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Institut f¨ ur Theoretische Physik, Universit¨ at Heidelbe rg, Philosophenweg 16-19, D-69120
Heidelberg, Germany
Email:s.koers atthphys.uni-heidelberg.de
Abstract: We construct new families of non-supersymmetric sourceles s type IIA AdS 4
vacua on those coset manifolds that also admit supersymmetr ic solutions. We investigate
the spectrum of left-invariant modes and ﬁnd that most, but n ot all, of the vacua are stable
under these ﬂuctuations. Generically, there are also no mas sless moduli.
∗Postdoctoral Fellow FWO – Vlaanderen.Contents
1. Introduction 1
2. Ansatz 3
3. Solutions 6
4. Stability analysis 11
5. Conclusions 15
A. SU(3)-structure 15
B. Type II supergravity 16
1. Introduction
The reasons for studying AdS 4vacua of type IIA supergravity are twofold: ﬁrst they are
examples of ﬂux compactiﬁcations away from the Calabi-Yau r egime, where all the moduli
can be stabilized at the classical level. Secondly, they can serve as a gravity dual in the
AdS4/CFT3-correspondence, which became the focus of attention due to recent progress
in the understanding of the CFT-side as a Chern-Simons-matt er theory describing the
world-volume of coinciding M2-branes [1].
Itismucheasiertoﬁndsupersymmetricsolutionsofsupergr avityasthesupersymmetry
conditions are simpler than the full equations of motion, wh ile at the same time there
are general theorems stating that the former – supplemented with the Bianchi identities
of the form ﬁelds – imply the latter [2, 3, 4, 5]. Although spec ial type IIA solutions
that came from the reduction of supersymmetric M-theory vac ua were already known (see
e.g. [6, 7, 8]), it was only in [3] that the supersymmetry cond itions for type IIA vacua with
SU(3)-structure were ﬁrst worked out in general. It was disc overed that there are natural
solutions to these equations on the four coset manifolds G/Hthat have a nearly-K¨ ahler
limit [9, 10, 11, 12, 13, 14] (solutions on other manifolds ca n be found in e.g. [3, 15, 16]).1
To be precise these are the manifolds SU(2) ×SU(2),G2
SU(3),Sp(2)
S(U(2)×U(1))andSU(3)
U(1)×U(1).2
These solutions are particularly simple in the sense that bo th the SU(3)-structure, which
determines the metric, as well as all the form ﬂuxes can be exp anded in terms of forms
which are left-invariant under the action of the group G. The supersymmetry equations
1For an early appearance of these coset manifolds in the strin g literature see e.g. [17].
2See [18] for a review and a proof that these are the only homoge neous manifolds admitting a nearly-
K¨ ahler geometry.
– 1 –of [3] then reduce to purely algebraic equations and can be ex plicitly solved. Nevertheless,
these solutions still have non-trivial geometric ﬂuxes as o pposed to the Calabi-Yau or torus
orientifolds of [15, 16]. Similarly to those papers it is pos sible to classically stabilize all
left-invariant moduli [14]. Inspired by the AdS 4/CFT3correspondence more complicated
type IIA solutions have in the meantime been proposed. The so lutions have a more generic
form for the supersymmetry generators, called SU(3) ×SU(3)-structure [19], and are not
left-invariant anymore [20, 21, 22, 23] (see also [24]). Sup ersymmetric AdS 4vacua in type
IIB with SU(2)-structure have also been studied in [25, 26, 2 7, 28] and in particular it has
been shown in [28] that also in this setup classical moduli st abilization is possible.
At some point, however, supersymmetry has to be broken and we have to leave
the safe haven of the supersymmetry conditions. In this pape r we construct new non-
supersymmetric AdS 4vacua without source terms. This means that the more complic ated
equations of motion of supergravity should be tackled direc tly3. In order to simplify the
equations we use a speciﬁc ansatz: we start from a supersymme tric AdS 4solution and scan
for non-supersymmetric solutions with the samegeometry (and thus SU(3)-structure), but
withdiﬀerent NSNS- and RR-ﬂuxes. Moreover, we expand these form ﬁelds in t erms of the
SU(3)-structure and its torsion classes. This may seem rest rictive at ﬁrst, but it works for
11D supergravity, where solutions like this have been found and are known as Englert-type
solutions [31, 32, 33] (see [34] for a review). To be speciﬁc, for each supersymmetric M-
theory solution of Freund-Rubin type (which means the M-the ory four-form ﬂux has only
legs along the external AdS 4space, i.e.F4=fvol4wherefis called the Freund-Rubin
parameter) it is possible to construct a non-supersymmetri c solution with the same inter-
nal geometry but with a diﬀerent four-form ﬂux. The modiﬁed fo ur-form of the Englert
solution has then a non-zero internal part: ˆF4∝η†γm1m2m3m4ηdxm1m2m3m4, whereηis
the 7D supersymmetry generator, and a diﬀerent Freund-Rubin parameterfE=−(2/3)f.
Also the Ricci scalar of the AdS 4space, and thus the eﬀective 4D cosmological constant,
diﬀers:R4D,E= (5/6)R4D. In type IIA with non-zero Romans mass (so that there is no lif t
to M-theory) non-supersymmetric solutions of this form hav e been found as well: for the
nearly-K¨ ahler geometry in [35, 29, 36] and for the K¨ ahler- Einstein geometry in [35, 20, 37].
In this paper we show that this type of solutions is not restri cted to these limits and sys-
tematically scan for them. Applying our ansatz to the coset m anifolds with nearly-K¨ ahler
limit, mentioned above, we ﬁnd that the most interesting man ifolds areSp(2)
S(U(2)×U(1))and
SU(3)
U(1)×U(1), on which we ﬁnd several families of non-supersymmetric AdS 4solutions. We
also ﬁnd some non-supersymmetric solutions in regimes of th e geometry that do not allow
for a supersymmetric solution.
These non-supersymmetric solutions are not necessarily st able. For instance, it is
known that if there is more than one Killing spinor on the inte rnal manifold (which holds
in particular for S7, the M-theory lift of CP3=Sp(2)
S(U(2)×U(1))), the Englert-type solution is
unstable [38]. We investigate stability of our solutions ag ainst left-invariant ﬂuctuations.
This means we calculate the spectrum of left-invariant mode s, and check for each mode
3Anotherroute would be toﬁndsome alternative ﬁrst-ordereq uations, which extendthe supersymmetry
conditions in that they still automatically imply the full e quations of motion in certain non-supersymmetric
cases, see e.g. [29, 30].
– 2 –whether the mass-squared is above the Breitenlohner-Freed man bound [39, 40]. This is not
a complete stability analysis in that there could still be no n-left-invariant modes that are
unstable. We do believe it provides a good ﬁrst indication. I n particular, we ﬁnd for the
type IIA reduction of the Englert solution on S7that the unstable mode of [38] is among
our left-invariant ﬂuctuations and we ﬁnd the exact same mas s-squared.
These non-supersymmetric AdS 4vacua are interesting, because, provided they are
stable, they should have a CFT-dual. For instance in [20] the CFT-dual for a non-
supersymmetric K¨ ahler-Einstein solution on CP3was proposed. Furthermore, for phe-
nomenologically more realistic vacua, supersymmetry-bre aking is essential. Really, one
would like to construct classical solutions with a dS 4-factor, which are necessarily non-
supersymmetric. Because of a series of no-go theorems – from very general to more speciﬁc:
[41, 42, 43, 44, 45] – this is a very non-trivial task. For pape rs nevertheless addressing this
problemsee[46,47,45,48,49,28]. Inthiscontext thelands capeofthenon-supersymmetric

