We study a duality, recently conjectured by Klebanov and Polyakov, between
higher-spin theories on AdS_4 and O(N) vector models in 3-d. These theories are
free in the UV and interacting in the IR. At the UV fixed point, the O(N) model
has an infinite number of higher-spin conserved currents. In the IR, these
currents are no longer conserved for spin s>2. In this paper, we show that the
dual interpretation of this fact is that all fields of spin s>2 in AdS_4 become
massive by a Higgs mechanism, that leaves the spin-2 field massless. We
identify the Higgs field and show how it relates to the RG flow connecting the
two CFTs, which is induced by a double trace deformation.Comment: 8 pages, latex; v2 references adde

A. Zaffaroni

Aharony

Anselmi

Berkooz

Engquist

Fradkin

Francia

Gubser

Haggi-Mani

Heidenreich

Klebanov

L. Girardello

Lang

Leonhardt

M. Porrati

Mikhailov

Petkou

Porrati

Sezgin

Sundborg

Vasiliev

Wilson

Witten

Physics Letters B

English

arXiv

AbstractWe study a duality, recently conjectured by Klebanov and Polyakov, between higher-spin theories on AdS4 and O(N) vector models in 3D. These theories are free in the UV and interacting in the IR. At the UV fixed point, the O(N) model has an infinite number of higher-spin conserved currents. In the IR, these currents are no longer conserved for spin s>2. In this Letter, we show that the dual interpretation of this fact is that all fields of spin s>2 in AdS4 become massive by a Higgs mechanism, that leaves the spin-2 field massless. We identify the Higgs field and show how it relates to the RG flow connecting the two CFTs, which is induced by a double trace deformation

Girardello, L.

Porrati, M.

Zaffaroni, A.

Elsevier - Publisher Connector 

3D interacting CFTs and generalized Higgs phenomenon in higher spin theories on AdS 

