Loop corrections to the antibrane potential
Received: March
Loop corrections to the antibrane potential
Iosif Bena 0 1
Johan Blaback 0 1
David Turton 0 1
0 CEA, CNRS , F91191 Gif sur Yvette , France
1 Institut de Physique Theorique, Universite Paris Saclay
Antibranes provide some of the most generic ways to uplift Antide Sitter ux compacti cations to de Sitter, and there is a growing body of evidence that antibranes placed in long warped throats such as the KlebanovStrassler warped deformed conifold solution have a branebranerepelling tachyon. This tachyon was rst found in the regime of parameters in which the backreaction of the antibranes is large, and its existence was inferred from a highly nontrivial cancellation of certain terms in the interbrane potential. We use a brane e ective action approach, similar to that proposed by Michel, Mintun, Polchinski, Puhm and Saad in [29], to analyze antibranes in KlebanovStrassler when their backreaction is small, and nd a regime of parameters where all perturbative contributions to the action can be computed explicitly. We nd that the cancellation found at strong coupling is also present in the weakcoupling regime, and we establish its existence to all loops. Our calculation indicates that the spectrum of the antibrane worldvolume theory is not gapped, and may generically have a tachyon. Hence uplifting mechanisms involving antibranes remain questionable even when backreaction is small.
Brane Dynamics in Gauge Theories; Dbranes; Flux compacti cations
1 Introduction 3 4 5
Discussion
A Conventions 2
From bulk solutions to worldvolume theories
2.1
2.2
2.3
The KlebanovStrassler background
The bosonic terms in the worldvolume theory
The fermionic terms in the worldvolume theory
Loop corrections and nonrenormalization theorems
Physical interpretation B Fermion masses from D5 polarization 1 5
1). The second is when the backreaction of the antibranes is
small in any region where supergravity can be trusted (gsN
1), however one does not
truncate to leading order in gsN . The third regime is when one truncates to leading order
in gsN
1; this is sometimes referred to as working in the gsN ! 0 limit.
The most commonly used systems for studying the physics of antibranes have D3, M2,
or D6 charges dissolved in ux, such as the KlebanovStrassler (KS) warped deformed
conifold background [1], the CveticGibbonsLuPope (CGLP) warped Stenzel background [2],
and the JanssenMeessenOrt n solution with
nite Romans mass [3]. The most precise
calculations of the physics of antibranes have been done in the rst (large backreaction)
regime. In this regime it was shown that antibrane solutions have a singularity [4{11] which
cannot be resolved by brane polarization when the antibranes are smeared and their
worldvolume is at [12{14], and moreover cannot be cloaked by a black hole horizon [15{18].1
1Other antibrane singularities such as those corresponding to antibranes with non at (Antide Sitter)
worldvolumes [19, 20] can be resolved by brane polarization. There are also antibrane singularities that
can be cloaked with a horizon [21]. However, the physics of these antibranes is very di erent from that
generically have a branebranerepelling tachyon on their worldvolume [24], which may be
responsible for the fact that their singularity cannot be cloaked by an event horizon.
The third regime of parameters described above corresponds to discarding all physics
beyond leading order in gsN
1. In this regime, one can study probe antiD3 branes in
the solution Sdual to the KS geometry. One
nds that the probe action describing the
polarization of these branes into D5 branes has a metastable minimum [25]; this result has
been extrapolated to the original KS regime to argue that antiD3 branes polarize into
NS5 branes and give rise to metastable KS minima [25]. However, as discussed in ref. [24],
such polarization can only be reliably described when gsN
1. Furthermore, as explained
in ref. [26], calculations that ignore subleading e ects in gsN can give misleading results
about metastable vacua: a brane con guration that appears metastable in the gsN ! 0
limit [26{28] can in fact correspond to a vacuum of a di erent theory, and this can only be
seen by studying the system at nite gsN .2
Hence, in order to investigate further whether antibranes may or may not be metastable
in long warped throats, the only regime amenable to calculations that remains to be
explored is the second one, 0 < gsN
1. In an interesting paper, Michel, Mintun, Polchinski,
Puhm, and Saad have argued [29] that in this regime, the correct way to describe one or
several antibranes in a background with positive charge dissolved in the uxes is to use a
socalled \brane e ective action"; this action is obtained by integrating out heavy degrees
of freedom to obtain an e ective eld theory (EFT) of light elds on the brane interacting
with supergravity elds [30, 31].
