions:
hφ(x
1
)φ(x
2
)φ(x
3
)φ(x
4
)i =
1
(x
2
12
x
2
34
)
∆
φ
+
1
(x
2
13
x
2
24
)
∆
φ
+
1
(x
2
14
x
2
23
)
∆
φ
,
=
1
(x
2
13
x
2
24
)
∆
φ
u
−∆
φ
+ 1 + v
−∆
φ
. (2.1)
Since this is the 4-pt correlator of a unitary CFT, it has a conformal block decomposi-
tion in every channel with positive conformal block coefficients. The operators appearing in
– 4 –