Physics Letters B 561 (2003) 289–293
www.elsevier.com/locate/npe
3D interacting CFTs and generalized Higgs phenomenon in higher
spin theories on AdS
L. Girardello a, M. Porrati b,c, A. Zaffaroni a
a Università di Milano-Bicocca and INFN, Piazza della Scienza 3, I-20126 Milano, Italy
b Department of Physics, New York University, 4 Washington Pl., New York, NY 10012, USA
c Scuola Normale Superiore, Piazza dei Cavalieri 7, I-56126 Pisa, Italy
Received 24 February 2003; accepted 28 March 2003
Editor: L. Alvarez-Gaumé
Abstract
We study a duality, recently conjectured by Klebanov and Polyakov, between higher-spin theories on AdS4 and O(N) vector
models in 3D. These theories are free in the UV and interacting in the IR. At the UV fixed point, the O(N) model has an infinite
number of higher-spin conserved currents. In the IR, these currents are no longer conserved for spin s > 2. In this Letter, we
show that the dual interpretation of this fact is that all fields of spin s > 2 in AdS4 become massive by a Higgs mechanism, that
leaves the spin-2 field massless. We identify the Higgs field and show how it relates to the RG flow connecting the two CFTs,
which is induced by a double trace deformation.
 2003 Published by Elsevier Science B.V. Open access under CC BY license.1. Introduction
Theories with an infinite number of massless higher-
spin gauge fields (HS) have a long story. Recently,
they have been reexamined by several authors [1].
One of the reasons for this resurgence of interest is
that these theories are candidates for a semi-classical
treatment of the small tension limit of string the-
ory. The important observation that higher-spin the-
ories can be consistently formulated in anti-de Sitter
space [2] also suggests that they are useful in the con-
E-mail addresses: girardello@mib.infn.it (L. Girardello),
massimo.porrati@nyu.edu (M. Porrati),
alberto.zaffaroni@mib.infn.it (A. Zaffaroni).0370-2693 2003 Published by Elsevier Science B.V.
doi:10.1016/S0370-2693(03)00492-1
Open access under CCtext of the AdS/CFT correspondence. Weakly coupled
gauged theories contain an infinite number of almost-
conserved currents that may be described by a dual
HS theory in anti-de Sitter space. While our under-
standing of the description of the weak coupling limit
of four-dimensional YM in terms of higher-spin theo-
ries is still elusive, some progress has been made for
certain three-dimensional conformal field theories [3].
The specific example of Ref. [3] deals with three-
dimensional O(N) vector models. The singlet sector
of the O(N) theories contains an infinite number of
conserved currents of even spin in the large N limit.
In [3], it was conjectured that the singlet sector of this
theory is dual to that of the higher spin theories in
AdS4 studied by Vasiliev [4], which consists of a single
 BY license.
290 L. Girardello et al. / Physics Letters B 561 (2003) 289–293Regge trajectory of even spin.1 Two different three-
dimensional conformal field theories were considered
in [3], the free O(N) model and the infrared fixed
point that can be obtained by perturbing the free theory
with a relevant double trace operator. Following the
general description of RG flows induced by double-
trace operators [6–8], the authors of [3] also conjec-
tured that both theories are described by the same
Vasiliev Lagrangian, but with a difference consisting
in the choice of boundary condition for a certain field.
As shown in [7], and further studied in [9,10], the dual
description of the RG flow induced by double-trace
deformation of the boundary CFT is unusual. Instead
of changing the 4D background, this flow leaves the
geometry unchanged, at least at tree level in the bulk
theory, but it changes the boundary behavior of a cer-
tain bulk field.
This raises the question that we want to address in
this Letter. When the bulk Lagrangian is dual to the
O(N) model at the IR fixed point, the higher spin cur-
rents are conserved only in the large N limit. Because
of the standard relation between conformal dimension
in 3D, and mass in AdS4, an infinite number of higher
spins should become massive when 1/N corrections
are included. All spins should instead remain massless
in the description of the free UV theory. This raises
an interesting puzzle. As we said, the two CFTs are
described by the same bulk Lagrangian and they only
differ by a choice of boundary condition of a certain
field. How can such a change of boundary conditions
in a (scalar) field induce masses for all particles of spin
