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JHEP01(2019)196

Published for SISSA by Springer

Received: November 21, 2018

Accepted: December 25, 2018

Published: January 25, 2019

All tree-level correlators in AdS

×S

supergravity:

hidden ten-dimensional conformal symmetry

Simon Caron-Huot and Anh-Khoi Trinh

Department of Physics, McGill University,

3600 Rue University, Montr´eal, QC H3A 2T8, Canada

E-mail: schuot@physics.mcgill.ca, anh-khoi.trinh@mail.mcgill.ca

Abstract: We study correlators of four protected (half-BPS) operators in strongly coupled

supersymmetric Yang-Mills theory. These are dual to tree-level supergravity amplitudes

on AdS

×S

for various spherical harmonics on the ﬁve-sphere. We use conformal ﬁeld

theory methods, in particular a recently obtained Lorentzian inversion formula, to analyti-

cally bootstrap these correlators. The extracted 1/N

double-trace anomalous dimensions

conﬁrm a simple pattern recently conjectured by Aprile, Drummond, Heslop and Paul. We

explain this pattern by an unexpected ten-dimensional conformal symmetry which appears

to be enjoyed by tree-level supergravity (or a suitable subsector of it). The symmetry

combines all spherical harmonics into a single ten-dimensional object, and yields compact

expressions for the leading logarithmic part of any half-BPS correlator at each loop order.

Keywords: AdS-CFT Correspondence, Conformal Field Theory, 1/N Expansion, Super-

symmetric Gauge Theory

ArXiv ePrint: 1809.09173

Open Access,

 The Authors.

Article funded by SCOAP

https://doi.org/10.1007/JHEP01(2019)196

JHEP01(2019)196

spin two or less [1]. While the gravity side of the correspondence typically remains more

straightforward to use, new analytic bootstrap techniques are becoming applicable and

should be vigorously pursued being potentially more general. They appear especially ad-

vantageous for observables like four-point correlation functions, which can now be studied

to unprecedented precision.

In this paper, we study the four-point function of general half-BPS states in maximally

supersymmetric super Yang-Mills theory, or N = 4 SYM for short, dual to type IIB

superstring theory on AdS

×S

geometry. Through the AdS/CFT correspondence, half-

BPS operators are dual to S

spherical harmonics (Kaluza-Klein modes) of the graviton,

and we will consider the general correlator of four such spherical harmonics. This provides

an essential ingredient for studying the duality at loop level, where all modes can run inside

the loop, and also provides an explicit model for how local physics in an internal manifold

gets encoded from the CFT perspective.

Four-point correlators have been much studied since the early days of the AdS/CFT

correspondence. Initially, techniques were developed for computing tree-level Witten dia-

grams in position space, which led to the ﬁrst complete results for the correlators of four

dilatons or stress tensor multiplets [2, 3]. In the case of the SYM model, part of these

correlators are ﬁxed by non-renormalization theorem and agree with calculations at weak

‘t Hooft coupling [4, 5], but there is in addition an unprotected part which contains inter-

esting dynamical information. For example, a suitable limit reveals the ﬂat space S-matrix

of the underlying ten-dimensional superstring theory [6–8]. This program was gradually

extended to higher spherical harmonics [9–13].

Meanwhile, by analyzing the operator product expansion (OPE), it was discovered

how to exploit the large gap property of holographic theories, that is the fact that all

non-protected single-traces get lifted from the spectrum. This led to various conjectures

for correlators [14, 15]. However, it took almost a decade for a conjecture to appear in

the general case [16], thanks to a remarkably simple pattern which was observed in Mellin-

space [7, 17]. The corresponding position space calculation of tree-level Witten diagrams

was only very recently completed after a longstanding eﬀort [18, 19], which conﬁrmed the

conjecture in all considered examples.

In this paper we will revisit correlators of half-BPS operators from the perspective of

the analytic bootstrap, which oﬀers a constructive method to build correlators from their

light-cone singularities [20–23]. This can be formulated nonperturbatively as a Kramers-

Kronig type dispersion relation, known as the Lorentzian inversion formula, which explicitly

reconstructs OPE data from a suitably deﬁned absorptive part [24–26]. For holographic

theories at tree-level, the absorptive part is a ﬁnite sum of conformal blocks, thus reducing

the computation in a given theory to essentially a group-theoretic exercise (see also [27, 28]).

The result is unique up to AdS contact interactions, whose size can be bounded by con-

sidering the Regge limit [24, 29]. These are further restricted by supersymmetry and in

fact absent in N = 4 SYM as we will see. The method, therefore, rigorously and unam-

biguously determines the correlators. Interestingly, in analogy with the unitarity method

for one-loop S-matrices [30], “squaring” this tree-level four-point data gives suﬃcient in-

formation to go to one-loop [31], as has now been successfully carried out for low spherical

harmonics [32–34].

– 2 –

JHEP01(2019)196

Study of N = 4 SYM speciﬁcally is further motivated by the possibility of ﬁnding un-

expected structures. In the planar limit, this theory is controlled by a rich and remarkable

integrable system, making it a rather unique nontrivial four-dimensional quantum ﬁeld the-

ory in which the spectrum can be computed exactly at ﬁnite ‘t Hooft coupling, see [35, 36].

