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 –