Institut f¨ ur Theoretische Physik, Universit¨ at Heidelbe rg, Philosophenweg 16-19, D-69120

Heidelberg, Germany

Email:s.koers atthphys.uni-heidelberg.de

Abstract: We construct new families of non-supersymmetric sourceles s type IIA AdS 4

vacua on those coset manifolds that also admit supersymmetr ic solutions. We investigate

the spectrum of left-invariant modes and ﬁnd that most, but n ot all, of the vacua are stable

under these ﬂuctuations. Generically, there are also no mas sless moduli.

∗Postdoctoral Fellow FWO – Vlaanderen.Contents

1. Introduction 1

2. Ansatz 3

3. Solutions 6

4. Stability analysis 11

5. Conclusions 15

A. SU(3)-structure 15

B. Type II supergravity 16

1. Introduction

The reasons for studying AdS 4vacua of type IIA supergravity are twofold: ﬁrst they are

examples of ﬂux compactiﬁcations away from the Calabi-Yau r egime, where all the moduli

can be stabilized at the classical level. Secondly, they can serve as a gravity dual in the

AdS4/CFT3-correspondence, which became the focus of attention due to recent progress

in the understanding of the CFT-side as a Chern-Simons-matt er theory describing the

world-volume of coinciding M2-branes [1].

Itismucheasiertoﬁndsupersymmetricsolutionsofsupergr avityasthesupersymmetry

conditions are simpler than the full equations of motion, wh ile at the same time there

are general theorems stating that the former – supplemented with the Bianchi identities

of the form ﬁelds – imply the latter [2, 3, 4, 5]. Although spec ial type IIA solutions

that came from the reduction of supersymmetric M-theory vac ua were already known (see

e.g. [6, 7, 8]), it was only in [3] that the supersymmetry cond itions for type IIA vacua with

SU(3)-structure were ﬁrst worked out in general. It was disc overed that there are natural

solutions to these equations on the four coset manifolds G/Hthat have a nearly-K¨ ahler

limit [9, 10, 11, 12, 13, 14] (solutions on other manifolds ca n be found in e.g. [3, 15, 16]).1

To be precise these are the manifolds SU(2) ×SU(2),G2

SU(3),Sp(2)

S(U(2)×U(1))andSU(3)

U(1)×U(1).2

These solutions are particularly simple in the sense that bo th the SU(3)-structure, which

determines the metric, as well as all the form ﬂuxes can be exp anded in terms of forms

which are left-invariant under the action of the group G. The supersymmetry equations

1For an early appearance of these coset manifolds in the strin g literature see e.g. [17].

