arXiv:1702.08619v1 [hep-th] 28 Feb 2017
Spinning Witten diagrams
Charlotte SLEIGHT
Massimo TARONNA1
Universit´e Libre de Bruxelles and International Solvay Institutes ULB-Campus Plaine CP231, 1050 Brussels, Belgium
[email protected],
[email protected]
Abstract: We develop a systematic framework to compute the conformal partial wave expansions (CPWEs) of tree-level four-point Witten diagrams with totally symmetric external fields of arbitrary mass and integer spin in AdSd+1 . Employing this framework, we determine the CPWE of a generic exchange Witten diagram with spinning exchanged field. As an intermediate step, we diagonalise the linear map between spinning three-point conformal structures and spinning cubic couplings in AdS. As a concrete application, we compute all exchange diagrams in the type A higher-spin gauge theory on AdSd+1 , which is conjectured to be dual to the free scalar O (N ) model. We furthermore explicitly show that our formalism and results carry over directly to geodesic Witten diagrams. Given a CFTd , our results provide the complete holographic reconstruction of all cubic couplings involving totally symmetric fields in the putative dual theory on AdSd+1 .
1
Postdoctoral Researcher of the Fund for Scientific Research-FNRS Belgium.
Contents 1 Introduction
1
2 Conformal Partial Waves 2.1 The Conformal Partial Wave Expansion 2.2 Spinning Conformal Partial Waves 2.2.1 External scalar operators 2.2.2 Spinning Conformal Partial Waves 2.2.3 Spinning Conserved Conformal Partial Waves
2 2 3 3 5 6
3 CPWE of Spinning Witten Diagrams 3.1 Spinning three-point Witten diagrams 3.1.1 Building blocks of cubic vertices 3.1.2 Spinning Witten diagrams from a scalar seed 3.1.3 A natural basis of cubic structures in AdS/CFT 3.2 Spinning bulk-to-bulk propagators 3.2.1 Massive case 3.2.2 Massless case 3.3 CPWE of spinning exchange diagrams 3.3.1 Natural basis of conformal partial waves in AdS/CFT 3.3.2 Generic spinning exchange diagram 3.4 Spinning exchanges in the type A higher-spin gauge theory 3.4.1 Off-shell cubic couplings 3.4.2 Four-point exchange diagrams
7 9 9 10 12 15 15 16 18 18 19 22 22 24
4 Spinning Geodesic Witten diagrams
26
A Conventions, notations and ambient space
30
B The improved current
31
C Trace of the currents
32
D Seed bulk integrals
34
–i–
1
Introduction
Conformal field theories (CFT) are among the most well studied examples of quantum field theories (QFT), and are also among the few which admit a simple non-perturbative definition. This is owing to the fact that conformal invariance fixes all 2pt and 3pt correlation functions up to numerical coefficients and spectrum, usually referred to as CFT data. Associativity of the conformal operator algebra then allows to reconstruct, in principle, all higher point correlation functions at the non-perturbative level. This intrinsic simplicity triggered the pioneering works [1–6] centred on the idea that symmetry and quantum mechanics alone should suffice to fix the dynamics of a QFT. This is known as the Bootstrap Program. This approach proved to be very successful in the 80’s in the context of 2d CFTs [7], but remained dormant for CFTs in d > 2 until very recently with the emergence of new analytic and numerical methods [8–15]. These have led to striking new numerical results for 3d CFTs [16, 17], and have further triggered new analytic results for the conformal bootstrap in various limits [18–22]. CFTs also play a pivotal role in the holographic dualities, and are conjectured to be dual to gravitational theories living in a higher-dimensional anti-de Sitter (AdS) space [23–25]. From a bottom up perspective, AdS/CFT maps bulk and boundary consistency into each other, repackaging the various kinematic building blocks in terms of bulk or boundary degrees of freedom. To some extent, without imposing any additional constraint, this is a kinematic rewriting of the same physical object in two different bases. It was further shown in [26, 27] that, in the large N limit, standard Feynman diagram expansion in the bulk does repackage solutions to the bootstrap at leading order in N1 . In particular, this repackaging is in terms of Witten diagrams. From the bulk perspective, the latter play the role of the building blocks in terms of which the observables of the theory are expressed – in direct analogy with S-matrix elements. In this holographic picture, the main physical consistency requirement is the emergence of bulk locality, see e.g. [26, 28–37] for an incomplete list of works in this direction. Holography thus naturally provides a reformulation of the bootstrap problem in terms of different types of building blocks, which have a neat physical interpretation. The link between these two pictures is the main subject of the present work, in which we explicitly diagonalise the map between spinning three-point conformal structures and CPWE expansion on the boundary, and the local spinning bulk cubic couplings and Witten diagrams in the bulk. At the level of four-point functions this draws upon the link between the shadow formalism and the split representation of AdS harmonic functions (see, for instance, [36, 38]). At the level of threepoint functions, given a CFTd our results provide the complete holographic reconstruction of all cubic couplings involving totally symmetric fields in the putative dual theory on AdSd+1 . A key motivation behind this work is that such a bulk repackaging of CFT objects may give new insights into the bootstrap program, potentially providing novel methods to solve the crossing equations. Furthermore, this may also shed light on the quest for understanding quantum gravity and which CFTs admit a well-defined gravitational dual. In the process of diagonalising the map between boundary OPE coefficients and bulk cubic couplings, we identify the corresponding bases of bulk and boundary 3pt and 4pt structures, which, in this sense, appear to be naturally selected by holography. This allows us to study treelevel four-point exchange amplitudes involving totally symmetric fields of arbitrary mass and
–1–
spin, and derive their conformal partial wave expansion (CPWE). As a concrete application of our formalism, we determine all four-point exchange diagrams in the type-A higher-spin gauge theory on AdSd+1 , whose cubic couplings have been recently been established in metric-like form in [39, 40]. To conclude we demonstrate that our formalism and results carry over directly to geodesic Witten diagrams [41]. The outline is as follows: Section §2 we review the CPWE in the standard setting of CFT, with a particular focus on the shadow formalism. In Section §3 we detail the parallel story in the bulk. In particular, how the harmonic function decomposition of four-point Witten diagrams provides the link with the shadow formalism via the split representation. In §3.1 we review the computation [39] of generic spinning three point Witten diagrams, and present a convenient explicit diagonal form of the linear map between three-point conformal structures and local bulk cubic couplings. In §3.3 we apply the latter results to compute the CPWE of a generic spinning exchange Witten diagram in AdSd+1 , and furthermore in §3.4 consider exchange diagrams in the concrete setting of the type A minimal higher-spin gauge theory. In §4 we extend our formalism and result to geodesic Witten diagrams. Various technical details are relegated to the appendices.
2
Conformal Partial Waves
2.1
The Conformal Partial Wave Expansion
The CPWE of correlation functions of primary operators in CFT is a decomposition into contributions from each conformal multiplet.1 As a simple illustrative example, let us first consider the CPWEs of correlation functions involving scalar primary operators Oi . For a four-point function expanded in the s-channel,2 this reads X (2.1) hO1 (y1 ) O2 (y2 ) O3 (y3 ) O4 (y4 )i = cO1 O2 O∆,s cO∆,s O3 O4 W∆,s (yi ) . O∆,s
The functions W∆,s are the conformal partial waves. These are purely kinematical objects, fixed completely by conformal symmetry and only depend on the representations of the primary operators O∆,s and Oi under the conformal group. Each conformal partial wave in the expansion (2.1) is weighted by the coefficients of the operator O∆,s in the O1 × O2 and O3 × O4 OPEs. The CPWE thus effectively disentangles the dynamical information, which depends on the theory under consideration, from the universal information dictated by conformal symmetry. Owing to these defining features, the CPWE expansion has turned out to be a powerful tool. This is highlighted, for instance, by its pivotal role in the successes ([10, 11, 16, 42, 43], to name a few) of the conformal bootstrap program [3, 5]. But in spite of this, explicit formulas for conformal partial waves are scarce. For the scalar case (2.1) closed form expressions are 1
I.e. each conformal partial wave re-sums the contribution of the primary operator + all of its descendants in the correlator, and is thus labelled by the dimension ∆ and spin s of the primary operator. 2 We use sans-serif font to denote the expansion channels, to be distinguished from the spin, s.
–2–
only available in even dimensions [8, 9], while in other cases CPWs are inferred via indirect methods, such as: recursion relations [13, 44–48] and efficient series expansions [49–51]. In the following section we review another indirect approach, which is convenient for the CPWE of correlators involving operators with spin – as well as their Witten diagram counterparts. This is underpinned by the shadow formalism of Ferrara, Gatto, Grillo, and Parisi [1, 4, 52, 53], and leads to an expression for conformal blocks for operators in arbitrary Lorentz representations as an integral of three-point conformal structures [13, 54–56]. 2.2
Spinning Conformal Partial Waves
To a given primary operator O∆,s , can be associated a dual (or shadow) operator3 Z s 1 1 0 ˆz 0 O∆,s y 0 ; z 0 , ˜ O∆,s (y; z) = κ∆,s d/2 dd y 0 z · I(y − y ) · ∂ π (y − y 0 )2d−2∆
(2.4)
of the same spin and scaling dimension d − ∆. The normalisation κ∆,s =
Γ (d − ∆ + s) 1 , d (∆ − 1)s Γ ∆− 2
(2.5)
ensures that applying (2.4) twice gives the identity. The key observation of the shadow approach to conformal partial waves is that the integral Z 1 ˜∆,s (y) , (2.6) P∆,s = κd−∆,s d/2 dd y O∆,s (y) |0ih0|O π projects onto the contribution of the conformal families of O∆,s and its shadow to a given fourpoint function. This is illustrated for the simplest case of scalar correlators in the following, before moving on to correlators of spinning operators. 2.2.1
External scalar operators
Restricting, for now, to the case of external scalar operators (2.1), when projecting onto the s-channel we have hO1 (y1 ) O2 (y2 ) P∆,s O3 (y3 ) O4 (y4 )i
(2.7) O∆,s
= cO1 O2 O∆,s c
O3 O4 W∆,s (yi )
˜ O
+ cO1 O2 O˜∆,s c
O3 O4 Wd−∆,s (yi ) ,
which implies the following integral representation cO1 O2 O∆,s cO∆,s O3 O4 W∆,s (yi ) + shadow (2.8) Z 1 ˜∆,s (y) O3 (y3 ) O4 (y4 )i, = κd−∆,s d/2 dd y hO1 (y1 ) O2 (y2 ) O∆,s (y)ihO π 3
Iµν (y) is the inversion tensor Iµν (y) = δµν −
2yµ yν ; y2
z1 · I (y) · z2 = z1 · z2 − 2
z1 · y z 2 · y . y2
(2.2)
The Thomas derivative [57] (see also [58]) ∂ˆzi = ∂zi −
1 zi ∂z2 , d − 2 + 2z · ∂z
accounts for tracelessness, i.e. z 2 = 0.
–3–
(2.3)
for the total contribution as a product of two three-point functions. Stripping off the dynamical data leaves a universal integral expression for the sum of a conformal partial wave and its shadow, dictated purely by conformal symmetry and the operator representations: W∆,s (yi ) + shadow = κd−∆,s
(2.9) γτ,s γ¯τ,s π d/2
Z
˜∆,s (y) O3 (y3 ) O4 (y4 )ii, dd y hhO1 (y1 ) O2 (y2 ) O∆,s (y)iihhO
where γτ,s
Γ = Γ
τ3 −τ4 +τ d 2 − 2 , τ3 −τ4 +τ +s 2
γ¯τ,s
Γ = Γ
τ4 −τ3 +τ d 2 − 2 . τ4 −τ3 +τ +s 2
(2.10)
The notation hh•ii denotes the kinematical part of the three-point function that is fixed by conformal symmetry. I.e. removal of the overall coefficient,4 hO1 (y1 ) O2 (y2 ) O∆,s (y)i = cO1 O2 O∆,s hhO1 (y1 ) O2 (y2 ) O∆,s (y)ii ˜∆,s (y) O3 (y3 ) O4 (y4 )i = c ˜ ˜∆,s (y) O3 (y3 ) O4 (y4 )ii, hO hhO O∆,s O3 O4
(2.12a) (2.12b)
which, for unit two-point function normalisation, is the removal of the OPE coefficients. Details on the above steps where given by Dolan and Osborn in [13] section 3 and [8]. An integral expression for a single, non-shadow, conformal partial wave can be obtained by introducing a contour integral5 Z ∞ ∆ − d2 dν W∆,s (yi ) = W (y ) + W (y ) , (2.13) d d i i +iν,s −iν,s 2 2 2 2π −∞ ν 2 + ∆ − d 2 and inserting (2.9) into the integrand. The CPWE (2.1) can then be re-cast as a contour integral [58, 59], hO1 (y1 ) O2 (y2 ) O3 (y3 ) O4 (y4 )i Z XZ ∞ = dν cs (ν) dd y hhO1 (y1 ) O2 (y2 ) O d s
2 +iν,s
−∞
(2.14) (y)iihhO d
2 −iν,s
(y) O3 (y3 ) O4 (y4 )ii,
˜d where for ease of notation we defined O d −iν,s = O +iν,s . The real function cs (ν) encodes 2 2 the dynamical information, with poles that carry the contribution from each spin-s conformal multiplet. For example, a contribution from a conformal multiplet [∆, s] manifests itself in cs (ν) with a pole at d2 + iν = ∆, with residue giving the OPE coefficients cO1 O2 O∆,s cO∆,s O3 O4 ∆ − d2 κd−∆,s γτ,s γ¯τ,s d + ... , cs (ν) = (2.15) d 2π d/2+1 2 − ∆ + iν 2 − ∆ − iν 4
Using the definition (2.4) one finds cO˜∆,s O3 O4 = γτ,s γ¯τ,s cO∆,s O3 O4 ,
(2.11)
which is the origin of the factors (2.10) in the expression (2.9). 5 The conformal partial wave W d ±iν,s (yi ) decays exponentially for Im (ν) → ∓∞. In applying the residue 2 theorem to obtain the LHS from the RHS, for W d ±iν,s we close the ν-contour in the lower/upper half plane 2 respectively.
