JHEP02(2007)008
Published by Institute of Physics Publishing for SISSA
Received: August 25, 2006
Revised: December 6, 2006
Accepted: January 12, 2007
Published: February 2, 2007
Non-supersymmetric membrane flows from fake
supergravity and multi-trace deformations
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 use fake supergravity as a solution generating tech nique to obtain a con-
tinuum of non-supersymmetric asymptotically AdS
4
×S
7
domain wall solutions of eleven-
dimensional supergravity with non-trivial scalars in the SL(8, R)/ SO(8) coset. Th ese solu-
tions are continuously conn ected to the supersymmetric domain walls describing a uniform
sector of the Coulomb branch of the M2-brane theory. We also provide a general argu-
ment that under certain conditions identifies the fake superpotential with the exact large-N
quantum effective potential of th e dual theory, describing a marginal multi-trace deforma-
tion. This identification strongly motivates further study of fake supergravity as a solution
generating method and it allows us to interpret our non-supersymmetric solutions as a
family of marginal triple-trace deformations of th e Coulomb branch that completely break
supersym metry and to calculate th e exact large-N anomalous dimensions of the opera-
tors involved. The hologr ap hic one- and two-point fun ctions for these solutions ar e also
computed.
Keywords: Gauge-gravity correspondence, AdS-CFT Correspondence, Field Theories in
Lower Dimensions, M-Theory.
c
° SISSA 2007 http://jhep.sissa.it/archive/papers/jhep022007008/jhep022007008.pdf
JHEP02(2007)008
Contents
1. Introduction a nd summary of results 1
2. The SL(N, R)/ SO(N) sector of gauged maximal supergravity and it s
higher-dimensional origin 5
3. All asymptotically AdS Poincar´e domain walls of the SL(N, R)/ SO(N)
sector 8
4. The non-supersymmetric Poincar´e domain walls in D = 4 11
4.1 Domain walls with a single scalar 12
4.2 Exact closed form solutions 15
5. Exact non-supersymmetric membrane flows 18
6. Holographic one-point functions 21
6.1 ∆ = ∆
+
25
6.2 ∆ = ∆
−
26
7. Holographic two-p oint functions 29
8. The fake superpotential a s a quantum effective potential and multi-trace
deformations 34
8.1 Triple-trace deformation of the Coulomb branch 36
A. Explicit form of the domain wall metric for W (φ; α), to first order in
α −α
o
and for general k 39
B. Uplifting the MTZ black hole to eleven dimensions 39
C. Computation of the holographic two-point functions 41
D. Multi-trace deformations in the large-N limit and the AdS/CFT corre-
spondence 42
1. Introduction and summary of results
The study of domain wall solutions of various supergravity theories has been strongly
motivated in r ecent years by the role these play in a variety of physical contexts, from the
AdS/CFT correspondence, where they describe an RG flow of the conformal field theory
– 1 –
JHEP02(2007)008
residing on the conformal boundary of AdS, to ‘Brane World’ scenarios and cosmological
models (see [1] for an extensive review of domain walls of N = 1 supergravity in four
dimensions). Although, when they arise as solutions to a particular supergravity theory,
such domain walls are often supersymmetric, this need not be the case. Indeed, many
non-supers ymmetric gravitational theories admit domain wall solutions as well. In this
paper, however, we will emphasize the fact that true supergravity theories also admit non-
supersym metric domain wall solutions, which can be physically important.