higher than 2? The answer to this question turns out to
be surprisingly similar to a case recently studied by
one of the authors [11]; namely, a graviton coupled to
conformal matter in AdS4. There, one can show that,
when matter is given non-standard boundary condi-
tions, it can form a bound state that acts as the Gold-
stone vector for the spin 2 field. In other words, in that
case the graviton gets a mass through a one-loop ef-
fect. In this Letter, we show that a similar mechanism
can give mass to all higher-spin fields in the dual of
the O(N) model at the IR fixed point. The mecha-
nism is intrinsically one-loop in the bulk theory. That
explains naturally why the masses of the higher-spin
fields are O(1/N). Differently from the case studied
1 Further work on the subject can be found in [5].in Ref. [11], here the boundary conditions of the bulk
fields leave the spin-2 field massless, at the fixed points
of the double-trace RG flow. We finally show that mass
generation can only occur, for spin s > 2, when the
AdS4 theory is dual to the O(N) model at the IR fixed
point.
2. The AdS/HS correspondence
To be self-contained, in this section we briefly
review the details of the correspondence conjectured
in [3]. The O(N) model is formulated in terms of a
three-dimensional scalar transforming in the vectorial
representation of O(N), with Lagrangian:
(1)L=
∫
d3x
[
∂φa∂φa + λ
2N
(
φaφa
)2]
,
where a = 1, . . . ,N . The theory has two fixed points.
There is an ultraviolet, free fixed point at λ = 0, and
an interacting infrared fixed point [12]. The free UV
theory has an infinite number of conserved currents.
Restricting to operators that are singlets of O(N) and
single trace,2 we find a conserved current for each
even spin. We can schematically write it as
(2)Jµ1,...,µs = φa(
←→
∂ )sφa − traces.
The IR theory is instead interacting and the currents in
Eq. (2) are not conserved. We can reasonably assume
that the only conserved current in the IR CFT is the
stress-energy tensor. Among the non-singlet operator
there are also other conserved currents; the Noether
currents of the (global) O(N) symmetry. However, it
is known [12] that the currents in Eq. (2), for s  2,
also have canonical dimension in the large N limit.3
Therefore, forN =∞, the IR theory too has an infinite
number of conserved currents. The currents do acquire
anomalous dimensions at order 1/N . The conjecture
formulated in [3] states that the singlet sector of both
theories has, in the large N limit, a dual description
in terms of a minimal bosonic HS theory containing
one massless gauge field for each even spin [4]. In this
correspondence, N must be identified with the inverse
cosmological constant of the HS theory.
2 We follow the misuse of the term single trace introduced in [3].
3 For more recent references on the subject see [13].
L. Girardello et al. / Physics Letters B 561 (2003) 289–293 291The description of the two fixed points differs only
by a choice of boundary conditions. Let Σ be the
bulk scalar dual to (φaφa). φaφa has dimension 1 in
the UV and dimension 2 + O(1/N) in the IR [12].
Σ is a scalar field with mass m2 = −2 at large N
(in units of the cosmological constant).Σ is, therefore,
a conformally coupled scalar field. The UV and IR
conformal dimensions of the operator correspond,
respectively, to the two roots of the equation
(3)m2 =∆(∆− 3).
The quantization of Σ is subtle, because both roots
∆± of the previous equation satisfies the unitary
bound in three dimensions. The analysis of such
cases has been performed in [14]. The bulk theory
corresponding to the assignment ∆=∆+ differs from
the other, ∆= ∆−, by boundary conditions. Namely,
if Σ has asymptotic behavior
(4)Σ ∼ αz∆− + βz∆+, ∆+ >∆−,
where z is the AdS radial coordinate, the two possible
quantizations are obtained by interchanging the role
of α and β [14]. In the UV, (φaφa) has dimension
∆− = 1 and it is quantized with boundary condition
α = 0. In the IR (φaφa) has dimension ∆+ = 2 in
the large N limit, and it is quantized with boundary
condition β = 0.
In [3] the interesting observation was made that the
two CFTs are connected by a RG flow induced by the
double-trace operator (φaφa)2, which, in the UV, is a
dimension-two relevant operator. An analysis of flows
induced by double-trace operators was carried out
in [6–8]. The deformation by a double-trace operator
only modifies the boundary condition for the bulk field
Σ dual to (φaφa). According to [7], the boundary