–4–
where the ... denote possible contributions from other spin-s multiplets in the spectrum. The contour integral form (2.14) of the CPWE admits a direct generalisation to four-point correlators involving operators with spin. The only difference with respect to the scalar case is that, in general, there is more than one conformal partial wave associated to each conformal multiplet. This is a consequence of the non-uniqueness of tensor structures compatible with conformal symmetry in three-point functions with more than one spinning operator. It is for this reason that external spinning operators are easily accommodated for in the integral form (2.13) of the conformal partial wave, which we discuss in the following. 2.2.2
Spinning Conformal Partial Waves
The integral representation (2.13) of conformal partial waves carries over straightforwardly to CPWEs of four-point functions containing operators with spin. In this case, however, since the structure of three-point functions with more than one operator of non-zero spin is not unique, generally there is more than one conformal partial wave associated to the contribution of a given conformal multiplet. The number of independent structures that may appear in a conformal three-point function with operators of spins s1 -s2 -s3 is [60] N (s1 , s2 , s3 ) =
(s1 + 1) (s1 + 2) (3s2 − s1 + 3) p (p + 2) (2p + 5) 1 − (−1)p − − , 6 24 16
(2.16)
where s1 ≤ s2 ≤ s3 and p ≡ Max (0, s1 + s2 − s3 ). For correlation functions with two scalar operators there is just a single structure compatible with conformal symmetry, N (0, 0, s) = 1, in accordance with the uniqueness of conformal partial waves with external scalar operators that we previously observed. Three-point functions involving two spinning operators have N (s1 , s2 , s3 ) > 1. A general three-point function of spinning operators in a parity-even theory takes the form6 hO∆1 ,s1 (y1 )O∆2 ,s2 (y2 )O∆3 ,s3 (y3 )i X Ys1 −n2 −n3 Y2s2 −n3 −n1 Y3s3 −n1 −n2 Hn1 1 Hn2 2 Hn3 3 = cns11,s,n22,s,n33 1 τ1 +τ2 −τ3 τ2 +τ3 −τ1 τ3 +τ1 −τ2 , (2.17) 2 ) 2 ) 2 ) 2 2 2 ni (y12 (y23 (y31 with theory-dependent OPE coefficients cns11,s,n22,s,n33 . The six three-point conformally covariant building blocks are given by (i ∼ = i + 3) Yi = Hi = 6
zi · yi(i+1) zi · yi(i+2) − , 2 2 yi(i+1) yi(i+2) 1 2 y(i+1)(i+2)
To be more precise,
P ni
=
(2.18)
2zi+1 · y(i+1)(i+2) zi+2 · y(i+2)(i+1) zi+1 · zi+2 + 2 y(i+1)(i+2)
! .
min{s −n3 ,s3 } min{s2 −n P1 ,s2 } min{s1P P3 ,s3 −n2 }
.
n3 =0
n2 =0
n1 =0
Let us also note that it is from this that one obtains the counting (2.16):
P ni
–5–
1 = N (s1 , s2 , s3 ).
(2.19)
A conformal partial wave with spinning external operators is thus labelled by two threecomponent vectors n = (n1 , n2 , n) and m = (m, m3 , m4 ), n,m W∆,s (yi ) + shadow
= κd−∆,s
γτ,s γ¯τ,s π d/2
(2.20) Z
˜∆,s (y)O∆ ,s (y3 )O∆ ,s (y4 )ii(m) , dd y hhO∆1 ,s1 (y1 )O∆2 ,s2 (y2 )O∆,s (y)ii(n) hhO 3 3 4 4
where, by applying the definition (2.12) of the operation hh•ii, hhO∆1 ,s1 (y1 )O∆2 ,s2 (y2 )O∆3 ,s3 (y3 )ii(n) =
Y1s1 −n2 −n Y2s2 −n−n1 Y3s−n1 −n2 Hn1 1 Hn2 2 Hn3 2 ) (y12
τ1 +τ2 −τ 2
2 ) (y23
τ2 +τ −τ1 τ +τ1 −τ2 2 ) 2 2 (y31
.
(2.21)
In the same way, the shadow contribution can be projected out by introducing a contour integral as in (2.13). 2.2.3
Spinning Conserved Conformal Partial Waves
Conservation of external operators places additional constraints on conformal partial waves, which is a consequence of the conservation conditions on three-point functions of conserved operators [61, 62]. The latter relates the coefficients cns11,s,n22,s,n33 in a general spinning three-point function (2.17) amongst each other, reducing the number of independent forms to [60] N (s1 , s2 , s3 ) = 1 + min {s1 , s2 , s3 } ,
(2.22)
when each operator in the three-point function is conserved. The general form for a three-point function of conserved operators in d > 3 is given as a generating functional by [63, 64],7 1+min(s1 ,s2 ,s3 ) 2
hJs1 (y1 )Js2 (y2 )Js3 (y3 )i =
X
ckJs Js Js 2 F1 1 2 3
k=0
×
1 d 1 Λ − k, −k, 3 − − 2k, − 2 2 2 2 H1 H22 H23
eY1 +Y2 +Y3 0 F1 (d − 2, − 12 H1 )0 F1 (d − 2, − 21 H2 )0 F1 (d − 2, − 12 H3 ) d d d 2 ) 2 −1 (y 2 ) 2 −1 (y 2 ) 2 −1 (y12 23 31
Λ2k , (2.23)
with Λ = Y1 Y2 Y3 +
1 [Y1 H1 + Y2 H2 + Y3 H3 ] , 2
(2.24)
and k takes both integer and half integer values. The OPE coefficients ckJs Js Js are not fixed 2 3 1 by current conservation and depend on the theory. The above counting implies, for instance, that conserved conformal partial waves representing the contribution of a conserved primary operator are labelled by two half-integers k ∈ {0, 1/2, 1, ..., 1 + min (s1 , s2 , s) /2} and k˜ ∈ {0, 1/2, 1, ..., 1 + min (s, s3 , s4 ) /2}, where (s1 , s2 , s3 , s4 ) are the spins of the external conserved operators, ˜
k,k W(s (yi ) + shadow 1 ,s2 |s|s3 ,s4 ) Z γτ,s γ¯τ,s ˜ = κd−∆,s d/2 dd y hhJs1 (y1 )Js2 (y2 )Js (y)ii(k) hhJ˜s (y)Js3 (y3 )Js4 (y4 )ii(k) , (2.25) π 7
To extract the explicit structure of the correlator from the generating function form (2.23) one expands and n2 n3 1 collects monomials of the form Y1s1 −n2 −n3 Y2s2 −n1 −n3 Y3s−n1 −n2 Hn 1 H2 H3 .
–6–
where Js is the exchanged spin-s conserved current and (k)
hhJs1 (y1 )Js2 (y2 )Js (y3 )ii ×
= 2 F1
1 Λ d 1 − k, −k, 3 − − 2k, − 2 2 2 H21 H22 H23
eY1 +Y2 +Y3 0 F1 (d − 2, − 21 H1 )0 F1 (d − 2, − 21 H2 )0 F1 (d − 2, − 12 H3 ) d d d 2 ) 2 −1 (y 2 ) 2 −1 (y 2 ) 2 −1 (y12 23 31
Λ2k . (2.26)
Conservation of higher-spin currents is a powerful constraint, with the presence of a single exactly conserved current of spin s > 2 in the spectrum implying (in d ≥ 3) that the theory is a free one [65–69].8 In this case the label k of each independent structure in (2.23) denotes the spin of the free conformal representation [64]. An example which we employ later on is the free scalar, where k = 0 (the scalar singleton) and the corresponding conserved three-point structure can be conveniently expressed in terms of Bessel functions9 ! 1 d−2 d Q3 − √ 4 Γ( d−2 ) J 4 −1 q 2 qi Y1s1 Y2s2 Y3s3 d i=1 2 i −2 2 2
hJs1 (y1 )Js2 (y2 )Js3 (y3 )i = c0Js
1 Js2 Js3
,
2 )d/2−1 (y 2 )d/2−1 (y 2 )d/2−1 (y12 23 31
(2.28) where qi = 2Hi ∂Yi+1 · ∂Yi+2 . The OPE coefficients were worked out in [39] to be √ c0Js Js Js 1 2 3
3
=
Nd.o.f. c0s1 c0s2 c0s3 ,
2 c0si
=
d−2 2 )Γ(si + d − d−2 2 Nd.o.f. si ! Γ(si + d−3 2 )Γ( 2 )
π 27−d−si Γ(si +
3)
.
(2.29)
CPWE of Spinning Witten Diagrams
The integral representation of the CPWE is most suitable for establishing CPWEs of Witten diagrams, as it arises naturally from their harmonic function decomposition (see [36] for a detailed review): The analogue of the CPWE expansion in the bulk is the decomposition into partial waves of the AdS isometry group. I.e. in terms of harmonic functions with energy and spin quantum 8 9
Assuming a single stress tensor. To see this one employs the identity x2 . Γ (α + 1) x−α Jα (2x) = 2−α 0 F1 α + 1; − 4
–7–
(2.27)
numbers,10
, (3.2) To make contact with the CPWE on the boundary, one notes that harmonic functions factorise [70] Z ν2 dd y Π d +iν,k (x1 , u1 ; y, ∂ˆz )Π d −iν,k (y, z; x2 , u2 ) , (3.3) Ων,k (x1 , u1 ; x2 , u2 ) = 2 2 πk! d2 − 1 k ∂AdS into a product of two boundary-to-bulk propagators of dimensions d2 ± iν and the same spin k. We see that, like for conformal partial waves (§2 equation (2.14)), each bulk partial wave factorises into a product of two three-point Witten diagrams,
. (3.4) Evaluating the bulk integrals yields a decomposition of the Witten diagram into products of three-point conformal structures on the boundary – i.e. the integral representation (2.14) of the conformal partial wave expansion. So far this approach has been applied to compute the CPWEs of tree-level Witten diagrams with only external scalars. This includes: The exchange of a massive spin-s field and the graviton exchange [38]; the exchange of spin-s gauge field on AdSd+1 [71] and contact diagrams for a general quartic scalar self-interaction [30]. Spinning Exchange Witten Diagrams In this section we generalise the aforementioned results, to include all possible four-point exchange diagrams involving totally symmetric fields of arbitrary integer spin and mass – both 10
The harmonic function Ων,s−2k is a symmetric and traceless (in both sets of indices) spin s−2k Eigenfunction of the Laplacian, + d2 + iν d2 − iν + s − 2k Ων,s−2k = 0, (3.1) which is divergence-free, ∇ · Ων,s−2k = 0.
–8–
internally and externally.11 To wit, we decompose into conformal partial waves the following general exchange of a spin-s field of mass m2 R2 = ∆ (∆ − d) − s in AdSd+1
,
(3.5)
between external fields of spin si and mass m2i R2 = ∆i (∆i − d) − si . The first step is to obtain the decomposition (3.2) of the exchange diagram. This is achieved by expressing the bulk-to-bulk propagator of the exchanged field in a basis of harmonic functions [38, 71, 75], which we review for massive fields in §3.2.1 and for massless fields in §3.2.2. This leads to the decomposition (3.4) of the exchange diagram (3.5) into products of tree-level three-point Witten diagrams, whose evaluation leads to the sought-for conformal partial wave expansion via identification with the integral form (2.20) of the conformal partial waves. 3.1
Spinning three-point Witten diagrams
In the light of the decomposition (3.2) of Witten diagrams, a key step to obtain CPWEs of spinning diagrams is therefore the evaluation of tree-level three-point Witten diagrams involving fields of arbitrary integer spin and mass. For parity even theories, this was carried out in [39] in general dimensions, whose results we review here and also further supplement with new ones. 3.1.1
Building blocks of cubic vertices
Employing the ambient space formalism (reviewed in appendix §A), a convenient basis of on-shell cubic vertices between totally symmetric fields ϕsi of spins si and mass m2i R2 = ∆i (∆i − d) − si is given by Isn11,s,n22,s,n3 3 = Y1s1 −n2 −n3 Y2s2 −n3 −n1 Y3s3 −n1 −n2
(3.6)
× H1n1 H2n2 H3n3 ϕs1 (X1 , U1 ) ϕs1 (X1 , U1 ) ϕs2 (X2 , U2 ) ϕss (X3 , U3 )
Xi =X
,
which is parameterised by the six basic contractions Y1 = ∂U1 · ∂X2 ,
Y2 = ∂U2 · ∂X3 ,
Y3 = ∂U3 · ∂X1 ,
(3.7a)
H1 = ∂U2 · ∂U3 ,
H 2 = ∂ U3 · ∂ U1 ,
H3 = ∂U1 · ∂U2 .
(3.7b)
11
For other works on spinning exchange diagrams, see: [72, 73] in the context of higher-spin gauge theories and more recently [74] in the context of the geodesic Witten diagram decomposition of standard Witten diagrams – see also §4.
–9–
Recall that, in accordance with standard AdS/CFT lore, the basis elements (3.6) are in oneto-one correspondence with the independent three-point conformal structures (2.21). The most general cubic vertex thus takes the form (c.f. footnote 6 for the sum over ni ) X gsn11,s,n22,s,n3 3 Isn11,s,n22,s,n3 3 . (3.8) Vs1 ,s2 ,s3 = ni
The choice of basis (3.6) is convenient for three main reasons: 1. Simplicity: The basis is built from the (commuting) ambient partial derivatives as opposed to the (non-commuting) AdS covariant derivatives. 2. Ease of manipulation and computation: This is a consequence of the above simplicity. One important example is given by integration by parts in the ambient formalism. While this is in general more involved compared to standard integration by parts directly on the AdS manifold, the basis (3.6) makes integration by parts as simple as in flat space. See [76] for details on integration by parts in the ambient space framework, and in particular for the basis (3.6). 3. Physical interpretation: Any vertex expressed in terms of covariant derivatives can straightforwardly be cast in terms of the basis (3.6), and vice versa, using (see §A) ∇A = PAB
∂ XB − ΣAB , ∂X B X2
(3.9)
where
∂ ∂ ∂ = UA − UB , (3.10) B B ] ∂U ∂U A ∂U is the spin connection in the ambient generating function formalism. See appendix B of [39] for more details about radial reduction. ΣAB = U[A
3.1.2
Spinning Witten diagrams from a scalar seed
Another virtue of the ambient space formalism is that Witten diagrams with spinning external legs can be seamlessly generated from those with only external scalars (which are comparably straightforward to evaluate) via the application of appropriate differential operators in the boundary variables. The ease of this approach to spinning Witten diagrams within the ambient framework is owing in particular to the homogeneity of the ambient representatives in both the bulk and boundary coordinates. The implication of this observation for the three-point Witten diagram generated by the basis vertex (3.6) is that it can be re-expressed in the form
,
– 10 –
(3.11)
for some homogeneous differential operator Fsn11,s,n22,s,n3 3 (Zi , Pi , ∂Pi ), acting on the diagram gen0,0,0 erated by the coupling I0,0,0 between scalars of some mass m ˜ 2i . The latter is a well known integral which is straightforward to evaluate [77], which we review in §D. Naturally, since the action of Fsn11,s,n22,s,n3 3 increases the spin of the external legs, it will be a non-trivial function of Zi . The decomposition (3.11) of the spinning three-point Witten diagram can straightforwardly be obtained by noting that spinning bulk-to-boundary propagators have an analogous differential relationship to scalar bulk-to-boundary propagators [36]12 K∆,s (X, U ; P, Z) =
1 (DP (Z; U ))s K∆,0 (X; P ) , (∆ − 1)s
(3.14)
with differential operator ∂ ∂ ∂ + (P · U ) Z · . DP (Z; U ) = (Z · U ) Z · −P · ∂Z ∂P ∂P
(3.15)
Ambient partial derivatives of spinning bulk-to-boundary propagators, which arise naturally from the basis (3.6), can readily be expressed in a similar form: (Uj · ∂X )n K∆,s (X, Ui ; P, Z) =
1 (DP (Z; U ))s (Ui · ∂X )n K∆,0 (X; P ) (∆ − 1)s
(3.16)
with (Ui · ∂X )n K∆,0 (X; P ) = 2n (∆)n (Ui · P )n K∆+n,0 (X; P ) .
