JHEP05(2007)075
Published by Institute of Physics Publishing for SISSA
Received: April 10, 2007
Accepted: May 17, 2007
Published: May 22, 2007
Multi-trace deformations in AdS/CFT: exploring the
vacuum structure of the deformed CFT
Ioannis Papadimitriou
DESY Theory Group,
Notkestrasse 85, D-22603 Hamburg, Germany, and
Center for Mathematical Physics,
Bundesstrasse 55, D-20146 Hamburg, Germany
E-mail: ioannis.papadimitriou@desy.de
Abstract: We present a general and systematic treatment of multi-trace deformations in
the AdS/CFT correspondence in the large N limit, pointing out and clarifying subtleties
relating to the formulation of the boundary value problem on a conformal boundary. We
then apply this method to study multi-trace deformations in the presence of a scalar VEV,
which requires the coupling to gravity to be taken into account. We show that supergravity
solutions subject to ‘mixed’ boundary conditions are in one-to-one correspondence with
critical points of the holographic effective action of the dual theory in the presence of a
multi-trace deformation, and we find a number of new exact analytic solutions involving
a minimally or conformally coupled scalar field satisfying ‘mixed’ boundary conditions.
These include the generalization to any dimension of the instanton solution recently found
in hep-th/0611315. Finally, we provide a systematic method for computing the holographic
effective action in the presence of a multi-trace deformation in a derivative expansion away
from the conformal vacuum using Hamilton-Jacobi theory. Requiring that this effective
action exists and is bounded from below reproduces recent results on the stability of the
AdS vacuum in the presence of ‘mixed’ boundary conditions.
Keywords: Gauge-gravity correspondence, AdS-CFT Correspondence.
c
° SISSA 2007 http://jhep.sissa.it/archive/papers/jhep052007075/jhep052007075.pdf
JHEP05(2007)075
Contents
1. Introduction and summary of results 1
2. Multi-trace deformations in QFTs with a large N limit 3
3. The boundary value problem for the Klein-Gordon operator in AlAdS
spaces 5
3.1 The variational problem in the presence of a conformal boundary 7
3.2 Boundary conditions 10
3.3 Solution of the boundary value problem 11
3.4 The on-shell action and the AdS/CFT dictionary 12
4. Toy model 14
4.1 General solution with linear boundary conditions 14
4.2 Vacua with non-linear boundary conditions 16
5. Effective action from Hamilton’s characteristic function 18
6. Minimal coupling 22
6.1 Two-derivative effective action for conformal boundary conditions 23
6.2 Minisuperspace approximation 28
6.3 The ‘2/3’ potential 30
7. Conformal coupling 33
7.1 Two-derivative effective action for conformal boundary conditions 35
7.2 Minisuperspace approximation 36
7.3 Instantons 41
A. The variational problem and Hamilton-Jacobi equations 45
1. Introduction and summary of results
Multi-trace deformations have been studied extensively in the context of the AdS/CFT
correspondence in the large N limit, both classically [1 – 7] and at the one-loop level [8 –
10]. Most of this work, however, has focused on the effect of multi-trace deformations on the
conformal vacuum, in which case the back-reaction to the geometry can be ignored. If the
deforming operator though is allowed to acquire a non-zero VEV, then the back-reaction
can no longer be ignored and the coupling to gravity must be taken into account. Only
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JHEP05(2007)075
recently have multi-trace deformations in the presence of a scalar VEV been considered,
mainly in the context of Designer Gravity [11 – 21].
In order for a CFT to admit multi-trace deformations it must contain operators with
low enough dimension. For double- or higher-trace deformations built out of a single opera-
tor, for example, not to be irrelevant, the operator must have conformal dimension ∆ ≤ d/2
in d dimensions. For scalar operators, for example, this means that the operator must have
the ‘non-standard’ ∆
−
dimension. This constraint, together with unitarity, which imposes
a lower bound on the dimension ∆, severely restricts the CFTs admitting multi-trace de-
formations. The possibilities are further narrowed if one insists that the undeformed CFT
be supersymmetric. Since the AdS/CFT dictionary relates multi-trace deformations in the
large N limit to a choice of boundary conditions for the dual bulk supergravity fields [1],
these restrictions on the conformal dimension of the operator translate into a condition
on the mass of the dual supergravity fields for them to admit the necessary generalized
boundary conditions. We are then interested in gauged supergravities that admit AdS
vacua and have fields with mass close to the Breitenlohner-Freedman bound [22].