It is important to determine what, if anything, of this structure persists beyond the planar

limit. Correlation functions at strong coupling oﬀer a clean, natural environment in which

to discuss this question. Furthermore, we expect the correlators in this limit to provide im-

portant cross-checks and guidance as integrability-based methods are now being developed

to tackle the 1/N expansion [37, 38], as well as novel AdS-based techniques [39–41].

Hints of a hidden symmetry in correlators have been uncovered recently [42], who stud-

ied the anomalous dimension matrix governing the mixing between double-trace operators

built from diﬀerent S

spherical harmonics, starting from the conjectured Mellin-space for-

mula of [16]. Amazingly, the eigenvalues of the anomalous dimension matrix are simple

rational numbers, for which the authors of ref. [42] conjectured a general formula.

This paper is organized into two parts. First in section 2 we review the three ideas

behind our calculations: the Operator Product Expansion, the supersymmetric Ward iden-

tities satisﬁed by N = 4 SYM correlators, and the Lorentzian inversion formula. As a

warm-up, we then apply these formulas at order 1/N

in section 3. Including the identity

in two cross-channels yields double-trace data in the direct channel. While conceptually

straightforward, the supersymmetric OPE diﬀers from the usual generalized free ﬁeld result

in simple but important ways. Section 4 contains our main computations: the tree-level

correlator (order 1/N

) at strong ‘t Hooft coupling. The full result is determined by just

the singular part arising from half-BPS single-trace operators. While their couplings are

well-known and could be obtained from non-renormalization theorems [4], we show that

crossing symmetry in fact suﬃces to bootstrap them from scratch. This computation yields

double trace mixing matrices, and we conﬁrm the conjecture of [42] in many examples.

In a second part of this paper, section 5, we explore the remarkable structure found in

this result. We will attribute the simplicity of the eigenvalues to an unexpected SO(10,2)

symmetry, representing an eﬀective conformal invariance of ten-dimensional supergravity

at tree-level. This symmetry will be used to unify all spherical harmonics into a single ten-

dimensional object. Proof of the symmetry at the level of four-point correlators amounts to

showing that this generating function correctly predicts the singular part of each correlator;

we check this for a large number of examples. We also show how the symmetry readily

implies the conjectured Mellin-space formula of [16], and use it to obtain the leading-

logarithmic term at each order in 1/N . Implications are discussed in the conclusion.

2 Generalities

At any coupling, N = 4 SYM contains a special class of operators, which are half-BPS

and thus annihilated by the maximal possible number of supercharges. They transform

as Lorentz scalars and traceless symmetric tensors with respect to the SO(6)

' SU(4)

global symmetry (Dynkin label



0, p, 0



). In the Lagrangian description of N = 4 SYM,

they are described as symmetrical traceless polynomials in the 6 adjoint scalar ﬁelds φ

– 3 –

JHEP01(2019)196

the theory. Denoting x

the spacetime coordinates and introducing null six-vectors y

parametrize the SO(6)

dependence, they can be written as

(x, y) ∝ Tr



(φ

)



+ O(1/N

) corrections, (2.1)

which will be normalized in this paper so that



, y

)







(2.2)

where x

= (x

− x

)

, y

= y

· y

. Due to the BPS condition, the scaling dimension of

is exactly p, where p ≥ 2. At strong ‘t Hooft coupling, they admit a dual description

as Kaluza-Klein modes of the graviton on AdS

×S

. According to the gravity analysis, all

other operators in this limit are either multi-trace composites built of products of the O

or are heavier than the string scale, ∆

∼

1/4

Four-point correlators depend on AdS

and S

cross-ratios:

u =

= z¯z, v =

= (1 − z)(1 − ¯z), (2.3)

σ =

= α¯α, τ =

= (1 − α)(1 − ¯α), (2.4)

up to an overall prefactor conventionally written as follows [43]:



, y

) ···O

, y

)



≡













−p





−p

× G

}

(z, ¯z, α, ¯α) . (2.5)

We will be studying the correlator in the regime appropriate to the gauge-gravity

duality, where the ’t Hooft coupling λ = g

Y M

N is large but ﬁnite, and work order by order

in the gravity loop expansion 1/c where c =

−1

}

= G

(0)

}

(1)

}

(2)

}

+ . . . (2.6)

The Operator Product Expansion (OPE) oﬀers a series expansion of the correlator

around kinematic limits. In its most straightforward form, where we do not try to exploit

supersymmetry, it involves standard four-dimensional conformal blocks, times S

spherical

harmonics for each R-symmetry representation which can be exchanged. The conformal

blocks for a given spin and dimension are written as

r,s

`,∆

(z, ¯z) =

z¯z

¯z −z



r,s

∆−`−2

(z)k

r,s

∆+`

(¯z) − k

r,s

∆+`

(z)k

r,s

∆−`−2

(¯z)



, (2.7)

r,s

(z) = z



h +

, h +

; 2h, z



, (2.8)

with r = p

≡ p

− p

, s = p

. Since SO(6)

and the SO(4,2) Lie algebra are analytic

continuations of each other, the S

spherical harmonics admit identical expressions up to

reversal of some quantum numbers:

r,s

m,n

(α, ¯α) = (−1)

−r,−s

m,−n

(α, ¯α), (2.9)

– 4 –

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