(3.17)
This further illustrates the convenience of the choice of basis (3.6). Employing the expression for spinning bulk-to-boundary propagators (3.16) one then obtains
Fsn11,s,n22,s,n3 3 =
2s˜1 +˜s2 +˜s3 (∆1 )s˜3 (∆2 )s˜1 (∆3 )s˜2 (∆1 − 1)s1 (∆2 − 1)s2 (∆3 − 1)s3 (˜ s1 )! (˜ s2 )! (˜ s3 )! ¯ s˜2 H ¯ s˜3 H ¯ s˜1 Ds1 Ds2 Ds3 U ¯ 1 · P1 × H1n1 H2n2 H3n3 H 1 2 3 P1 P2 P3
(3.18) s˜3
¯2 · P2 U
s˜2
¯3 · P3 U
s˜1
¯i where for concision we defined s˜i = si − ni−1 − ni+1 and introduced the auxiliary vector U ¯ which enters the contraction Hi = ∂Ui−1 · ∂U¯i+1 . The mass of each scalar entering the seed vertex on the RHS of (3.11) is given in terms of the quantum numbers of the original spinning fields on the LHS: ˜ i (∆ ˜ i − d) m ˜ 2i R2 = ∆ 12
with
˜ i = ∆i + si+2 − ni − ni+1 . ∆
(3.19)
By evaluating the action of the differential operator one recovers the standard expression [78] K∆,s (X, U ; P, Z) = (U · P · Z)s
C∆,s , (−2X · P )∆
C∆,s =
(s + ∆ − 1) Γ (∆) . 2π d/2 (∆ − 1) Γ ∆ + 1 − d2
This also dictates the normalisation of the dual operator two-point function at large Nd.o.f s C∆,s 2z1 · y12 z2 · y21 hO∆,s (y1 ; z1 ) O∆,s (y2 ; z2 )i = z · z + . 1 2 2 2 ∆ y12 (y12 )
– 11 –
(3.12)
(3.13)
What remains to obtain the result for the spinning Witten diagram in the LHS of (3.11) is to simply insert the result (D.5) for the scalar seed on the RHS and then act with the differential operator (3.18). Denoting the amplitude by Asn11,s,n22,s,n33;τ1 ,τ2 ,τ3 , this procedure yields13 1 ,n2 ,n3 An s1 ,s2 ,s3 ;τ1 ,τ2 ,τ3 (y1 , y2 , y3 ) = P3
X
3 Y
(−1)si −ni −δi +αi +βi 2si −ni −γi −δi −ωi
α,β,δ,ω,γ i=1
×
ni !(αi + βi )!(si − ni+1 − ni−1 )! γi !δi !αi !ωi !(βi + δi+1 − ni+1 + 1)!
(αi + βi + ∆i )si +δi(i+1) −γi+1 −ni+1 −ωi+1 −∆i (αi + βi − γi+1 − γi−1 − ωi+1 + 1)!(si − αi − ni+1 − ni−1 − ωi−1 + 1)!(ni+1 + ni−1 − βi − δi+1 − δi−1 + 1)!
× Hγ11 +δ1 +ω1 Hγ22 +δ2 +ω2 Hγ33 +δ3 +ω3 Y1s1 −γ2 −γ3 −δ2 −δ3 −ω2 −ω3 Y2s2 −γ1 −γ3 −δ1 −δ3 −ω1 −ω3 Y3s3 −γ1 −γ2 −δ1 −δ2 −ω1 −ω2 ,
where i ∼ = i + 3. The pre-factor is given by X 1 1 P3 = Γ ( τ2α + sα − nα ) − 16 π d (y12 )δ12 (y23 )δ23 (y31 )δ31 α
! d 2
3 Y Γ(∆i − 1)(∆i + si − 1) , Γ ∆i + 1 − d2 i=1
where 1 δ(i−1)(i+1) = (τi−1 + τi+1 − τi ) , 2
τi = ∆i − si .
(3.21)
While the basis (3.6) is convenient as a means to evaluate spinning Witten diagrams, the resulting one-to-one map (3.21) between the bulk basis elements (3.6) and the canonical basis (2.21) of three-point conformal structures is rather involved. In the following section we introduce an alternative bulk and boundary pair of bases, through which the aforementioned bulk-to-boundary mapping simplifies dramatically and moreover allows to elegantly re-sum the expression (3.21). 3.1.3
A natural basis of cubic structures in AdS/CFT
Let us motivate this alternative basis with a simple example. As observed in [39], the amplitude generated by the highest derivative basis vertex s1 s2 s3 Is0,0,0 = Y Y Y ϕ (X , U ) ϕ (X , U ) ϕ (X , U ) ϕ (X , U ) , (3.22) s 1 1 s 1 1 s 2 2 s 3 3 s 1 1 2 1 2 3 1 ,s2 ,s3 Xi =X
admits a very simple re-summation in terms of Bessel functions Bsi ;τi As0,0,0 (y1 , y2 , y3 ) = 1 ,s2 ,s3 ;τ1 ,τ2 ,τ3 δ 12 (y12 ) (y23 )δ23 (y31 )δ31 " 3 # Y δ(i+1)(i−1) −1 δ(i+1)(i−1) 1 − δ(i+1)(i−1) √ 4 2 × 2 Γ qi2 J(δ(i+1)(i−1) −2)/2 ( qi ) Y1s1 Y2s2 Y3s3 , (3.23) 2 i=1
13
The summation symbol is defined as: X α,β,δ,ω,γ
≡
sκ −kκ X
kκ X nκ X
ακ−1 +βκ−1 ακ−1 +βκ−1
ακ =0 βκ =0 δκ =0
– 12 –
X
X
ωκ =0
γκ =0
,
(3.20)
where we recall that qi = 2Hi ∂Yi+1 ∂Yi−1 and the overall coefficient is given by Bsi ;τi
1 = Γ 16π d
τ1 + τ2 + τ3 − d + 2(s1 + s2 + s3 ) 2
3 Y (−2)si Γ si + δi(i+1) Γ si + δ(i−1)i Γ(si + τi − 1) × . (3.24) d Γ s + τ − + 1 Γ δ Γ(2s + τ − 1) i i i i (i+1)(i−1) 2 i=1 Such three-point conformal structures are for instance generated in free scalar CFTs (see e.g. (2.28) for the case of three-point functions of conserved operators). Given the simplicity and compactness of the three-point conformal structure (3.23) generated by the basis vertex (3.22), it is temping to consider the following basis of conformal structures, " 3 # Y δ(i+1)(i−1) +n −1 δ Hn1 1 Hn2 2 Hn3 3 (i+1)(i−1) i 2 2 [[O∆1 ,s1 (y1 )O∆2 ,s2 (y2 )O∆3 ,s3 (y3 )]] ≡ Γ + ni 2 (y12 )δ12 (y23 )δ23 (y31 )δ31 i=1 " 3 # Y 1−ni − δ(i+1)(i−1) √ 2 4 × J(δ(i+1)(i−1) +2ni −2)/2 ( qi ) Y1s1 −n2 −n3 Y2s2 −n3 −n1 Y3s3 −n1 −n2 qi (n)
i=1
(3.25)
in the view of simplifying the map between bulk and boundary structures. Indeed, working iteratively one finds that the conformal structure (3.25) is generated by the bulk vertex14 X 3 Isn11,s,n22,s,n3 3 = Csn11,s,n22,s,n3 ;m I m1 ,m2 ,m3 , (3.26) 1 ,m2 ,m3 s1 ,s2 ,s3 mi
with coefficients
3 Csn11,s,n22,s,n3 ;m 1 ,m2 ,m3
3 Csn11,s,n22,s,n3 ;m = 1 ,m2 ,m3
given by
d−2(s1 +s2 +s3 −1)−(τ1 +τ2 +τ3 ) 2 m1 +m2 +m3 3 Y ni mi
×
2
i=1
mi
(ni + δ(i+1)(i−1) − 1)mi , (3.27)
In particular, denoting the three-point amplitude generated by each basis element (3.26) by Asn11,s,n22,s,n33;τ1 ,τ2 ,τ3 , we have:15 Ans11,s,n22,s,n33;τ1 ,τ2 ,τ3 (y1 , y2 , y3 ) = B(si ; ni ; τi ) [[O∆1 ,s1 (y1 )O∆2 ,s2 (y2 )O∆3 ,s3 (y3 )]](n) , 14 15
For concision we define
P mi
=
(3.28)
min{sP 1 ,s2 ,n3 } min{s1 −n 3 ,s3 −n2 ,n1 } P3 ,s3 ,n2 } min{s2 −nP m3 =0
n2 =0
n1 =0
Note, the vertices constructed here should not be confused with those written down in [79] in the context of geodesic Witten diagrams. In section §4 we also consider geodesic Witten diagrams, and in particular show that from the result (3.28) the corresponding geodesic integral follows immediately.
– 13 –
with the coefficient B(si ; ni ; τi ) given by 1 +s2 +s3 ) B(si ; ni ; τi ) = π −d (−2)(s1 +s2 +s3 )−(n1 +n2 +n3 )−4 Γ τ1 +τ2 +τ3 −d+2(s 2 τi +τi−1 −τi+1 τi +τi+1 −τi−1 3 Γ s −n Γ s + n − n + Γ(si + ni+1 + ni−1 + τi − 1) Y i i+1 i−1 i i+1 + ni−1 + 2 2 × . Γ si + τi − d2 + 1 Γ 2ni + τi+1 +τ2i−1 −τi Γ(2si + τi − 1) i=1 (3.29)
Given a CFTd , the result (3.28) provides the complete holographic reconstruction of all cubic couplings involving totally symmetric fields in the putative dual theory on AdSd+1 .16 Relation between bulk basis To conclude it is useful to spell out the explicit dictionary between the building blocks (3.6), which allow to straightforwardly evaluate spinning Witten diagrams, and the basis (3.26) introduced in the previous section, which give a simple form for spinning three-point amplitudes. Given a coupling of the form X Vs1 ,s2 ,s3 = gsn11,s,n22,s,n3 3 Isn11,s,n22,s,n3 3 , (3.30) ni
the problem is to determine the explicit form of the coefficient g˜sn11,s,n22,s,n3 3 in the basis: X g˜sn11,s,n22,s,n3 3 Isn11,s,n22,s,n3 3 , Vs1 ,s2 ,s3 =
(3.31)
ni
with Isn11,s,n22,s,n3 3 and Isn11,s,n22,s,n3 3 given in (3.6) and (3.26), respectively. Working iteratively, one arrives at the following expression for the coefficient g˜sn11,s,n22,s,n3 3 as a function of the coefficients gsn11,s,n22,s,n3 3 in the original basis:17 " # 3 X Y (2n + δ − 1) mi g i jk m ,m ,m 1 2 3 P , (−1)ni +mi m g˜sn11,s,n22,s,n3 3 = i ( d2 +1+ α (mα −sα − τ2α ))m1 +m2 +m3 2 (ni + δjk − 1)m +1 ni mi
i=1
i
(3.33) which is the inverse of the map (3.26). Notice that the new basis (3.25) generalises to non-conserved operators the basis (2.26) of three-point conserved conformal structures. In this regard, our basis (3.25) seems to be naturally selected by free singleton CFTs. 16
See [30, 31, 39, 80–82] for previous works on the holographic reconstruction of cubic couplings, and also [30] for quartic couplings. 17 For concision we define: Min{s1 ,s2 } Min{s3 ,s1 −m3 } Min{s2 −m3 ,s3 −m2 }
X mi
=
X
X
X
m3 =n3
m3 2=n2
m1 =n1
– 14 –
(3.32)
3.2
Spinning bulk-to-bulk propagators
In this section we review previous works on the harmonic function decomposition of bulk-tobulk propagators for totally symmetric fields of arbitrary mass and integer spin [71].18 Up to cubic order in perturbations about the AdS background, a spin-s field of mass m2 R2 = ∆ (∆ − d) − s is governed by an effective Euclidean action of the form
Z Sm2 ,s [ϕs ] = s! AdS
1 ϕs (x, ∂u ) − m2 + ... ϕs (x, u) + ϕs (x, ∂u ) Js (x, u) + O ϕ4 , (3.34) 2
where the source Js in the cubic interaction term is quadratic in the perturbations. The ... denote terms which depend on the off-shell completion, which we discuss case-by-case in the sequel. Upon varying the action, the corresponding bulk-to-bulk propagator satisfies an equation of the form 1 − m2 + ... Πm2 ,s (x1 ; x2 ) = −δ d+1 (x1 , x2 ) , (3.35) where for convenience we suppressed the index structure, for now. To determine the propagator as a decomposition in harmonic functions, one can consider an ansatz of the form Πm2 ,s (x1 , u1 ; x2 , u2 ) =
(3.36)
bs/2c s−2k Z ∞ X X k=0 l=0
−∞
dν gk,l (ν) u21
k
u22
k
(u1 · ∇1 )l (u2 · ∇2 )l Ων,s−2k−l (x1 , u1 ; x2 , u2 ) .
The functions gk,l (ν) are fixed by requiring that the equation of motion (3.35) is satisfied. We first review the solution for massive spinning fields before moving on to the massless case, where one has the additional requirement of gauge invariance. 3.2.1
Massive case
The Lagrangian formulation for freely propagating totally symmetric massive fields of arbitrary spin was first considered by Singh and Hagen in the 70’s [95, 96].19 In order for the Fierz-Pauli physical state conditions [101–103] − m2 ϕs (x, u) = 0,
(∂u · ∇) ϕs (x, u) = 0,
(∂u · ∂u ) ϕs (x, u) = 0,
(3.37)
to be recovered upon varying the action, the field content consists of the traceless field ϕs , and additional traceless auxiliary fields of ranks s − 2, s − 3, ..., 0 which vanish on-shell.20 The complete off-shell form of the free Lagrangian is involved, and is moreover currently unavailable in its entirety on an AdS background. On the other hand, the terms which have not yet been identified explicitly are those which vanish on-shell (i.e. the ... in (3.34)) and thus 18
For earlier works spinning bulk-to-bulk propagators, see [83, 84] by B. Allen for the graviton and (massive and massless) vector propagators (also [38, 85–88]); for higher spin see [38, 72, 73, 89–94]. 19 See also [97–100]. 20 See [104, 105] for an alternative formulation of the free massive Lagrangian in terms of curvatures, free from such auxiliary fields. Their removal, however, comes at the price of introducing non-localities.