We will focus on domain walls preserving Poincar´e invariance in d = D −1 dimensions,
where D is the spacetime dimension where the given gravitational theory lives. Such
domain walls take the form
1
ds
2
D
= dr
2
+ e
2A(r)
η
ij
dx
i
dx
j
, φ
I
= φ
I
(r), (1.1)
where η = diag (−1, 1, . . . , 1) is the Minkows ki metric in d dimensions. Sin ce only the
metric and a number of scalar fields are involved in these s olutions, they can generically
be described by an effective gravitational theory with an action of the f orm
S =
Z
M
d
D
x
√
−g
µ
1
2κ
2
D
R −
1
2
G
IJ
(φ)g
µν
∂
µ
φ
I
∂
ν
φ
J
− V (φ)
¶
, (1.2)
where κ
2
D
= 8πG
D
is the effective gravitational constant and G
IJ
is a generic (Riemannian)
metric on the scalar manifold. Such theories arise naturally as consistent truncations of
various gauged supergravities, in which case the scalar potential is generated by the non-
trivial gauging of (some of) the isometries of the scalar manifold. Generically, however, this
effective description will only be valid locally in the moduli s pace of a given supergravity
theory [3]. Here we are inter ested in the application of domain walls to the AdS/CFT
correspondence and so we assume that the metric (1.1) is asymptotically AdS, which is
equivalent to the statement that A(r) ∼ r as r → ∞. This in turn implies that the scalar
potential V (φ) has at least one stable fixed point at φ
I
= φ
I
∗
such that V (φ
∗
) < 0. By a
reparameterization of the scalar manifold we can set φ
I
∗
= 0. If this potential arises from
some gauged supergravity, this fixed point corresponds to the maximally symmetric AdS
D
vacuum.
The equations of motion following from the action (1.2) are Einstein’s equations
R
µν
−
1
2
Rg
µν
= κ
2
D
T
µν
, (1.3)
with the stress tensor given by
T
µν
= G
IJ
(φ)∂
µ
φ
I
∂
ν
φ
J
− g
µν
µ
1
2
G
IJ
(φ)g
ρσ
∂
ρ
φ
I
∂
σ
φ
J
+ V (φ)
¶
, (1.4)
and
∇
µ
¡
G
IJ
(φ)∂
µ
φ
J
¢
−
1
2
∂G
LM
∂φ
I
g
µν
∂
µ
φ
L
∂
ν
φ
M
−
∂V
∂φ
I
= 0. (1.5)
1
More general domain walls with a different isometry do ex ist, as is discussed e.g. in [2], but we will not
discuss them here.
– 2 –
JHEP02(2007)008
Substituting the domain wall ansatz (1.1) into the equations of motion one obtains the
following equations for the war p factor A(r) and the scalar fields φ
I
(r):
˙
A
2
−
κ
2
d(d −1)
³
G
IJ
(φ)
˙
φ
I
˙
φ
J
− 2V (φ)
´
= 0,
¨
A + d
˙
A
2
+
2κ
2
d − 1
V (φ) = 0,
G
IJ
(φ)
¨
φ
J
+
∂G
IJ
∂φ
K
˙
φ
K
˙
φ
J
−
1
2
∂G
LM
∂φ
I
˙
φ
L
˙
φ
M
+ d
˙
AG
IJ
(φ)
˙
φ
J
−
∂V
∂φ
I
= 0, (1.6)
where the dot den otes the derivative with respect to the radial coordinate r.
It is important to distinguish between two types of solutions of these second order
equations. Following [4] we will call a ‘BPS domain wall’ any domain wall of the form (1.1)
which satisfies the first order equations
˙
A = −
κ
2
d − 1
W (φ),
˙
φ
I
= G
IJ
(φ)
∂W
∂φ
J
, (1.7)
for some function W (φ) of the scalar fields such that the scalar potential can be expressed
as
V (φ) =
1
2
µ
G
IJ
(φ)
∂W
∂φ
I
∂W
∂φ
J
−
dκ
2
d −1
W
2
¶
. (1.8)
Note that the first order equations (1.7) together with (1.8) ensure that the second or-
der equations (1.6) are automatically satisfied. Given the expression (1.8) for the scalar
potential in terms of the function W , the first order equations (1.7) can be derived ´a la Bo-
gomol’nyi by extremizing the energy functional E[A, φ] that has (1.6) as its Euler-Lagrange
equations [4, 5]. In the context of gauged supergravity, a function W (φ) satisfying (1.8)
arises naturally as the superpotential, W
o
(φ), w hich enters the gravitino and dilatino vari-
ations
δψ
µ
= D
µ
ε −
κ
2
2(d − 1)
W
o
(φ)γ
µ
ε, (1.9)
δχ
I
=
µ
γ
µ
∂
µ
φ
I
+ G
IJ
(φ)
∂W
o
∂φ
J
¶
ε. (1.10)
It follows that the domain walls defined by the s uperpotential W
o
(φ) are supersymmetric
solutions of the particular gauged supergravity. Crucially, however, equation (1.8) does not
define the function W (φ) uniquely and hence there may generically exist other fun ctions
W (φ) satisfying (1.8) in addition to W
o
(φ).