Both the maximal gauged supergravities in four and five dimensions contain scalars
with the right mass, and indeed black hole solutions with scalar hair that satisfy gener-
alized boundary conditions were found numerically in [12], following earlier work in three
dimensions [11]. Smooth instantons and gravitational soliton solutions of N = 8 D = 4
gauged supergravity with generalized boundary conditions were also found numerically
in [13], and shown to be related to a Big Crunch geometry. More recently, exact solutions
of N = 8 D = 4 gauged supergravity obeying generalized boundary conditions were found
analytically in [23] and [24] and uplifted to eleven dimensions. The AdS/CFT identifies
these solutions with ‘vacua’ or ‘states’ in the dual deformed CFT. In particular, the ex-
trema of the large N quantum effective action for the VEV of the deforming operator are in
one-to-one correspondence with bulk solutions satisfying the relevant boundary conditions.
These bulk solutions then provide a window into the vacuum structure of the deformed
theory.
A very interesting question, in particular, is whether the conformal vacuum - which
generically remains a vacuum of the deformed theory - is stable or not under certain
boundary conditions. The instantons found in [12] and [24] show that it is not, under
the particular AdS-invariant boundary conditions that these instantons satisfy, since these
mediate the tunneling of the conformal vacuum to an instability region. This, of course,
does not contradict any of the well known stability theorems [25 – 27], because these apply
only to certain special boundary conditions. The question of stability with more general
boundary conditions corresponding to multi-trace deformations has been addressed recently
in the context of Designer Gravity [18 – 21]. The approach followed is a generalization of the
spinorial argument of [28], but as in the earlier work [26, 27] no supersymmetry is required.
The argument only relies on the existence ‘fake Killing spinors’, which themselves can be
constructed from a ‘fake superpotential’. Non-perturbative stability then follows from the
existence of a suitable ‘fake superpotential’.
However, the AdS/CFT correspondence allows us to address the problem of non-
perturbative stability from a completely different point of view. Namely, if we knew the
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JHEP05(2007)075
effective action of the dual theory, then we would be able to address the question of stabil-
ity/instability directly. We will show that the effective action can be computed holograph-
ically in a derivative expansion using Hamilton-Jacobi theory [29]. Requiring that this
effective action exists and it is stable reproduces all known stability results, including the
recent results in the case of generalized boundary conditions. This agreement can be traced
to the fact that both arguments require global existence of a suitable ‘fake superpotential’.
In the latter case, however, this is interpreted as Hamilton’s characteristic function, which
allows us to immediately generalize these results to other systems, such as conformally
coupled scalars.
The paper then is organized as follows. In section 2 we review a general description of
multi-trace deformations in the large N limit, which relies on large N factorization. This
will make manifest the correspondence between multi-trace deformations on the boundary
and boundary conditions in the bulk in section 3, where we revisit the boundary value
problem and the possible boundary conditions for the Klein-Gordon operator in asymp-
totically locally AdS spaces. In particular, we present a general systematic method to
address multi-trace deformations and to properly account for the fact that the boundary
is a conformal boundary - as opposed to a hard boundary. As we show, this automati-
cally removes the divergences associated with the infinite volume of the space. Although
we present these results for scalar fields, they immediately generalize to any field admit-
ting boundary conditions corresponding to multi-trace deformations. In section 4 then we
demonstrate the general method in the case of a free massive scalar field in a fixed AdS
background, reproducing in a concise way a number of known results. We then move on in
section 5 to include gravity and we describe in detail our method for computing the holo-
graphic effective action of the dual theory in a systematic way based on Hamilton-Jacobi
theory. This method is then applied to the cases of scalars minimally and conformally
coupled to gravity in sections 6 and 7 respectively, which contain our main results. In sec-
tion 6 we generalize the non-supersymmetric Poincar´e domain wall solutions found in [23]
to arbitrary dimension, while the same is done for the instanton solution found in [24] in
section 7. Moreover, we find all possible domain wall solutions - both flat and curved -
for the conformally coupled scalar in any dimension, and we show that this completely
determines the two-derivative effective action of the dual theory. Some technical results
regarding the variational problem for minimally and conformally coupled scalars, as well
as the Hamilton-Jacobi method for these systems, are collected in the appendix.