– 15 –
only generate contact terms in exchange amplitudes. The latter are not universal contributions, as they are highly dependent on the field frame. For our purposes it is therefore not necessary to keep track of such terms,21 and we can solve the following equation for the massive spin-s bulk-to-bulk propagator 1 − m2 Πm2 ,s (x1 , u1 ; x2 , u2 ) = − {(u1 · u2 )s } δ d+1 (x1 , x2 ) ,
(3.38)
where the notation {•} signifies a traceless projection. Since in this case the field is traceless, the following ansatz can be considered for the bulk-to-bulk propagator Πm2 ,s (x1 , w1 ; x2 , w2 ) =
s Z X l=0
∞
dν gl (ν) (w1 · ∇1 )l (w2 · ∇2 )l Ων,s−l (x1 , w1 ; x2 , w2 ) ,
(3.39)
−∞
where the null auxiliary vectors wi2 = 0 enforce tracelessness. Substituting into the equation of motion (3.38), one finds [71] gl (ν) =
d 2
1 2 − ∆ + ν 2 − l + l(d + 2s − ` − 1) ×
(3.40)
l! (d + 2s − 2l − 1)l
d 2
d 2
+ s − l − 21 l . + s − l + iν l d2 + s − l − iν l
2l (s − l + 1)l
Before moving on to consider the massless case, let us briefly highlight some generic features of the propagator (3.39): • The traceless and transverse part of the propagator (corresponding to l = 0 in (3.39)) Z ∞ dν TT Πm2 ,s (x1 ; x2 ) = Ων,s (x1 ; x2 ) , (3.41) 2 2 −∞ d − ∆ + ν 2 is universal, and encodes the propagating degrees of freedom. • The remaining contributions from harmonic functions of spin < s (the l > 0 in (3.39)) are purely off-shell, and generate only contact terms in exchange amplitudes. 3.2.2
Massless case
On contrast to the massive case discussed in the previous section, the construction of free Lagrangians for massless fields is somewhat simplified owing to the additional guidance provided by gauge invariance. Recalling that the concept of masslessness in AdS is slightly deformed owing to the background curvature, requiring gauge invariance of the Fierz-Pauli system (3.37) under the gauge transformation δξ ϕs (x, u) = (u · ∇) ξs−1 (x, u) , (3.42) 21
When it is feasible we do keep track of contact terms, such as for the massless case introduced in the following section.
– 16 –
fixes ∆ = s + d − 2 in the mass m2 R2 = ∆ (∆ − d) − s. The complete off-shell Lagrangian form was determined by Fronsdal in the 70’s [106], and reads Z s! (2) ϕs (x; ∂u ) Gs (x; u) , (3.43) SFronsdal [ϕs ] = 2 AdSd+1 where Gs is the corresponding spin-s generalisation of the linearised Einstein tensor 1 2 Gs (x; u) = 1 − u ∂u · ∂u Fs (x; u, ∇, ∂u ) ϕs (x, u) , 4
(3.44)
with Fs the so-called Fronsdal operator 1 Fs (x, u, ∇, ∂u ) = − m − u (∂u · ∂u ) − (u · ∇) (∇ · ∂u ) − (u · ∇)(∂u · ∂u ) . 2 2
2
(3.45)
The latter is fixed by invariance under linearised spin-s gauge transformations (3.42) with symmetric and traceless rank s − 1 gauge parameter ξs−1 .22 The Bianchi identity (∂u · ∇) Gs (x, u) = 0
(3.46)
requires that the field ϕs is double-traceless23 (∂u · ∂u )2 ϕs (x, u) = 0.
(3.47)
To determine the bulk-to-bulk propagator one needs to invert the equation of motion with source 1 (3.48) (1 − u2 ∂u · ∂u )Fs (x; u, ∇, ∂u ) ϕs (x, u) = −Js (x, u) , 4 where from gauge-invariance it follows that Js is conserved on-shell, (∂u · ∇) Js ≈ 0.24 For treelevel diagrams involving a single exchange, this inversion is independent of the off-shell gauge fixing of the exchanged field, since the exchanged field couples to on-shell external legs. In this context, the bulk-to-bulk propagator can be determined disregarding terms proportional 22
Alternative formulations have been developed which eliminate this algebraic trace constraint on the gauge parameter, however they come at the price of introducing introducing non-localities [107] or auxiliary fields [72, 108, 109]. 23 Note that the double-trace of ϕs is gauge invariant owing to the tracelessness of the gauge parameter. Forgoing the double-traceless constraint (3.47) without introducing auxiliary fields (apart from deforming the Bianchi identity (3.46) and thus requiring a modification of the action (3.43)) would lead to the propagation of non-unitary modes, which one may try to kill by imposing appropriate boundary conditions. This has been shown to be possible in flat space [109] though it is not yet clear if this approach can be extended to AdS space-times, or if it is compatible with introducing a source. For this reason we stick to the standard Fronsdal formulation (3.43) with double-trace constraint (3.47). 24 To be more precise, consistency with higher-spin symmetry (3.42) requires that Js has vanishing doubletrace, and moreover is conserved up to pure trace terms, (∂u · ∇) Js (x, u) ≈ O u2 . (3.49) As we shall demonstrate explicitly in §3.4.1 (supplemented by §B), improvement terms (which do not contribute to on-shell vertices) can be added to the Js such that it is exactly conserved.
– 17 –
to gradients [72, 73, 93] – both in the equation of motion and in the solution. To wit, one may solve25 (1 − m2 ) − u21 (∂u1 · ∂u1 ) Πs (x1 , u1 ; x2 , u2 ) 1 2 u ∂u 4 1 1
= −(1 −
(3.51) nn oo (u1 · u2 )s δ d+1 (x12 ) , · ∂u1 )−1
up to gradient terms. It is then sufficient to make an ansatz that is free from gradient terms, Πs (x1 , u1 ; x2 , u2 ) =
bs/2c Z ∞ X k=0
−∞
dν gk (ν) u21
k
u22
k
Ων,s−2k (x1 , u1 ; x2 , u2 ).
(3.52)
Plugging the ansatz into (3.51) fixes the functions gs,k (ν) [71] 1 , + s − 2)2 + ν 2 (1/2) (s − 2k + 1)2k gs,k (ν) = − 2k+3 k−1 d 2 · k! ( 2 + s − 2k)k ( d2 + s − k − 3/2)k ( d2 + s − 2k + iν)/2 k−1 ( d2 + s − 2k − iν)/2 k−1 , × ( d2 + s − 2k + 1 + iν)/2 k ( d2 + s − 2k + 1 − iν)/2 k gs,0 (ν) =
( d2
(3.53) k 6= 0.
As for the massive propagators in the previous section, the k = 0 term is the traceless and transverse part of the propagator which encodes the propagating degrees of freedom, while those for k > 0 generate purely contact terms in exchange amplitudes. 3.3
CPWE of spinning exchange diagrams
In this section we put together the results of the preceding sections to determine CPWEs of tree-level four-point exchange Witten diagrams with fields of arbitrary mass and integer spin on the internal and external legs. 3.3.1
Natural basis of conformal partial waves in AdS/CFT
To this end, it is useful to briefly discuss the integral form (2.20) of spinning conformal partial waves in terms of the natural AdS/CFT basis (3.25) of three point conformal structures. Employing this new basis, the spinning conformal partial waves of §2.2.2 read n,m W∆,s (yi ) + shadow
= κd−∆,s 25
γτ,s γ¯τ,s π d/2
(3.54) Z
˜∆,s (y)O∆ ,s (y3 )O∆ ,s (y4 )]](m) , dd y [[O∆1 ,s1 (y1 )O∆2 ,s2 (y2 )O∆,s (y)]](n) [[O 3 3 4 4
The symbol {{•}} indicates a double-traceless projection: (∂u · ∂u )2 {{f (u, x)}} = 0,
and
{{f (u, x)}} = f (u, x)
– 18 –
iff
(∂u · ∂u )2 f (u, x) = 0.
(3.50)
where for convenience we repeat here the form of the basis elements (3.25) [[O∆1 ,s1 (y1 )O∆2 ,s2 (y2 )O∆3 ,s3 (y3 )]]
(n)
" 3 # Y δ(i+1)(i−1) +n −1 δ Hn1 1 Hn2 2 Hn3 3 (i+1)(i−1) i 2 2 + ni ≡ Γ 2 (y12 )δ12 (y23 )δ23 (y31 )δ31 i=1 (3.55)
" ×
3 Y
1−ni 2
qi
−
δ(i+1)(i−1) 4
#
√ J(δ(i+1)(i−1) +2ni −2)/2 ( qi ) Y1s1 −n2 −n3 Y2s2 −n3 −n1 Y3s3 −n1 −n2
i=1
In combination with the basis (3.26) of bulk cubic vertices, once the harmonic function decomposition of a given spinning Witten diagram is known the choice of basis (3.54) of spinning conformal partial waves makes its CPWE follow almost automatically. We demonstrate this explicitly in the following section. 3.3.2
Generic spinning exchange diagram
We consider a generic tree-level four-point exchange of a spin-s field of mass m2 between fields of spin si and mass m2i . This is depicted for the s-channel below,
.
(3.56)
At this level the cubic vertices Vs,si ,sj are kinematic, and are not constrained by any consistency condition aside from the necessary requirement of respecting the AdS isometry. Expanded in the natural basis (3.26), they read X ni ,nj ,n ni ,nj ,n Vs,si ,sj = gsi ,sj ,s Isi ,sj ,s , (3.57) ni n ,n ,n
with arbitrary couplings gsii,sjj,s . The harmonic function decomposition decomposition follows upon insertion of the massive spin-s bulk-to-bulk propagator (3.39)
. (3.58)
– 19 –
What remains to determine the CPWE is to evaluate the three-point Witten diagrams on the RHS, which can be carried out seamlessly by employing the tools developed in §3.1. For this generic case we focus on the part of the exchange which encodes the propagating degrees of freedom. These are carried by the traceless and transverse part of the bulk-to-bulk propagator (3.41), and accordingly we focus on the l = 0 contribution in the harmonic function decomposition (3.58). The latter is factorised into three-point Witten diagrams of the form:
,
(3.59)
by virtue of the split representation (3.3) of the harmonic function. The amplitudes (3.59) can be straightforwardly given in any basis of three-point conformal structures using the results of §3.1. However, choosing to expand the couplings (3.57) in the natural AdS/CFT basis (3.26) of cubic vertices gives the following simple and compact form X As;τ ± (yi , yj , y) = gsn Ans;τ ± (yi , yj , y) (3.60) n
=
X
gsn B(s; n; τ ± ) [[O∆i ,si (yi )O∆j ,sj (yj )O d ±iν,s (y)]](n) 2
n
where we defined the vectors s = (si , sj , s), n = (ni , nj , n) and τ ± = (τi , τj , d2 ± iν − s).26 Using the integral representation of the conformal partial waves (3.54), one can then immediately write down the CPWE of the s-channel exchange (3.56)27 28 Z ∞ dν ν2 s As1 ,s2 |s,m2 |s3 ,s4 = + (y1 , y2 , y) A − (y3 , y4 , y) + ... s3,4 ;τ3,4 2 π As1,2 ;τ1,2 −∞ ν 2 + d − ∆ 2 Z ∞ X (y ) + ... . (3.61) dν cn,m (ν) W dn,m = +iν,s i −∞
26
n,m
2
n ,n ,n
In particular, gsn = gsii,sjj,s and An s;τ is the amplitude (3.28) with labels s = (si , sj , s), n = (ni , nj , n) and τ = (τi , τj , d2 ± iν − s). ± 27 Here, si,j = (si , sj , s), τi,j = (τi , τj , d2 ± iν − s), n = (n1 , n2 , n) and m = (m3 , m4 , m). 28 The ... denote contact terms generated by the l > 0 contributions in the harmonic function decomposition (3.58).
– 20 –
with29 d
cn,m (ν) =
−gsn1,2 gsm3,4
+ + B(s12 ; n; τ1,2 )B(s34 ; m; τ3,4 ) 2π 2 −1 Γ (iν + 1) ν . 2 iν + s + d2 − 1 Γ iν + d2 − 1 ν 2 + d2 − ∆
(3.63)
This is the contour integral form (2.14) of the conformal partial wave expansion, reviewed in §2.2.1. Recall that the functions cn,m (ν) encode the contribution from spin-s operators: A pole at scaling dimension d2 + iν = λ in the lower-half ν-plane signifies a contribution from the conformal multiplet [λ, s], whose residue gives the corresponding OPE coefficient. Separating out such poles into a function pn,m (ν), cn,m (ν) = ¯cn,m (ν) pn,m (ν) ,
(3.64)
we have pn,m (ν) =
1 ν2 +
d 2
−∆
2 Γ
×Γ
2(s1 +n−n2 )+τ1 +τ2 +s−( d 2 +iν ) 2
2(s3 +m−m4 )+τ3 +τ4 +s−( d 2 +iν ) 2
2(s2 +n−n1 )+τ1 +τ2 +s−( d 2 +iν ) Γ 2
2(s4 +m−m3 )+τ3 +τ4 +s−( d 2 +iν ) Γ . (3.65) 2
There are two types of contributions, in accord with the standard lore on CPWEs of Witten diagrams [27, 36, 86, 87, 110–116]: 1. Single-trace: This is the universal contribution to an exchange diagram, corresponding to the exchange of the bulk single-particle state. Accordingly, it is generated by the polefactor in the traceless and transverse part of the bulk-to-bulk propagator (3.41), which carries the propagating degrees of freedom. This translates into a pole at d2 + iν = ∆ in (3.65), which coincides with the scaling dimension of the spin-s single-trace operator O∆,s that is dual to the exchanged spin-s single-particle state of mass m2 R2 = ∆ (∆ − d) − s in the bulk. 2. Double-trace: The remaining contributions originate from contact terms, arising from the collision of the two points that are integrated over the entire volume of AdS. This generates 2-particle states in the bulk, which are dual to double-trace operators on the conformal boundary. Accordingly, the corresponding poles are encoded in the factors (3.29) arising from the integration over AdS. In the pole-function (3.65) these are the 29
To obtain this expression we used that − B(s34 ; m; τ3,4 )=
iκd−iν,s γ d +iν−s,s γ¯ d +iν−s,s 2
2
2νπ d/2 C d +iν,s 2
– 21 –
+ B(s34 ; m; τ3,4 ).
(3.62)
origin of the two sets of Gamma function poles (p=0,1,2,3,...) 1.
d + iν 2
d + iν 2
− s = τ1 + τ2 + 2 (s1 + n − n2 + p) ,
d + iν 2
− s = τ1 + τ2 + 2 (s2 + n − n1 + p) (3.66a)
2.