2
This has been termed fake supergravity and in
this context any function W (φ) that solves (1.8) is called a fake superpotential [2, 3, 6, 7].
In [7] it was shown that any BPS Poincar´e domain wall of th e form (1.1), defined by a func-
tion W (φ) which is not necessarily the true superpotential of a given gauged supergravity,
2
Note, however, that not every function W (φ) that satisfies (1.8) is acceptable, since it will not generically
correspond to an asymptotically AdS domain wall. We will discuss in detail the cond itions W (φ) must satisfy
below. See also [2, 4].
– 3 –
JHEP02(2007)008
is ‘supersymmetric’ in the sense that one can always find Killing spinors, at least locally.
In [7] this was considered as an indication that any fu nction W (φ) that solves (1.8) (and
possibly subject to su itable boundary conditions) may be the true superpotential of some
supergravity theory, even though there is no systematic way to find which is the relevant,
known or u nknown, theory [8]. Despite the elegance of this statement, it is difficult in
practice to confirm or refute it. We will adopt a rather different point of view here, how-
ever. Namely, we will confine ourselves to a particular gauged supergravity, with a certain
superpotential W
o
(φ). Clearly, any BPS domain wall defined by a solution W (φ) 6= W
o
(φ)
of (1.8) is not supersymmetric in this context. We will nevertheless continue to call such
solutions ‘BPS’ since they satisfy the first order equations (1.7). They are s till special
solutions because they allow for the definition of fake Killing spinors via (1.9) with W
o
(φ)
replaced by the fake superpotential W (φ) [2].
3
The existence of fake Killing spinors im-
plies, in particular, non-perturbative gravitational stability, at least in the absence of naked
singularities [9, 4, 2].
If the scalar potential cannot be written in the form (1.8), however, there can still
exist domain wall solutions of the form (1.1) that solve the second order equations (1.6).
We will refer to such solutions as ‘non-BPS domain walls’.
4
We will not consider further
such domain walls here since we are interested in scalar potentials that arise from gauged
supergravities, and such potentials are guaranteed to be expressible in the form (1.8) since
this is at least possible using the true superpotential W
o
(φ).
5
Although one often views fake supergravity as an effective subsector of some gauged
supergravity, by identifying both the scalar potential and the fake superpotential of fake
supergravity with the true potential and superpotential respectively of the gauged su per-
gravity [4, 3, 6], we will instead treat fake supergravity as a powerful solution generating
technique for non-supersymmetric solutions of a given gauged s upergravity. In particular,
we will treat (1.8) as a first order non-linear differential equation for th e fake superpotential
W (φ) [10, 11, 2, 12] (see also [13] where a very similar perspective is adopted). For s calar
potentials arising from some gauged supergravity this equation admits at least one solution,
namely the true superpotential of the theory. Our aim here will be to determine all so-
lutions of (1.8) satisfying appropriate boundary conditions. Each solution W (φ) 6= W
o
(φ)
defines a non-supersymmetric domain wall solution of the given gauged supergravity, and
therefore describes a non-supersymmetric RG flow of the dual field theory.
The paper is organized as follows. In the next s ection we will discuss a common sub-
sector of gauged maximal supergravities in dimensions D = 4, 5, 7 with the scalar fields
parameterizing an SL(N, R)/ SO(N) coset, where N = 8, 6, 5 respectively. The complete
non-linear ansatz for uplifting any solution of this subsector to eleven-dimensional or Type
IIB supergravity is known [14, 15], and all supersymmetric Poincar´e domain walls, describ-
3
Note that our fake superp otential differs by a factor of −
2(d−1)
κ
2
relative to the fake superpotential
defined in [2]. The supercovariant derivative is uniquely determined, however, by the requirement that it
reduces to that of pure AdS , namely (D
µ
+
1
2l
γ
µ
)ε, when the scalar fields vanish.
4
Note though the analysis of [7], which suggests that any potential that admits domain wall solutions
can be written in the form (1.8) and so there are no ‘non-BPS’ domain walls.
5
Generically the superpotential W
o
(φ) will be a matrix, however, instead of a scalar quantity. See e.g. [2].
– 4 –