condition on Σ to be imposed along the flow is:
(5)α = λβ.
We see that in the two limiting cases, λ = 0 and λ =
∞, we recover the two different boundary conditions
describing, respectively, the UV and the IR fixed
points.
3. A generalized Higgs effect
It is intriguing to notice that the same Lagrangian
gives a semi-classical description of two differentfixed points, one of which is free while the other is
interacting. In particular, an obvious question can be
raised. Since the currents in Eq. (2) are not conserved
in the IR at finite N , we expect that the corresponding
higher spin fields in the bulk acquire a mass of order
1/N , when quantum (loop) corrections are included.
We want now to prove that this is indeed the case.
Namely, that in the bulk Lagrangian describing the IR
fixed point, a generalized Higgs effect may take place,
which gives mass to all fields of spin greater than
two. We will also show that no Higgs effect is expect
in the Lagrangian describing the UV fixed point, and
that boundary conditions alone are responsible for the
different behaviors of the two theories.
In AdS, a spin s field can acquire a mass by “eat-
ing” a single massive field of spin s − 1, by a Higgs-
like mechanism. To describe this phenomenon prop-
erly, recall that the representations of the 3D confor-
mal group, SO(3,2), which is also the isometry group
of AdS4, are labeled by their quantum numbers un-
der the maximal compact subgroup SO(3) × U(1):
the spin s and the conformal weight ∆. A represen-
tation D(∆, s) satisfies a shortening condition when
∆= s + 1, and corresponds to a conserved current in
the CFT and a massless field in AdS4. A massive spin s
representation of the conformal group decomposes in
the massless limit as [15]
(6)D(∆, s)∆→s+1−→ D(s + 1, s)⊕D(s + 2, s − 1).
The representation D(s + 2, s − 1) is the Goldstone
field. Since in AdS4 the energy spectrum is discrete,
two gauge fields can form a bound state with the
quantum number of the Goldstone field (even when
they are free!). In the presence of an appropriate
trilinear coupling, a spin s field can then acquire
mass through radiative corrections. Let us stress that
this phenomenon cannot occur in flat space where
the spectrum is continuous. This analysis was already
performed in [11] in the case of a graviton in AdS4,
coupled to a conformal scalar.
Let us denote with Ws ≡ Wµ1,...,µs the spin s
gauge field. In a Lagrangian as those proposed by
Vasiliev [4], we expect many trilinear couplings be-
tween gauge fields of different spin. Some involve the
field Σ and some do not. Those without Σ can be
schematically written asWs∂k(Ws1Ws2)with s1+s2+
k = s, with derivatives arbitrarily distributed among
the gauge fields. They cannot be responsible for the
292 L. Girardello et al. / Physics Letters B 561 (2003) 289–293Higgs mechanism. Even if the product of representa-
tions of spin s1 and s2 contains a mode in the repre-
sentation D(s + 2, s− 1), the latter would have wrong
parity for being a Goldstone field; let us see why. Since
we are interested in a one loop effect, we can neglect
1/N corrections to the dimensions of our fields. All
the Ws are thus massless in the large N limit and,
therefore, have dimension s + 1. The ground state of
the would-be Goldstone representation has conformal
dimension s+ 2. It is obtained from the lowest weight
state of D(si + 1, si ), which has dimension si + 1,
by acting on it with k raising operators of the group
SO(3,2). Since the parity of a genuine spin si field is
(−1)si and the parity of a raising operator is −1, this
mode has parity P = (−1)s1+s2+k = (−1)s , which is
the wrong one for a spin (s − 1) gauge field.
The field Σ is what describes the RG flow, and
it is, moreover, the only one to change under it, to
leading order in 1/N . So, we expect it to appear in the
couplings needed to give mass to our high-spin fields.
Recall that Σ only has different boundary conditions
at the two fixed points of the RG flow. We can easily
write a trilinear coupling of the form
(7)Wµ1,...,µsWµ1,...,µs−2∂µs−1∂µsΣ,
where, for simplicity, we chose a specific distributions
for the derivatives. Such coupling can be certainly
reconstructed from the three point function of free
fields in the CFT. It is also reminiscent of the equation
for the conservation of the currents in Eq. (2). To
see this, write the Lagrangian Eq. (1) in terms of an
auxiliary field σ ,
(8)L=
∫
d3x
[
∂φa∂φa + σ (φaφa)− Nσ 2
2λ
]
,
and use the equations of motions σ = λ(φaφa)/N ,✷φa = σφa . Then, the divergence of the current can