− s = τ3 + τ4 + 2 (s3 + m − m4 + p) ,
d + iν 2
− s = τ3 + τ4 + 2 (s4 + m − m3 + p) , (3.66b)
corresponding to contributions from the two families [O∆1 ,s1 O∆2 ,s2 ]s and [O∆3 ,s3 O∆4 ,s4 ]s of spin-s double-trace operators, respectively. In the bulk, these correspond to 2-particle states created, respectively, by ϕs1 with ϕs2 , and ϕs3 with ϕs4 . Let us briefly comment on the l > 0 contributions to the harmonic function decomposition (3.58). As explained earlier these are purely contact terms, and likewise generate doubletrace contributions [O∆1 ,s1 O∆2 ,s2 ]s−l and [O∆3 ,s3 O∆4 ,s4 ]s−l to the CPWE, but of lower spin s − l. 3.4
Spinning exchanges in the type A higher-spin gauge theory
So far our dialogue has not been restricted to any particular theory of spinning fields. In recent years, a lot of interest has been generated in theories of higher-spin gauge fields, owing in part to the conjectured duality [117–122] between higher-spin gauge theories on AdS backgrounds and free CFTs. In this section we apply the tools and results of the preceding sections to compute all four-point exchange Witten diagrams in the simplest higher-spin gauge theory for d > 2, which is known as the type A minimal higher-spin theory expanded about AdSd+1 [123].30 This theory is conjectured to be dual to the (singlet sector of the) free scalar O (N ) model in d-dimensions [120, 121]. The spectrum consists of a tower of totally symmetric even spin gauge fields (one for each even spin s = 2, 4, 6, ...) and a parity even scalar of mass m20 R2 = −2(d − 2), which sits in the higher-spin multiplet. Before moving to the computation of the exchange amplitudes, we first review the result for the metric-like cubic couplings established in [39]. 3.4.1
Off-shell cubic couplings
The off-shell cubic couplings of the type A minimal higher-spin theory on AdSd+1 were determined in [39], for de Donder gauge.31 For tree-level exchanges we only require couplings with a single field – the one that is exchanged – off-shell. For a spin-s field ϕs in de Donder gauge, 30
See [30, 36, 39, 71, 124–128] for other results on Witten diagrams in higher-spin gauge theories. Note that although the result (3.67) for the complete cubic couplings was fixed using the holographic duality, it was later verified [40] that the result solves the Noether procedure – i.e. requiring that each cubic coupling is local, the cubic vertices coincide with those that would be obtained without employing holography. The result built upon the covariant classification [76, 129–132] of cubic interactions in AdSd+1 . 31
– 22 –
its interaction with two on-shell fields of spins s1 and s2 reads 1 2 2 Vs1 ,s2 ,s = gs1 ,s2 ,s 1 − (d − 2 + Yi ∂Yi ) ∂Y3 ∂U3 Y1s1 Y2s2 Y3s 2
(3.67)
× ϕs1 (X1 , U1 ) ϕs2 (X2 , U2 ) ϕs (X3 , U3 ) . The coupling constants gs1 ,s2 ,s , for canonically normalised kinetic terms, are given by [39] d−3
gs1 ,s2 ,s
3d−1+s1 +s2 +s
2 1 π 4 2 =√ N Γ(d + s1 + s2 + s − 3)
s
2
Y Γ(s + d−1 2 ) Γ (s + 1) i=1
s
Γ(si + d−1 2 ) . Γ (si + 1)
(3.68)
The first term in (3.67) is the traceless and transverse part of the vertex, which is non-trivial on-shell. The second term accounts for the off-shell de Donder field ϕs , and accordingly is proportional to its trace. For the four-point exchange of a spin-s gauge field, we massage the vertices (3.67) into the form Vs1 ,s2 ,s (X) = s!Js|s1 ,s2 (X, ∂U ) ϕs (X, U ) , (3.69) with the spin-s current Js|s1 ,s2 bi-linear in ϕs1 and ϕs2 . This is an exercise of integration by parts in ambient space, and gives min(s1 ,s2 ) X (−2)k Γ(s +s +s+d−3) Γ(s +1) Γ(s +1) 1 2 1 2 Js|s ,s = H3k Y¯1s1 −k Y¯2s2 −k Y¯3s + ... , 1
k!
2
Γ(s1 +s2 +s+d−3−k) Γ(s1 −k+1) Γ(s2 −k+1)
k=0
(3.70) where the ... are terms that constitute the completion with ϕs off shell, which are reinstated below. For convenience above we defined the contractions Y¯1 = Y1 ,
Y¯2 = −∂U2 · ∂X1 ,
Y¯3 = 21 ∂U3 · (∂X1 − ∂X2 ) ,
H3 = ∂U1 · ∂U2 .
(3.71)
In its present form, the complete current (3.70) is not exactly conserved. Indeed, recall that for a doubly-traceless Fronsdal field ϕs , higher-spin symmetry at the linearised level only requires that it is conserved up to traces (c.f. footnote 24). On the other hand, as emphasised in §3.2.2, the manifest trace form (3.52) of the bulk-to-bulk propagators requires the use of exactly conserved currents. In appendix §B we show the details of how the current (3.70) can be improved such that it satisfies exact conservation. Here we just state that it can be attained by taking on-shell non-trivial part of (3.70) and dressing each term with a differential operator [s/2] X H3k Y¯1s1 −k Y¯2s2 −k Y¯3s → αn(k) (∂Y2¯3 )n (∂U2 3 )n H3k Y¯1s1 −k Y¯2s2 −k Y¯3s , (3.72) n=0
where αn(k) =
2n 1 Γ(3 + k − s1 − s2 − d2 + n) 1 . 2 n! Γ(3 + k − s1 − s2 − d2 )
– 23 –
(3.73)
3.4.2
Four-point exchange diagrams
Consider the four-point exchange of a spin-s gauge field between gauge fields of spin si in the s-channel. The manifest trace form of the bulk-to-bulk propagator (3.52) gives the harmonic function decomposition
, (3.74) where the operator Jsi is the spin-si conserved current in the free scalar O (N ) model dual to the spin-si gauge field ϕsi in the bulk. The notation J (k) denotes the k-th trace of the conserved current J, which arise from the trace structure of the bulk-to-bulk propagator contact terms. The explicit form of J is given in §3.4.1, while its k-th trace is derived in §C. To determine the CPWE, we therefore need to evaluate three-point Witten diagrams of the form,
,
(3.75)
which, employing the tools introduced in §3.1 entails expressing the cubic couplings in the basis (3.26). We focus first on the k = 0 contribution, which encodes the propagating degrees of freedom. As we saw for the massive exchanges in §3.3.2, this is generated by the traceless and transverse part of the bulk-to-bulk propagator (3.52). Accordingly, only the on-shell non-trivial (traceless and transverse) part of the cubic couplings (3.67) contribute, whose explicit form we give here for convenience: VsT1T,s2 ,s (X) = gs1 ,s2 ,s Y1s1 Y2s2 Y3s ϕs1 (X1 , U1 ) ϕs2 (X2 , U2 ) ϕs (X3 , U3 ) . (3.76) Xi =X
Nicely, this is already in the natural AdS/CFT basis (3.26) and the amplitudes (3.75) for k = 0 can be immediately written down by employing the result (3.28) As;τ ± (yi , yj , y) = gsi ,sj ,s B (s; 0; τ ) [[Jsi (yi ) Jsj (yj ) O d
2 ±iν,s
– 24 –
(y)]]0 ,
(3.77)
where here τ ± = d − 2, d − 2, d2 ± iν − s and n = (0, 0, 0). Following the discussion of §3.3.2 for the generic case, one then obtains that the k = 0 term in the harmonic function decomposition (3.74) of the exchange diagram yields the following contributions to its CPWE Z ∞ s dν cs (ν) W d0,0 As1 ,s2 |s|s3 ,s4 = (y ) + ... (3.78) +iν,s i −∞
2
with d
B(s12 ; 0; τ + )B(s34 ; 0; τ + ) 2π 2 −1 Γ (iν + 1) ν , cs (ν) = −gs1 ,s2 ,s gs3 ,s4 ,s 2 iν + s + d2 − 1 Γ iν + d2 − 1 ν 2 + s + d2 − 2
(3.79)
where the pole function (3.65) in this case is given by 1 2(s1 +d−2)+s−( d2 +iν ) 2(s2 +d−2)+s−( d2 +iν ) Γ ps (ν) = 2 Γ 2 2 ν 2 + s + d2 − 2 2(s3 +d−2)+s−( d2 +iν ) 2(s4 +d−2)+s−( d2 +iν ) ×Γ Γ . (3.80) 2 2 In the following we discuss in detail the particular contributions. Single-trace In line with the discussion of the generic case in §3.3.2, the pole factor in the traceless and transverse part of the bulk-to-bulk propagator (3.52) generates a pole in (3.80) at d2 + iν = s + d − 2, which is the scaling dimension of the dual spin-s conserved current Js in the free scalar O (N ) model. Furthermore, notice that for d2 +iν = s+d−2 the three-point conformal structure generated by the k = 0 amplitude (3.77) coincides with the three-point conserved structure (2.28) in free scalar theories. The corresponding spin-s conformal partial wave thus coincides with the conserved conformal partial wave Ws0,0 in the set (2.25), which represents the contribution 1 ,s2 |s|s3 ,s4 from the conserved operator Js to the four-point function hJs1 Js2 Js3 Js4 i in free scalar theories. With the result (2.29) of all single-trace conserved current OPE coefficients in free scalar theories, in this case we can confirm the standard expectation that the single-trace contribution to an exchange Witten diagram coincides with the contribution from the same single-trace operator in the CPWE of the dual CFT four-point function, when expanded in the same channel. Indeed, using that [39, 40]32 B(sij ; 0; d − 2, d − 2, d − 2) gsi ,sj ,s p = cJsi Jsj Js , Csi +d−2,si Csj +d−2,sj Cs+d−2,s
(3.81)
we have (closing the contour in the lower-half ν-plane) 0,0 d − 2πiRes cs (ν) W d +iν,s (yi ) , 2 + iν = s + d − 2 = cJs1 Js2 Js cJs Js3 Js4 Ws0,0 (yi ) , 1 ,s2 |s|s3 ,s4 2
(3.82) as expected. 32
Here we divide by the normalisation of the bulk-to-boundary propagators (3.12) to give unit normalisation to the dual single-trace operator two point functions (3.13).
– 25 –
Double-trace In this case the two sets of Gamma function poles 1. 2.
d + iν − s = 2 (d − 2) + 2 (p + s1 ) , 2 d + iν − s = 2 (d − 2) + 2 (p + s3 ) , 2
d + iν − s = 2 (d − 2) + 2 (p + s2 ) , 2 d + iν − s = 2 (d − 2) + 2 (p + s4 ) , 2
(3.83a) (3.83b)
with p = 0, 1, 2, 3, ... , correspond to contributions from the two families [Js1 Js2 ]s and [Js3 Js4 ]s of spin-s double-trace operators build from single-trace conserved currents. k > 0 contributions Similarly, being contact, the k > 0 contributions to the CPWE of the exchange (3.74) are from double-trace operators [Js1 Js2 ]s−2k and [Js3 Js4 ]s−2k of lower spin s − 2k. Computing the corresponding three-point Witten diagrams (3.75) for k > 0 is a lot more involved, as it (k) requires to compute the k-th trace of the currents Js|s . We give a recipe for computing them i ,sj in §C, but stop short of evaluating the corresponding Witten diagrams; given that the s1 -s2 s3 -s4 quartic contact vertex is currently unfixed in metric-like form for the type A minimal higher-spin theory, the k > 0 contributions are anyway highly dependent on the choice of field frame. We non-the-less point out that there are simplifications for particular combinations of the external spins, such as for a single spinning external field (e.g. s1 -0-0-0) and also for a single spinning external field either side of the exchange (e.g. s1 -0-s3 -0 in the s-channel), where the three-point bulk integrals for k > 0 are of the same type as in the k = 0 case.
4
Spinning Geodesic Witten diagrams
Figure 1.
Conformal partial wave expansions of standard Witten diagrams typically receive a slew of contributions from composite primary operator conformal multiplets, which arise from local contact terms. Taking the example of a single-exchange diagram considered in §3, on top of the contribution from the single-trace operator dual to the exchanged single-particle state, one finds a multitude of double-trace operator contributions generated from the collision of the two
– 26 –
points that are integrated over in the bulk. The latter can be regarded as effective local quartic contact interactions, and indeed: the CPWE of quartic contact Witten diagrams generated by local quartic vertices contain contributions from an infinite number of double-trace operators, as shown explicitly for quartic scalar self-interactions in [26, 30, 41]. An AdS-covariant deformation of the integration over AdS that prevents the proliferation of double-trace contributions is given by instead integrating the bulk cubic vertices over geodesics γ12 and γ34 connecting the boundary points hosting the external operators, as depicted in figure 1.33 In restricting to geodesics, one prohibits the collision of integration points. In other words, up to normalisation one has
(4.1) To evaluate the “geodesic Witten diagram” on the RHS of (4.1), like for the standard exchange Witten diagrams, one employs the harmonic function decomposition (3.36) of the bulk-to-bulk propagator: As reviewed in §3, the split representation (3.3) of the harmonic function provides an explicit link with conformal partial waves in their integral form (2.13). Owing to the separated integration points we require only the traceless and transverse part (3.41) of the bulk-to-bulk propagator, since the remaining terms in the propagator are generated by local source terms. To wit,
(4.2) 33
This approach has been taken in a number of works, such as [133–148] in AdS3 for Virasoro and WN conformal partial waves, and [41, 74, 79, 149] for conformal partial waves in general dimensions. The very recent works [74, 79] in particular also consider geodesic Witten diagrams with totally symmetric external fields.
– 27 –
The only non-trivial step is then to evaluate three-point the geodesic integrals:
.
(4.3)
In this section we show that the result for (4.3) follows immediately from the results [39] previously established for the corresponding standard three-point Witten diagrams, which were reviewed in §3.1. This essentially boils down to the fact that the differential relationship (3.11), between a spinning standard three-point Witten diagram and a standard scalar seed, also holds for geodesic three-point integrals:
. (4.4) 0,0,0 0,0,0 can be performed The geodesic three-point integral γi,j A0,0,0;τ for the vertex I0,0,0 1 ,τ2 ,τ3 using similar techniques to those for the integration over the full volume of AdS, which are reviewed in §D. For the geodesic integral, the main ingredient is the parametrisation for the geodesic connecting boundary points P1 and P2 :
X M (λ) =
e−λ P1M + eλ P2M . (−2P1 · P2 )1/2
(4.5)
Employing the Schwinger-parameterised form (D.1) of the bulk-to-boundary propagators, one obtains γ1,2
3 ∞Y
Z C∆i ,0 dti ∆i (y1 , y2 , y3 ) = t dλ e2(t1 P1 +t2 P2 +t3 P3 )·X(λ) Γ (∆ ) t i i 0 γ 12 i=1 Z ∞ Y 3 3 Y Γ(∆1 )Γ(∆2 ) C∆i ,0 dmi δ(i+1)(i−1) /2 = mi exp −mi P(i+1)(i−1) . (4.6) ∆1 +∆2 −∆3 mi Γ( ) i=1 Γ (∆i ) 0 2 i=1
A0,0,0 0,0,0;τ1 ,τ2 ,τ3
Z
Comparing with the scalar seed (D.5) for standard three-point Witten diagrams, we see that both AdS and geodesic integrals are proportional to the same scalar three-point conformal
– 28 –
structure – as it should be by AdS covariance – with the difference encoded purely in the overall coefficient. The implication is that the result for a given geodesic three-point Witten diagram can be obtained from the result for the standard three-point Witten diagram, and vice versa, simply by accounting for the difference in overall coefficient. In particular, the geodesic integral for a generic element Isn11,s,n22,s,n3 3 of the basis (3.6) of cubic vertices (depicted in (4.4)) has the same form (3.21) as its standard three-point Witten diagram but with overall coefficient P3 → Pg3 : ! 3 Y Γ(∆ − 1)(∆ + s − 1) 1 1 i i i Pg3 = 3d 2 )δ12 (y 2 )δ23 (y 2 )δ31 d (y Γ ∆ + 1 − 2 i 8π 31 23 12 2 i=1 ×
Γ(−n1 − n2 + s1 + s3 + τ1 )Γ(−n2 − n3 + s1 + s2 + τ2 ) . (4.7) Γ 21 (−2n2 + 2s1 + τ1 + τ2 − τ3 )
This completes the evaluation of the general spinning three-point geodesic integral (4.3). As a final comment, it is useful to give the ratio between Witten and geodesic prefactors which is given at fixed ni as: Pg3 2 2Γ(s1 + s3 + τ1 − n1 − n2 )Γ(s1 + s2 + τ2 − n2 − n3 ) P = d/2 . d P3 π Γ s1 − n2 + δ12 Γ α (sα + τα − nα ) − 2
(4.8)
Interestingly the above ratio at fixed spins and twists depends only on ni , meaning that the result for the geodesic integration is a very simple deformation of the diagonal map for the standard AdS volume integral spelled out in (3.28). Furthermore, this observation allows us to straightforwardly diagonalise the linear map between three-point geodesic Witten diagrams and three-point conformal structures. In particular, when integrating over a geodesic γij connecting the boundary points Pi and Pj , one recovers a result similar to (3.28) for standard AdS Witten diagrams: γi,j n1 ,n2 ,n3 As1 ,s2 ,s3 ;τ1 ,τ2 ,τ3
(y1 , y2 , y3 ) = B(si , ni , δjk ) [[O∆1 ,s1 (y1 )O∆2 ,s2 (y2 )O∆3 ,s3 (y3 )]](n) .