2. Multi-trace deformations in QFTs with a large N limit
In a quantum field theory with a standard large N limit, large N factorization allows for
a universal description of generic multi-trace deformations. As we now briefly review, the
effect of such a deformation can most naturally be described in terms of the generating
functional of the deforming operator and its Legendre transform [7].
Let O(x) be a local, generically composite, gauge-invariant and single-trace operator
transforming in some representation of the relevant rank N group. For concreteness we take
this to be the adjoint representation and we normalize the operator such that hOi = O(N
0
)
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JHEP05(2007)075
as N → ∞. The dynamics of O(x) is encoded in the generating functional of connected
correlators, W [J], which can be represented as a path integral over the fundamental degrees
of freedom, {φ}, of the theory, weighted by the action S[φ], as
e
−W [J]
=
Z
[dφ]e
−S[φ]−N
2
R
d
d
xJ(x)O(x)
. (2.1)
Since W [J] is O(N
2
) as N → ∞, it is convenient to work instead with w[J] ≡ N
−2
W [J].
In particular, the one-point function of O(x) in the presence of a source is given by
σ(x) ≡ hOi
J
=
δw[J]
δJ
. (2.2)
Alternatively, the dynamics can be encoded in the Legendre transform of the generating
functional, Γ[σ], given by
e
−Γ[σ]
=
Z
[dJ]e
−N
2
w[J]+N
2
R
d
d
xJ(x)σ(x)
. (2.3)
Γ[σ] is known as the effective action of the local operator O(x), or the generating functional
of 1PI diagrams. Again, it is useful to introduce the O(N
0
) quantity
¯
Γ[σ] = N
−2
Γ[σ], such
that
J(x) = −
δ
¯
Γ[σ]
δσ
. (2.4)
Suppose now that the action is deformed by a function, f (O), of the local operator
O(x) as S
f
[φ] = S[φ] + N
2
R
d
d
xf(O ). In the following we will only consider deformations
for which f (0) = 0. The question we want to address now is how this deformation modifies
the functionals w[J] and
¯
Γ[σ]. As we now show, large N factorization allows for a very
simple and universal answer in the large-N limit, which is summarized in table 1. Of
course, beyond the large N approximation, the answer to this question is non-universal
and much more involved, since the operator O(x) will generically mix with other operators
at 1/N order. We will only consider the leading large N behavior here.
Consider first the generating functional in the deformed theory, which is given by
e
−N
2
w
f
[J
f
]
=
Z
[dφ]e
−S[φ]−N
2
R
d
d
x(J
f
O+f(O))
=
Z
[dφ]e
−S[φ]−N
2
R
d
d
x(JO+f(O)−f
′
(σ)O)
N→∞
≈ e
−N
2
w[J]
e
−N
2
R
d
d
x(f(σ)−σf
′
(σ))
, (2.5)
where we introduced J ≡ J
f
+ f
′
(σ) in the second line in order to remove the linear term
from f(O) so that large N factorization can be used in the last step. This proves the result
shown in the third row of table 1. Similarly, the effective action in the deformed theory is
given by
e
−N
2
¯
Γ
f
[σ]
=
Z
[dJ
f
]e
−N
2
w
f
[J
f
]+N
2
R
d
d
xJ
f
σ
N→∞
≈
Z
[dJ]e
−N
2
w[J]
e
−N
2
R
d
d
x(f(σ)−σf
′
(σ))
e
N
2
R
d
d
x(J−f
′
(σ))σ
= e
−N
2
¯
Γ[σ]−N
2
R
d
d
xf(σ)
, (2.6)
– 4 –