2See [18] for a review and a proof that these are the only homoge neous manifolds admitting a nearly-

K¨ ahler geometry.

– 1 –of [3] then reduce to purely algebraic equations and can be ex plicitly solved. Nevertheless,

these solutions still have non-trivial geometric ﬂuxes as o pposed to the Calabi-Yau or torus

orientifolds of [15, 16]. Similarly to those papers it is pos sible to classically stabilize all

left-invariant moduli [14]. Inspired by the AdS 4/CFT3correspondence more complicated

type IIA solutions have in the meantime been proposed. The so lutions have a more generic

form for the supersymmetry generators, called SU(3) ×SU(3)-structure [19], and are not

left-invariant anymore [20, 21, 22, 23] (see also [24]). Sup ersymmetric AdS 4vacua in type

IIB with SU(2)-structure have also been studied in [25, 26, 2 7, 28] and in particular it has

been shown in [28] that also in this setup classical moduli st abilization is possible.

At some point, however, supersymmetry has to be broken and we have to leave

the safe haven of the supersymmetry conditions. In this pape r we construct new non-

supersymmetric AdS 4vacua without source terms. This means that the more complic ated

equations of motion of supergravity should be tackled direc tly3. In order to simplify the

equations we use a speciﬁc ansatz: we start from a supersymme tric AdS 4solution and scan

for non-supersymmetric solutions with the samegeometry (and thus SU(3)-structure), but

withdiﬀerent NSNS- and RR-ﬂuxes. Moreover, we expand these form ﬁelds in t erms of the

SU(3)-structure and its torsion classes. This may seem rest rictive at ﬁrst, but it works for

11D supergravity, where solutions like this have been found and are known as Englert-type

solutions [31, 32, 33] (see [34] for a review). To be speciﬁc, for each supersymmetric M-

theory solution of Freund-Rubin type (which means the M-the ory four-form ﬂux has only

legs along the external AdS 4space, i.e.F4=fvol4wherefis called the Freund-Rubin

parameter) it is possible to construct a non-supersymmetri c solution with the same inter-

nal geometry but with a diﬀerent four-form ﬂux. The modiﬁed fo ur-form of the Englert

solution has then a non-zero internal part: ˆF4∝η†γm1m2m3m4ηdxm1m2m3m4, whereηis

the 7D supersymmetry generator, and a diﬀerent Freund-Rubin parameterfE=−(2/3)f.

Also the Ricci scalar of the AdS 4space, and thus the eﬀective 4D cosmological constant,

diﬀers:R4D,E= (5/6)R4D. In type IIA with non-zero Romans mass (so that there is no lif t

to M-theory) non-supersymmetric solutions of this form hav e been found as well: for the

nearly-K¨ ahler geometry in [35, 29, 36] and for the K¨ ahler- Einstein geometry in [35, 20, 37].

In this paper we show that this type of solutions is not restri cted to these limits and sys-

tematically scan for them. Applying our ansatz to the coset m anifolds with nearly-K¨ ahler

limit, mentioned above, we ﬁnd that the most interesting man ifolds areSp(2)

S(U(2)×U(1))and

SU(3)

U(1)×U(1), on which we ﬁnd several families of non-supersymmetric AdS 4solutions. We

also ﬁnd some non-supersymmetric solutions in regimes of th e geometry that do not allow

for a supersymmetric solution.

These non-supersymmetric solutions are not necessarily st able. For instance, it is

known that if there is more than one Killing spinor on the inte rnal manifold (which holds

in particular for S7, the M-theory lift of CP3=Sp(2)

S(U(2)×U(1))), the Englert-type solution is

unstable [38]. We investigate stability of our solutions ag ainst left-invariant ﬂuctuations.

This means we calculate the spectrum of left-invariant mode s, and check for each mode

3Anotherroute would be toﬁndsome alternative ﬁrst-ordereq uations, which extendthe supersymmetry

conditions in that they still automatically imply the full e quations of motion in certain non-supersymmetric

cases, see e.g. [29, 30].

– 2 –whether the mass-squared is above the Breitenlohner-Freed man bound [39, 40]. This is not

a complete stability analysis in that there could still be no n-left-invariant modes that are

unstable. We do believe it provides a good ﬁrst indication. I n particular, we ﬁnd for the

type IIA reduction of the Englert solution on S7that the unstable mode of [38] is among

our left-invariant ﬂuctuations and we ﬁnd the exact same mas s-squared.

These non-supersymmetric AdS 4vacua are interesting, because, provided they are

stable, they should have a CFT-dual. For instance in [20] the CFT-dual for a non-

supersymmetric K¨ ahler-Einstein solution on CP3was proposed. Furthermore, for phe-

nomenologically more realistic vacua, supersymmetry-bre aking is essential. Really, one

would like to construct classical solutions with a dS 4-factor, which are necessarily non-

supersymmetric. Because of a series of no-go theorems – from very general to more speciﬁc:

[41, 42, 43, 44, 45] – this is a very non-trivial task. For pape rs nevertheless addressing this

problemsee[46,47,45,48,49,28]. Inthiscontext thelands capeofthenon-supersymmetric