(4.9)
In the above formula we conveniently re-defined the basis of bulk structures by the requirement that the same basis (3.25) of three-point conformal structures on the boundary is obtained. To this end we performed a change of basis for the bulk cubic couplings (3.26), which entails 3 34 a rescaling of the coefficients Csn11,s,n22,s,n3 ;m 1 ,m2 ,m3 : P d τα d Γ π2 α (sα − mα + 2 ) − 2 Γ (si − mj + δij ) γij n1 ,n2 ,n3 (C )s1 ,s2 ,s3 ;m1 ,m2 ,m3 = C n1 ,n2 ,n3 , 2 Γ(si + sk − mi − mj + τi )Γ(si + sj − mj − mk + τj ) s1 ,s2 ,s3 ;m1 ,m2 ,m3 (4.10) yielding the basis vertices X 1 ,m2 ,m3 (C γij )ns11,s,n22,s,n33;m1 ,m2 ,m3 Ism1 ,s , (4.11) (I γij )ns11,s,n22,s,n33 = 2 ,s3 mi
whose three-point geodesic integrals satisfy equation (4.9). The change of basis (4.10) allows us to extend all of the discussion and results in sections 3 to the case of three-point geodesic integration, giving the geodesic Witten diagram representation of a single spinning conformal partial wave on the boundary. 34
Notice that the asymmetry i ↔ j is expected from the cyclic structure of the basis of cubic couplings.
– 29 –
Acknowledgements C. S. and M. T. are grateful to M. Henneaux for useful discussions, and also A. Castro, E. Llabr´es and F. G. Rej´ on-Barrera in the context of geodesic Witten diagrams. C. S. also thanks D. Francia for useful correspondence. The research of M. T. is partially supported by the Fund for Scientific Research-FNRS Belgium, grant FC 6369 and by the Russian Science Foundation grant 14-42-00047 in association with Lebedev Physical Institute.
A
Conventions, notations and ambient space
In this work we employ the same conventions as in [39], which we gvery briefly review of here for completeness. For more details on the ambient space formalism, see for instance [76, 150, 151]. The ambient formalism is an indispensable framework for computations in AdSd+1 space. In this context, the latter is viewed as a hyperboloid embedded in an ambient (d + 2)-dimensional Minkowski space X 2 + R2 = 0 ,
X0 > 0 ,
(A.1)
where R is the AdS radius. In ambient light-cone coordinates (X + , X − , X i ) with X 2 = −X + X − + δij X i X j , the solution of the constraints (A.1) in the Poincar´e co-ordinates xµ = z, y i is given by R X A = (1, z 2 + y 2 , y i ) . (A.2) z Bulk fields In order to obtain a one-to-one correspondence between fields on AdS and those living in the higher-dimensional flat ambient space, one imposes constraints with defining the ambient space extensions of the AdSd+1 fields [89]. Such restrictions are usually given as homogeneity and tangentiality constraints. Employing a generating function formalism with intrinsic and ambient auxiliary vectors uµ and U A , a symmetric rank-s tensor ϕs (x, u) intrinsic to the AdS manifold is represented in ambient space by ϕs (x, u) =
1 ϕµ ...µ (x)uµ1 ...uµs s! 1 s
→
ϕs (X, U ) =
1 ϕA1 ...As (X)U A1 . . . U As . s!
(A.3)
subject to the following homogeneity and tangentiality conditions (X · ∂X − ∆)ϕs (X, U ) = 0 ,
(X · ∂U ) ϕs (X, U ) = 0 .
(A.4)
Nicely, the conditions (A.4) imply that on-shell 2 ∂X ϕA1 ...As = 0,
(A.5)
for the ambient representative of the AdS field ϕs of mass m2 R2 = ∆ (∆ − d) − s. Let us stress that in imposing tangentiality and homogeneity conditions (A.4) one is implicitly extending the AdS field to the full ambient space, where X 2 plays the role of the radial coordinate. This formalism is different from the manifestly intrinsic formalism (for instance used in [38]) where one never moves away from the AdS manifold X 2 = −R2 .
– 30 –
The ambient representative of the AdS covariant derivative ∇µ takes the simple form ∂ , ∂X B
(A.6)
∇ = P ◦ ∂ ◦ P.
(A.7)
∇A = PAB and acts via
Boundary fields The boundary of AdSd+1 is identified with the null rays P 2 = 0,
P ∼ λP,
λ 6= 0,
(A.8)
where P gives the ambient space embedding of the CFT coordinate y i . It is convenient to introduce the boundary analog of the auxiliary variables U A , which we refer to as ZA (y) and extend to ambient space the null CFT auxiliary variable z i . Working in light cone coordinates P A = (P + , P − , P i ), with the gauge choice P + = 1 one has P A (y) = (1, y 2 , y i )
Z A (y) = (0, 2y · z, z i ) .
and
(A.9)
A symmetric rank-s boundary operator O∆,s of scaling dimension ∆ is represented by:35 O∆,s (y, z) =
1 Oµ ...µ (y) z µ1 · · · z µs s! 1 s
→
O∆,s (P, Z) =
1 OA ...A (P ) Z A1 · · · Z As , s! 1 s (A.10)
where (P · ∂P − ∆)O∆,s (P, Z) = 0 ,
(P · ∂Z ) O∆,s (P, Z) = 0 ,
(A.11)
and, being restricted to the null cone (A.8), there is an extra redundancy OA1 ...As (P ) → OA1 ...As (P ) + P(A1 Λ A2 ...As ) , P
B
A1
ΛA1 ...As−1 = 0,
−(∆+1)
ΛA1 ...As−1 (λP ) = λ
ΛA1 ...As−1 (P ),
η
(A.12) A1 A2
ΛA1 ...As−1 = 0. (A.13)
The improved current
In this appendix we detail the improvement of the higher-spin currents (3.70) to make them exactly conserved. We begin with the traceless and transverse part of the current, min(s1 ,s2 )
JsT3T|s1 ,s2
=
X
Γ(s1 +1) Γ(s2 +1) (−2)k Γ(s1 +s2 +s3 +d−3) k k! Γ(s1 +s2 +s3 +d−3−k) Γ(s1 −k+1) Γ(s2 −k+1) H3
Y¯1s1 −k Y¯2s2 −k Y¯3s3 .
(B.1)
k=0
On-shell, each monomial in the above is conserved. One can therefore study the structure of the required improvements with ϕ3 off-shell for a given monomial fs(k) = H3k Y¯1s1 −k Y¯2s2 −k Y¯3s3 . 1 ,s2 ,s3 35
(B.2)
Note that here z denotes the auxiliary vector z i and should not be confused with the radial Poincar´e co-ordinate in (A.2).
– 31 –
The combination of different monomials in (B.1) above is necessary to achieve on-shell gauge invariance with respect to ϕ1 and ϕ2 which can be easily verified explicitly (see e.g. [76]). In order to proceed to find the conserved improvement, the doubly-traceless condition on ϕ3 together with the traceless condition on the corresponding gauge parameter needs to be dropped. Not doing so would only recover a current whose traceless part is conserved. We hence consider the following ansatz for the improvement, dressing each monomial (B.2) with trace operators [s3 /2] X Fs(k) = αn(k) (∂Y2¯3 )n (∂U2 3 )n fs(k) , (B.3) 1 ,s2 ,s3 1 ,s2 ,s3 n=0
where the derivative with respect to Y¯3 accounts for the fact that taking the trace lowers the (k) spin. The coefficients αn in the ansatz (B.3) are fixed by requiring gauge invariance of the vertex with ϕ3 off-shell and with traceless gauge parameter ξ3 Z dX (U3 · ∇3 ξ3 ) Fs(k) ϕ ϕ = 0. (B.4) 1 ,s2 ,s3 1 2 In the above the fields ϕ1 and ϕ2 are on-shell, while the integral sign (which in the following will be omitted for ease of notation) implies that the above identity holds modulo total derivatives. Employing the explicit form of the gradient operator: U3 · ∇3 = U3 · ∂X3 −
U3 · X3 (X3 · ∂X3 − U3 · ∂U3 ) , X32
(B.5)
we arrive to the following conservation condition: ∞ h X
i (k) (∂U2 3 )n ∂U3 · ∂X3 = 0 , 2(n + 1)αn+1 − 14 αn(k) [d + 2(s1 + s2 − k − n) − 4] (∂Y2¯3 )n fs(k) 1 ,s2 ,s3
n=0
(B.6) (k)
which leads to the solution for αn in the form 2n 1 Γ(3 + k − s1 − s2 − d2 + n) 1 (k) αn = . 2 n! Γ(3 + k − s1 − s2 − d2 )
C
(B.7)
Trace of the currents
In order to evaluate the current exchange (3.74) with the manifest trace form of the propagator (3.52), we are required to compute the n-th trace of the exactly conserved current derived in the previous section. The process can be simplified by noting that in the present context the traces are contracted with harmonic functions, where we encounter terms of the form 2 n 2 q (∂Y2¯3 )n fs(k) (∂ ) (u ) Ω , (C.1) ν,s −2q ,s ,s U 3 3 1 2 3 3 U3 =0
where u23 is the intrinsic symmetrised metric tensor written in generating function form, which can be re-expressed in the ambient formalism as u23 = U32 −
U3 · X3 . X32
– 32 –
(C.2)
q To evaluate the trace one commutes the U3 contained in the u23 to the far left hand side, where the condition U3 = 0 can be applied. We first commute the u23 past the ∂U2 3 , which, employing the tracelessness of the harmonic functions, reads (∂U2 3 )n (u23 )q Ων,s3 −2q = Aqn (u23 )q−n Ων,s3 −2q ,
Aqn = 22n
d 1 Γ(n−q) Γ(− 2 +n+q−s3 + 2 ) d 1 Γ(−q) Γ(− +q−s3 + ) 2
.
(C.3)
2
What remains is to evaluate terms of the form
. (∂Y2¯3 )n fs(k) (u23 )q−n Ων,s3 −2q 1 ,s2 ,s3 U3 =0
(C.4)
To this end, it is useful split Y¯3 as36 1 Y¯3 = (V31 − V32 ) , 2
V31 = ∂U3 · ∂X1 ,
V31 = ∂U3 · ∂X2 ,
(C.5)
and express any function of Y¯3 instead in terms of V31 and V21 . In this way, the action of some operator g Y¯3 on u3 can be expressed in the form A A = ∂X ∂ + ∂X ∂ g Y¯3 uA g (V31 , V32 ) , (C.6) 3 1 V31 2 V32 U3 =0
from which follows the general formula n g Y¯3 u23 = 2 [∂X1 · ∂X2 + X1 · ∂X1 X2 · ∂X2 ] ∂V31 ∂V32 U3 =0
o + X1 · ∂X1 (X1 · ∂X1 − 1)∂V231 + X2 · ∂X2 (X2 · ∂X2 − 1)∂V232 g (V31 , V32 ) , (C.7) which can be iteratively applied to evaluate the traces in (C.4). Now, since each current in the exchange is to be integrated over AdS, we can evaluate the above terms up to integrations by parts using X1 · ∂X1 = −(d − 2 + Y¯1 ∂Y¯1 + Y¯2 ∂Y¯2 + V31 ∂V31 + Q3 ∂Q3 ) , X2 · ∂X2 = −(d − 2 + Y¯1 ∂Y¯1 + Y¯2 ∂Y¯2 + V32 ∂V32 + Q3 ∂Q3 ),
(C.8) (C.9)
where 1 Q3 = − (X1 · ∂X1 + X1 · ∂X1 + ∆3 + d)(X1 · ∂X1 + X1 · ∂X1 − ∆3 ), 2
(C.10)
and ∆3 = d2 ± iν. After evaluating the action of the above operators one can integrate by parts to obtain an expression for the final form of the trace terms (C.1) in the form min(s1 ,s2 ) (k)
Js3 |s1 ,s2 · Π d ±iν,s3 −2k = 2
X
βsk,m H3m Y1s1 −m Y2s2 −m Y3s3 −2k , 3 |s1 ,s2
(C.11)
m=0
where, via integration by parts, we replaced Y¯2 → eλH3 ∂Y1 ∂Y2 , V31 → Y3 and V32 → −Y3 + ∂U3 · ∂X3 = −Y3 , where in the latter equality we used that the harmonic function is divergenceless. 36
Note that in fact V31 = Y3 , but for ease of notation in this section we employ the labelling V31 .
– 33 –
See [39, 76] for (in-context) reviews of integration by parts in the ambient space formalism, where in particular the parameter λ and its use is defined. The explicit form of the coefficients βsk,m is rather involved, and since the results of this 3 |s1 ,s2 work do not rely on the knowledge of the explicit expression for contact terms in exchange amplitudes (which are anyway highly dependent on the field frame), we do not present them here.
D
Seed bulk integrals
Our approach to evaluate spinning three-point Witten diagrams is underpinned by their differential relationship (3.11) with basic seed diagrams with external scalars [77]. The latter is the basic ingredient from which our results are generated, which we briefly review here. It is useful to employ the Schwinger-parameterised form for the propagator [152, 153] Z ∞ C∆,0 dt ∆ 2tP ·X K∆ (X; P ) = t e , (D.1) Γ (∆) 0 t which results in 0,0,0 A0,0,0;τ 1 ,τ2 ,τ3
Z (P1 , P2 , P3 ) = 0
3 ∞Y i=1
C∆i ,0 dti ∆i t Γ (∆i ) ti
Z
dXe2(t1 P1 +t2 P2 +t3 P3 )·X .