be rewritten, schematically, as
(9)∂µJµ,µ1,...,µs−1 ∼ Jµ1,...,µs−2∂µs−1σ |ST,
where the subscript means that the right-hand side is
projected on the symmetric-traceless part.
The coupling in Eq. (7) can give mass to the spin s
fields, by a one-loop diagram, only when the product
of the representations to which Wµ1,...,µs−2 and Σ
belong contains the Goldstone representation D(s +
2, s − 1). To leading order in 1/N , Σ has dimension∆= 1 in the UV, but dimension ∆= 2 in the IR, while
all the Ws have always dimension s + 1. We also have
[15]
D(s − 1, s − 2)⊕D(∆,0)
(10)=
∞∑
S=0
∞∑
n=0
D(∆+ S + s + n− 1, s + S − 2).
This equation shows that a mode D(s + 2, s − 1),
with the right quantum numbers to be the Goldstone,
appears for both values of ∆. However, it is easy
to check that the candidate Goldstone has the same
parity of the would-be massive field Ws only when
∆= 2.4 We conclude that, only when Σ is quantized
with conformal weight 2 in the large N limit, a Higgs
mechanism is possible.
We must also check that the graviton remains mass-
less: in a CFT, a singlet conserved current correspond-
ing to the stress-energy tensor always exists. It was
already noticed in [11] that the graviton coupled to a
conformal scalar can acquire mass only if the bound-
ary conditions on the scalar make it belong to the re-
ducible representationD(1,0) ⊕ D(2,0). In our case,
the scalar belongs to the D(1,0) in the UV, and to the
D(2,0) in the IR, so that no Higgs mechanism is ex-
pected. We can see this explicitly from the decompo-
sition
D(∆′,0)⊕D(∆,0)
(11)=
∞∑
S=0
∞∑
n=0
D(∆+∆′ + S + 2n,S),
that replaces Eq. (10) in the case s = 2. No Goldstone
representation D(4,1) is contained in this formula for
∆=∆′, and ∆,∆′ equal to either 1 or 2.
4. Conclusions
The duality between O(N) critical vector models
and HS theories á la Vasiliev is still in its infancy. A
challenge in establishing it firmly is that while the 3D
4 In the case ∆= 1, we create the lowest weights of D(s+2, s−
1) by applying two raising operators, L+i , to the product of the
lowest weight of D(s− 1, s − 2) and D(1,0), thus obtaining a field
of parity P = (−1)s . D(s + 2, s − 1) is thus a pseudo-spin (s − 1)
field.
L. Girardello et al. / Physics Letters B 561 (2003) 289–293 293CFT is considerably simpler than in the adjoint case,
the 4D AdS dual is much more complicated than semi-
classical supergravity. In this Letter we furthered the
study of that duality by showing how to explain a puz-
zling feature of the IR (interacting) fixed point of the
O(N) model. There, almost all higher-spin currents
that were conserved in the UV acquire anomalous di-
mensions. In the AdS dual, this means that almost all
massless fields of the HS theory become massive. To
interpret this effect as a Higgs phenomenon, one has
to explain how to reconcile it with the fact that the
(double trace) perturbation of the UV theory flowing
into the IR fixed point does not change the AdS back-
ground, to leading order in 1/N . In this Letter, we
showed that a radiative Higgs effect, where the Gold-
stone particle is composite, can solve this puzzle. We
performed a group theoretical analysis showing that
only particles with spin s > 2 can become massive,
and only at the IR fixed point. It would be interest-
ing and important to explicitly compute the one-loop
self-energy diagram for all particles in the dual HS the-
ory [4], to check this phenomenon explicitly and quan-
titatively. It may also be possible to extend our analysis
to a model that contains some (or all) the non-singlet
currents of O(N), or to other examples of RG flows in
3D, like those discussed in [16].
Acknowledgements
We thank D. Anselmi for fruitful conversations.
A.Z. thank the Scuola Normale Superiore in Pisa for
its kind hospitality while this work was done. L.G.
and A.Z. are partially supported by INFN and MURST
under contract 2001-025492, and by the European
Commission TMR program HPRN-CT-2000-00131.
M.P. is supported in part by NSF grant PHY-0070787.References
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3D interacting CFTs and generalized Higgs phenomenon in higher spin theories on AdS

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3-D Interacting CFTs and Generalized Higgs Phenomenon in Higher Spin
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3-D Interacting CFTs and Generalized Higgs Phenomenon in Higher Spin Theories on AdS

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