(D.2)
AdS
The integration over AdS is then straightforward to perform, and yields (see e.g. box 5.2 in [151]) Z 0
3 ∞Y i=1
dti ∆i t ti
Z
dXe2(t1 P1 +t2 P2 +t3 P3 )·X
(D.3)
AdS
−d +
d 2
=π Γ
P3
i=1 ∆i
!Z
2
0
3 ∞Y i=1
dti ∆i (−t1 t2 P12 −t1 t3 P13 −t2 t3 P23 ) e , t ti i
where Pij = −2Pi · Pj . Through the change of variables, r r r m2 m3 m1 m3 m1 m2 t1 = , t2 = , t3 = , m1 m2 m3
(D.4)
we then obtain the final result: 0,0,0 A0,0,0;τ (P1 , P2 , P3 ) 1 ,τ2 ,τ3 ! 3 ! P d 3 Y C∆ ,0 Z ∞ Y −d + 3i=1 ∆i dmi δ(i+1)(i−1) /2 π2 i Γ mi exp −mi P(i+1)(i−1) , = 2 2 Γ (∆i ) mi 0 i=1
i=1
(D.5) where i ∼ = i + 3 and δi(i+1) =
∆i + ∆(i+1) − ∆(i−1) , 2
– 34 –
(D.6)
The standard three-point conformal structure for scalar operators is obtained from (D.5) using the integral representation of the Gamma function 1
0,0,0 A0,0,0;τ (P1 , P2 , P3 ) = C (∆1 , ∆2 , ∆3 ; 0) 1 ,τ2 ,τ3
∆1 +∆3 −∆2 2
P13
∆2 +∆3 −∆1 2
P23
∆1 +∆2 −∆3 2
,
(D.7)
P12
where explicitly C (∆1 , ∆2 , ∆3 ; 0) =
1 d π2Γ 2
−d +
P3
i=1 ∆i
2
! C∆1 ,0 C∆2 ,0 C∆3 ,0
Γ
∆1 +∆2 −∆3 2
Γ ∆1 +∆23 −∆2 Γ ∆2 +∆23 Γ (∆1 ) Γ (∆2 ) Γ (∆3 )
(D.8) −∆ 1
.
References [1] S. Ferrara, A. F. Grillo, G. Parisi, and R. Gatto, The shadow operator formalism for conformal algebra. vacuum expectation values and operator products, Lett. Nuovo Cim. 4S2 (1972) 115–120. [Lett. Nuovo Cim.4,115(1972)]. [2] G. Mack and I. T. Todorov, Conformal-invariant green functions without ultraviolet divergences, Phys. Rev. D8 (1973) 1764–1787. [Phys. Rev.D6,1764(1973)]. [3] S. Ferrara, A. F. Grillo, and R. Gatto, Tensor representations of conformal algebra and conformally covariant operator product expansion, Annals Phys. 76 (1973) 161–188. [4] S. Ferrara, A. F. Grillo, G. Parisi, and R. Gatto, Covariant expansion of the conformal four-point function, Nucl. Phys. B49 (1972) 77–98. [Erratum: Nucl. Phys.B53,643(1973)]. [5] A. M. Polyakov, Nonhamiltonian approach to conformal quantum field theory, Zh. Eksp. Teor. Fiz. 66 (1974) 23–42. [6] S. Ferrara, A. F. Grillo, R. Gatto, and G. Parisi, Analyticity properties and asymptotic expansions of conformal covariant green’s functions, Nuovo Cim. A19 (1974) 667–695. [7] A. A. Belavin, A. M. Polyakov, and A. B. Zamolodchikov, Infinite Conformal Symmetry in Two-Dimensional Quantum Field Theory, Nucl. Phys. B241 (1984) 333–380. [8] F. A. Dolan and H. Osborn, Conformal four point functions and the operator product expansion, Nucl. Phys. B599 (2001) 459–496, [hep-th/0011040]. [9] F. A. Dolan and H. Osborn, Conformal partial waves and the operator product expansion, Nucl. Phys. B678 (2004) 491–507, [hep-th/0309180]. [10] R. Rattazzi, V. S. Rychkov, E. Tonni, and A. Vichi, Bounding scalar operator dimensions in 4D CFT, JHEP 12 (2008) 031, [arXiv:0807.0004]. [11] V. S. Rychkov and A. Vichi, Universal Constraints on Conformal Operator Dimensions, Phys. Rev. D80 (2009) 045006, [arXiv:0905.2211]. [12] F. Caracciolo and V. S. Rychkov, Rigorous Limits on the Interaction Strength in Quantum Field Theory, Phys. Rev. D81 (2010) 085037, [arXiv:0912.2726]. [13] F. A. Dolan and H. Osborn, Conformal Partial Waves: Further Mathematical Results, arXiv:1108.6194. [14] M. F. Paulos, JuliBootS: a hands-on guide to the conformal bootstrap, arXiv:1412.4127.
– 35 –
[15] D. Simmons-Duffin, A Semidefinite Program Solver for the Conformal Bootstrap, JHEP 06 (2015) 174, [arXiv:1502.0203]. [16] S. El-Showk, M. F. Paulos, D. Poland, S. Rychkov, D. Simmons-Duffin, and A. Vichi, Solving the 3D Ising Model with the Conformal Bootstrap, Phys. Rev. D86 (2012) 025022, [arXiv:1203.6064]. [17] F. Kos, D. Poland, D. Simmons-Duffin, and A. Vichi, Precision islands in the Ising and O(N ) models, JHEP 08 (2016) 036, [arXiv:1603.0443]. [18] A. L. Fitzpatrick, J. Kaplan, D. Poland, and D. Simmons-Duffin, The Analytic Bootstrap and AdS Superhorizon Locality, JHEP 12 (2013) 004, [arXiv:1212.3616]. [19] Z. Komargodski and A. Zhiboedov, Convexity and Liberation at Large Spin, JHEP 11 (2013) 140, [arXiv:1212.4103]. [20] L. F. Alday and A. Zhiboedov, Conformal Bootstrap With Slightly Broken Higher Spin Symmetry, JHEP 06 (2016) 091, [arXiv:1506.0465]. [21] L. F. Alday, Large Spin Perturbation Theory, arXiv:1611.0150. [22] R. Gopakumar, A. Kaviraj, K. Sen, and A. Sinha, Conformal Bootstrap in Mellin Space, Phys. Rev. Lett. 118 (2017), no. 8 081601, [arXiv:1609.0057]. [23] J. M. Maldacena, The Large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113–1133, [hep-th/9711200]. [Adv. Theor. Math. Phys.2,231(1998)]. [24] S. S. Gubser, I. R. Klebanov, and A. M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B428 (1998) 105–114, [hep-th/9802109]. [25] E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253–291, [hep-th/9802150]. [26] I. Heemskerk, J. Penedones, J. Polchinski, and J. Sully, Holography from Conformal Field Theory, JHEP 10 (2009) 079, [arXiv:0907.0151]. [27] S. El-Showk and K. Papadodimas, Emergent Spacetime and Holographic CFTs, JHEP 10 (2012) 106, [arXiv:1101.4163]. [28] M. Gary, S. B. Giddings, and J. Penedones, Local bulk S-matrix elements and CFT singularities, Phys. Rev. D80 (2009) 085005, [arXiv:0903.4437]. [29] A. L. Fitzpatrick and J. Kaplan, AdS Field Theory from Conformal Field Theory, JHEP 02 (2013) 054, [arXiv:1208.0337]. [30] X. Bekaert, J. Erdmenger, D. Ponomarev, and C. Sleight, Quartic AdS Interactions in Higher-Spin Gravity from Conformal Field Theory, JHEP 11 (2015) 149, [arXiv:1508.0429]. [31] E. D. Skvortsov and M. Taronna, On Locality, Holography and Unfolding, JHEP 11 (2015) 044, [arXiv:1508.0476]. [32] J. Maldacena, D. Simmons-Duffin, and A. Zhiboedov, Looking for a bulk point, JHEP 01 (2017) 013, [arXiv:1509.0361]. [33] M. Taronna, Pseudo-local Theories: A Functional Class Proposal, in Proceedings, International Workshop on Higher Spin Gauge Theories: Singapore, Singapore, November 4-6, 2015, pp. 59–84, 2017. arXiv:1602.0856. [34] X. Bekaert, J. Erdmenger, D. Ponomarev, and C. Sleight, Bulk quartic vertices from boundary four-point correlators, in Proceedings, International Workshop on Higher Spin Gauge Theories:
– 36 –
Singapore, Singapore, November 4-6, 2015, pp. 291–303, 2017. arXiv:1602.0857. [35] M. Taronna, A Note on Field Redefinitions and Higher-Spin Equations, EPJ Web Conf. 125 (2016) 05025, [arXiv:1607.0471]. [36] C. Sleight, Interactions in Higher-Spin Gravity: a Holographic Perspective. PhD thesis, Munich U., 2016. arXiv:1610.0131. [37] A. Belin, B. Freivogel, R. A. Jefferson, and L. Kabir, Sub-AdS Scale Locality in AdS3 /CFT2 , arXiv:1611.0860. [38] M. S. Costa, V. Gon¸calves, and J. Penedones, Spinning AdS Propagators, JHEP 09 (2014) 064, [arXiv:1404.5625]. [39] C. Sleight and M. Taronna, Higher Spin Interactions from Conformal Field Theory: The Complete Cubic Couplings, Phys. Rev. Lett. 116 (2016), no. 18 181602, [arXiv:1603.0002]. [40] C. Sleight and M. Taronna, Higher-Spin Algebras, Holography and Flat Space, JHEP 02 (2017) 095, [arXiv:1609.0099]. [41] E. Hijano, P. Kraus, E. Perlmutter, and R. Snively, Witten Diagrams Revisited: The AdS Geometry of Conformal Blocks, JHEP 01 (2016) 146, [arXiv:1508.0050]. [42] D. Poland and D. Simmons-Duffin, Bounds on 4D Conformal and Superconformal Field Theories, JHEP 05 (2011) 017, [arXiv:1009.2087]. [43] D. Poland, D. Simmons-Duffin, and A. Vichi, Carving Out the Space of 4D CFTs, JHEP 05 (2012) 110, [arXiv:1109.5176]. [44] M. S. Costa, J. Penedones, D. Poland, and S. Rychkov, Spinning Conformal Blocks, JHEP 11 (2011) 154, [arXiv:1109.6321]. [45] A. Castedo Echeverri, E. Elkhidir, D. Karateev, and M. Serone, Deconstructing Conformal Blocks in 4D CFT, JHEP 08 (2015) 101, [arXiv:1505.0375]. [46] L. Iliesiu, F. Kos, D. Poland, S. S. Pufu, D. Simmons-Duffin, and R. Yacoby, Bootstrapping 3D Fermions, JHEP 03 (2016) 120, [arXiv:1508.0001]. [47] A. Castedo Echeverri, E. Elkhidir, D. Karateev, and M. Serone, Seed Conformal Blocks in 4D CFT, JHEP 02 (2016) 183, [arXiv:1601.0532]. [48] M. Hogervorst, Dimensional Reduction for Conformal Blocks, JHEP 09 (2016) 017, [arXiv:1604.0891]. [49] M. Hogervorst and S. Rychkov, Radial Coordinates for Conformal Blocks, Phys. Rev. D87 (2013) 106004, [arXiv:1303.1111]. [50] M. Hogervorst, H. Osborn, and S. Rychkov, Diagonal Limit for Conformal Blocks in d Dimensions, JHEP 08 (2013) 014, [arXiv:1305.1321]. [51] M. S. Costa, T. Hansen, J. Penedones, and E. Trevisani, Radial expansion for spinning conformal blocks, JHEP 07 (2016) 057, [arXiv:1603.0555]. [52] S. Ferrara and G. Parisi, Conformal covariant correlation functions, Nucl. Phys. B42 (1972) 281–290. [53] S. Ferrara, A. F. Grillo, and G. Parisi, Nonequivalence between conformal covariant wilson expansion in euclidean and minkowski space, Lett. Nuovo Cim. 5S2 (1972) 147–151. [Lett. Nuovo Cim.5,147(1972)]. [54] D. Simmons-Duffin, Projectors, Shadows, and Conformal Blocks, JHEP 04 (2014) 146,
– 37 –
[arXiv:1204.3894]. [55] M. S. Costa and T. Hansen, Conformal correlators of mixed-symmetry tensors, JHEP 02 (2015) 151, [arXiv:1411.7351]. [56] F. Rejon-Barrera and D. Robbins, Scalar-Vector Bootstrap, JHEP 01 (2016) 139, [arXiv:1508.0267]. [57] T. Y. Thomas, On conformal geometry, Proceedings of the National Academy of Sciences of the United States of America 12 (1926), no. 5 352–359. [58] V. K. Dobrev, V. B. Petkova, S. G. Petrova, and I. T. Todorov, Dynamical Derivation of Vacuum Operator Product Expansion in Euclidean Conformal Quantum Field Theory, Phys. Rev. D13 (1976) 887. [59] V. K. Dobrev, G. Mack, V. B. Petkova, S. G. Petrova, and I. T. Todorov, eds., Harmonic Analysis: on the n-Dimensional Lorentz Group and Its Application to Conformal Quantum Field Theory, vol. 63 of Lecture Notes in Physics, Berlin Springer Verlag, 1977. [60] R. R. Metsaev, Cubic interaction vertices of massive and massless higher spin fields, Nucl. Phys. B759 (2006) 147–201, [hep-th/0512342]. [61] H. Osborn and A. C. Petkou, Implications of conformal invariance in field theories for general dimensions, Annals Phys. 231 (1994) 311–362, [hep-th/9307010]. [62] J. Erdmenger and H. Osborn, Conserved currents and the energy momentum tensor in conformally invariant theories for general dimensions, Nucl. Phys. B483 (1997) 431–474, [hep-th/9605009]. [63] Y. S. Stanev, Correlation Functions of Conserved Currents in Four Dimensional Conformal Field Theory, Nucl. Phys. B865 (2012) 200–215, [arXiv:1206.5639]. [64] A. Zhiboedov, A note on three-point functions of conserved currents, arXiv:1206.6370. [65] J. Maldacena and A. Zhiboedov, Constraining Conformal Field Theories with A Higher Spin Symmetry, J. Phys. A46 (2013) 214011, [arXiv:1112.1016]. [66] N. Boulanger, D. Ponomarev, E. D. Skvortsov, and M. Taronna, On the uniqueness of higher-spin symmetries in AdS and CFT, Int. J. Mod. Phys. A28 (2013) 1350162, [arXiv:1305.5180]. [67] V. Alba and K. Diab, Constraining conformal field theories with a higher spin symmetry in d=4, arXiv:1307.8092. [68] V. Alba and K. Diab, Constraining conformal field theories with a higher spin symmetry in d > 3 dimensions, JHEP 03 (2016) 044, [arXiv:1510.0253]. [69] D. Friedan and C. A. Keller, Cauchy Conformal Fields in Dimensions d > 2, Commun. Math. Phys. 348 (2016), no. 2 655–694, [arXiv:1509.0747]. [70] C. Fronsdal, Elementary particles in a curved space. ii, Phys. Rev. D 10 (Jul, 1974) 589–598. [71] X. Bekaert, J. Erdmenger, D. Ponomarev, and C. Sleight, Towards holographic higher-spin interactions: Four-point functions and higher-spin exchange, JHEP 03 (2015) 170, [arXiv:1412.0016]. [72] D. Francia, J. Mourad, and A. Sagnotti, Current Exchanges and Unconstrained Higher Spins, Nucl. Phys. B773 (2007) 203–237, [hep-th/0701163]. [73] D. Francia, J. Mourad, and A. Sagnotti, (A)dS exchanges and partially-massless higher spins,
– 38 –
Nucl. Phys. B804 (2008) 383–420, [arXiv:0803.3832]. [74] A. Castro, E. Llabr´es, and F. Rejon-Barrera, Geodesic Diagrams, Gravitational Interactions & OPE Structures, arXiv:1702.0612. [75] J. Penedones, High Energy Scattering in the AdS/CFT Correspondence. PhD thesis, Porto U., 2007. arXiv:0712.0802. [76] M. Taronna, Higher-Spin Interactions: three-point functions and beyond. PhD thesis, Pisa, Scuola Normale Superiore, 2012. arXiv:1209.5755. [77] D. Z. Freedman, S. D. Mathur, A. Matusis, and L. Rastelli, Correlation functions in the CFTd / AdSd+1 correspondence, Nucl. Phys. B546 (1999) 96–118, [hep-th/9804058]. [78] A. Mikhailov, Notes on higher spin symmetries, hep-th/0201019. [79] E. Dyer, D. Z. Freedman, and J. Sully, Spinning Geodesic Witten Diagrams, arXiv:1702.0613. [80] A. C. Petkou, Evaluating the AdS dual of the critical O(N) vector model, JHEP 03 (2003) 049, [hep-th/0302063]. [81] E. D. Skvortsov, On (Un)Broken Higher-Spin Symmetry in Vector Models, in Proceedings, International Workshop on Higher Spin Gauge Theories: Singapore, Singapore, November 4-6, 2015, pp. 103–137, 2017. arXiv:1512.0599. [82] D. J. Gross and V. Rosenhaus, The Bulk Dual of SYK: Cubic Couplings, arXiv:1702.0801. [83] B. Allen, The Graviton Propagator in Homogeneous and Isotropic Space-times, Nucl. Phys. B287 (1987) 743–756. [84] B. Allen and T. Jacobson, Vector Two Point Functions in Maximally Symmetric Spaces, Commun. Math. Phys. 103 (1986) 669. [85] M. Turyn, The Graviton Propagator in Maximally Symmetric Spaces, J. Math. Phys. 31 (1990) 669. [86] H. Liu and A. A. Tseytlin, On four point functions in the CFT / AdS correspondence, Phys. Rev. D59 (1999) 086002, [hep-th/9807097]. [87] E. D’Hoker and D. Z. Freedman, Gauge boson exchange in AdSd+1 , Nucl. Phys. B544 (1999) 612–632, [hep-th/9809179]. [88] E. D’Hoker, D. Z. Freedman, S. D. Mathur, A. Matusis, and L. Rastelli, Graviton and gauge boson propagators in AdSd+1 , Nucl. Phys. B562 (1999) 330–352, [hep-th/9902042]. [89] C. Fronsdal, Singletons and Massless, Integral Spin Fields on de Sitter Space (Elementary Particles in a Curved Space. 7., Phys. Rev. D20 (1979) 848–856. [90] T. Leonhardt, R. Manvelyan, and W. Ruhl, The Group approach to AdS space propagators, Nucl. Phys. B667 (2003) 413–434, [hep-th/0305235]. [91] T. Leonhardt, W. Ruhl, and R. Manvelyan, The Group approach to AdS space propagators: A Fast algorithm, J. Phys. A37 (2004) 7051, [hep-th/0310063]. [92] R. Manvelyan and W. Ruhl, The Off-shell behaviour of propagators and the Goldstone field in higher spin gauge theory on AdSd+1 space, Nucl. Phys. B717 (2005) 3–18, [hep-th/0502123]. [93] R. Manvelyan, K. Mkrtchyan, and W. Ruhl, Ultraviolet behaviour of higher spin gauge field propagators and one loop mass renormalization, Nucl. Phys. B803 (2008) 405–427, [arXiv:0804.1211].
– 39 –
[94] K. Mkrtchyan, Higher Spin Interacting Quantum Field Theory and Higher Order Conformal Invariant Lagrangians. PhD thesis, Yerevan Phys. Inst., 2010. arXiv:1011.0160. [95] L. P. S. Singh and C. R. Hagen, Lagrangian formulation for arbitrary spin. 1. The boson case, Phys. Rev. D9 (1974) 898–909. [96] L. P. S. Singh and C. R. Hagen, Lagrangian formulation for arbitrary spin. 2. The fermion case, Phys. Rev. D9 (1974) 910–920. [97] S. D. Rindani and M. Sivakumar, Gauge - Invariant Description of Massive Higher - Spin Particles by Dimensional Reduction, Phys. Rev. D32 (1985) 3238. [98] S. D. Rindani, D. Sahdev, and M. Sivakumar, Dimensional reduction of symmetric higher spin actions. 1. Bosons, Mod. Phys. Lett. A4 (1989) 265–273. [99] C. Aragone, S. Deser, and Z. Yang, Massive Higher Spin From Dimensional Reduction of Gauge Fields, Annals Phys. 179 (1987) 76. [100] Yu. M. Zinoviev, On massive high spin particles in AdS, hep-th/0108192. [101] P. A. M. Dirac, Relativistic wave equations, Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences 155 (1936), no. 886 447–459. [102] M. Fierz, Force-free particles with any spin, Helv. Phys. Acta 12 (1939) 3–37. [103] M. Fierz and W. Pauli, On relativistic wave equations for particles of arbitrary spin in an electromagnetic field, Proc. Roy. Soc. Lond. A173 (1939) 211–232. [104] D. Francia, Geometric Lagrangians for massive higher-spin fields, Nucl. Phys. B796 (2008) 77–122, [arXiv:0710.5378]. [105] D. Francia, Geometric massive higher spins and current exchanges, Fortsch. Phys. 56 (2008) 800–808, [arXiv:0804.2857]. [106] C. Fronsdal, Massless Fields with Integer Spin, Phys. Rev. D18 (1978) 3624. [107] D. Francia and A. Sagnotti, Free geometric equations for higher spins, Phys. Lett. B543 (2002) 303–310, [hep-th/0207002]. [108] D. Francia and A. Sagnotti, Minimal local Lagrangians for higher-spin geometry, Phys. Lett. B624 (2005) 93–104, [hep-th/0507144]. [109] D. Francia, On the Relation between Local and Geometric Lagrangians for Higher spins, J. Phys. Conf. Ser. 222 (2010) 012002, [arXiv:1001.3854]. [110] H. Liu, Scattering in anti-de Sitter space and operator product expansion, Phys. Rev. D60 (1999) 106005, [hep-th/9811152]. [111] D. Z. Freedman, S. D. Mathur, A. Matusis, and L. Rastelli, Comments on 4 point functions in the CFT / AdS correspondence, Phys. Lett. B452 (1999) 61–68, [hep-th/9808006]. [112] E. D’Hoker and D. Z. Freedman, General scalar exchange in AdSd+1 , Nucl. Phys. B550 (1999) 261–288, [hep-th/9811257]. [113] E. D’Hoker, D. Z. Freedman, S. D. Mathur, A. Matusis, and L. Rastelli, Graviton exchange and complete four point functions in the AdS / CFT correspondence, Nucl. Phys. B562 (1999) 353–394, [hep-th/9903196]. [114] L. Hoffmann, A. C. Petkou, and W. Ruhl, A Note on the analyticity of AdS scalar exchange graphs in the crossed channel, Phys. Lett. B478 (2000) 320–326, [hep-th/0002025].
– 40 –
[115] L. Hoffmann, A. C. Petkou, and W. Ruhl, Aspects of the conformal operator product expansion in AdS / CFT correspondence, Adv. Theor. Math. Phys. 4 (2002) 571–615, [hep-th/0002154]. [116] L. I. Uruchurtu, AdS/CFT for Four-Point Amplitudes involving Gravitino Exchange, JHEP 09 (2007) 086, [arXiv:0707.0424]. [117] P. Haggi-Mani and B. Sundborg, Free large N supersymmetric Yang-Mills theory as a string theory, JHEP 04 (2000) 031, [hep-th/0002189]. [118] B. Sundborg, Stringy gravity, interacting tensionless strings and massless higher spins, Nucl. Phys. Proc. Suppl. 102 (2001) 113–119, [hep-th/0103247]. [119] E. Witten, Spacetime Reconstruction, talk at J.H. Schwarz 60-th Birthday Conference (CalTech) (November 2001) [http://theory.caltech.edu/jhs60/witten/1.html]. [120] E. Sezgin and P. Sundell, Massless higher spins and holography, Nucl. Phys. B644 (2002) 303–370, [hep-th/0205131]. [Erratum: Nucl. Phys.B660,403(2003)]. [121] I. R. Klebanov and A. M. Polyakov, AdS dual of the critical O(N) vector model, Phys. Lett. B550 (2002) 213–219, [hep-th/0210114]. [122] R. G. Leigh and A. C. Petkou, Holography of the N = 1 higher spin theory on AdS4 , JHEP 06 (2003) 011, [hep-th/0304217]. [123] M. A. Vasiliev, Nonlinear equations for symmetric massless higher spin fields in (A)dS(d), Phys. Lett. B567 (2003) 139–151, [hep-th/0304049]. [124] S. Giombi and X. Yin, Higher Spin Gauge Theory and Holography: The Three-Point Functions, JHEP 09 (2010) 115, [arXiv:0912.3462]. [125] S. Giombi and X. Yin, Higher Spins in AdS and Twistorial Holography, JHEP 04 (2011) 086, [arXiv:1004.3736]. [126] C.-M. Chang and X. Yin, Higher Spin Gravity with Matter in AdS3 and Its CFT Dual, JHEP 10 (2012) 024, [arXiv:1106.2580]. [127] M. Ammon, P. Kraus, and E. Perlmutter, Scalar fields and three-point functions in D=3 higher spin gravity, JHEP 07 (2012) 113, [arXiv:1111.3926]. [128] E. Hijano, P. Kraus, and E. Perlmutter, Matching four-point functions in higher spin AdS3 /CFT2 , JHEP 05 (2013) 163, [arXiv:1302.6113]. [129] E. Joung and M. Taronna, Cubic interactions of massless higher spins in (A)dS: metric-like approach, Nucl. Phys. B861 (2012) 145–174, [arXiv:1110.5918]. [130] E. Joung, L. Lopez, and M. Taronna, Solving the Noether procedure for cubic interactions of higher spins in (A)dS, J. Phys. A46 (2013) 214020, [arXiv:1207.5520]. [131] E. Joung, L. Lopez, and M. Taronna, Generating functions of (partially-)massless higher-spin cubic interactions, JHEP 01 (2013) 168, [arXiv:1211.5912]. [132] E. Joung and M. Taronna, Cubic-interaction-induced deformations of higher-spin symmetries, JHEP 03 (2014) 103, [arXiv:1311.0242]. [133] T. Hartman, Entanglement Entropy at Large Central Charge, arXiv:1303.6955. [134] A. L. Fitzpatrick, J. Kaplan, and M. T. Walters, Universality of Long-Distance AdS Physics from the CFT Bootstrap, JHEP 08 (2014) 145, [arXiv:1403.6829]. [135] C. T. Asplund, A. Bernamonti, F. Galli, and T. Hartman, Holographic Entanglement Entropy from 2d CFT: Heavy States and Local Quenches, JHEP 02 (2015) 171, [arXiv:1410.1392].
– 41 –
[136] J. de Boer, A. Castro, E. Hijano, J. I. Jottar, and P. Kraus, Higher spin entanglement and WN conformal blocks, JHEP 07 (2015) 168, [arXiv:1412.7520]. [137] E. Hijano, P. Kraus, and R. Snively, Worldline approach to semi-classical conformal blocks, JHEP 07 (2015) 131, [arXiv:1501.0226]. [138] A. L. Fitzpatrick, J. Kaplan, and M. T. Walters, Virasoro Conformal Blocks and Thermality from Classical Background Fields, JHEP 11 (2015) 200, [arXiv:1501.0531]. [139] K. B. Alkalaev and V. A. Belavin, Classical conformal blocks via AdS/CFT correspondence, JHEP 08 (2015) 049, [arXiv:1504.0594]. [140] E. Hijano, P. Kraus, E. Perlmutter, and R. Snively, Semiclassical Virasoro blocks from AdS3 gravity, JHEP 12 (2015) 077, [arXiv:1508.0498]. [141] A. L. Fitzpatrick, J. Kaplan, M. T. Walters, and J. Wang, Hawking from Catalan, JHEP 05 (2016) 069, [arXiv:1510.0001]. [142] K. B. Alkalaev and V. A. Belavin, Monodromic vs geodesic computation of Virasoro classical conformal blocks, Nucl. Phys. B904 (2016) 367–385, [arXiv:1510.0668]. [143] A. L. Fitzpatrick and J. Kaplan, Conformal Blocks Beyond the Semi-Classical Limit, JHEP 05 (2016) 075, [arXiv:1512.0305]. [144] P. Banerjee, S. Datta, and R. Sinha, Higher-point conformal blocks and entanglement entropy in heavy states, JHEP 05 (2016) 127, [arXiv:1601.0679]. [145] M. Besken, A. Hegde, E. Hijano, and P. Kraus, Holographic conformal blocks from interacting Wilson lines, JHEP 08 (2016) 099, [arXiv:1603.0731]. [146] B. Carneiro da Cunha and M. Guica, Exploring the BTZ bulk with boundary conformal blocks, arXiv:1604.0738. [147] K. B. Alkalaev, Many-point classical conformal blocks and geodesic networks on the hyperbolic plane, JHEP 12 (2016) 070, [arXiv:1610.0671]. [148] A. L. Fitzpatrick, J. Kaplan, D. Li, and J. Wang, Exact Virasoro Blocks from Wilson Lines and Background-Independent Operators, arXiv:1612.0638. [149] M. Nishida and K. Tamaoka, Geodesic Witten diagrams with an external spinning field, arXiv:1609.0456. [150] M. Grigoriev and A. Waldron, Massive Higher Spins from BRST and Tractors, Nucl. Phys. B853 (2011) 291–326, [arXiv:1104.4994]. [151] C. Sleight, Lectures on Metric-like Methods in Higher Spin Holography, in 12th Modave Summer School in Mathematical Physics Modave, Belgium, September 11-17, 2016, 2017. arXiv:1701.0836. [152] J. Penedones, Writing CFT correlation functions as AdS scattering amplitudes, JHEP 03 (2011) 025, [arXiv:1011.1485]. [153] M. F. Paulos, Towards Feynman rules for Mellin amplitudes, JHEP 10 (2011) 074, [arXiv:1107.1504].
– 42 –