The exploration of ref. [29] leaves open the question of whether or not antibranes in the
KS solution have a branebranerepelling tachyon of the type found in [24]. Indeed, upon
examining the brane e ective action of antiD3 branes localized at the North Pole of the S3
at the bottom of the KS solution, one can easily see that all the terms of this action must
transform in representations of the SO(6) Rsymmetry group. For example, the interaction
potential between two branes is a combination of an SO(6) singlet and a term transforming
in the 200 [32], that furthermore must be invariant under the SO(3)
SO(3) symmetry
preserved by the background and one of the branes. The absence of a tachyon depends on
the exact balance of these terms: if the term in the 1 is stronger than the one in the 200
then there is no tachyon, but, if the term in the 200 is stronger than the term in the 1,
there will always exist a tachyon.
The purpose of this paper is to identify a regime of parameters in which the brane
e ective action describing localized antiD3 branes in the KS solution can be computed,
of antiD3 brane uplifting constructions [22]. For example, the antibranes with non at worldvolume only
polarize when their worldvolume cosmological constant is parametrically large [20]. Similarly, the cloaked
solutions of [21] have a very nongeneric type of transverse
uxes which allow them to evade the blackening
nogo theorem of [18], but it is hard to see how antibranes in generic transverse uxes could do the same [23].
2One can also see this from the fact that the action of the tunneling instanton diverges: when gsN > 0
the distance between the supersymmetric and nonsupersymmetric brane con gurations diverges at spatial
in nity; in the gsN
! 0 limit, this distance is nite but the tension of the branes diverges, and so the
tunneling process cannot take place [26].
{ 2 {
and to use it to evaluate the interbrane potential to all orders in perturbation theory. As
we will discuss in section 4, the diagrams in the brane e ective action approach of ref. [29]
correspond to string diagrams in the limit of massless closed strings. For example, at one
loop, the string diagram is an annulus. In the opposite eldtheory limit in which the open
strings become light, the same string diagram corresponds to a oneloop diagram in the
worldvolume gauge theory of the antiD3 branes. Similarly, the higherloop diagrams in
this theory correspond to limits of string diagrams with more than two boundaries. Since
this limit allows explicit computations to be performed, we work in the lowenergy gauge
theory on the branes.
We rst compute the treelevel action, including the terms that are induced by the
Upon placing antiD3 branes in a background with a
transverse threeform
ux, the fermions on the branes acquire a mass, proportional to the
value of the imaginary selfdual (ISD) component of the ux [33]. The ISD threeform
ux
also induces a scalar trilinear interaction [34, 35]. In addition, antiD3 branes placed in
transverse
uxes will generically also have treelevel scalar masses, that can be obtained
by expanding the brane potential to quadratic order.3
The treelevel fermion and scalar masses in the action of antiD3 branes placed in the
KS solution (or similar supersymmetric ISD backgrounds) have three important properties.
Firstly, of the six Hermitian scalars, three are massive with equal masses, while three are
massless. Secondly, of the four Weyl fermions, three are massive and one is massless, and
the masssquared of the three massive fermions is half that of the three massive scalars.
Thirdly, the scalar trilinear and the fermion mass term obey a very simple linear relation,
discussed in more detail in section 2.3.
The rst property follows from the fact that the functions entering in the KS solution
only depend on the radial coordinate . This implies that there is a at potential, and hence
no force preventing antibranes from moving along the three directions inside the large S3
at the bottom of the KS solution. The second property is even more intriguing, and is a key
feature of antiD3 branes in KS. There are several ways to see this; the most straightforward
way would be to match the multiple conventions for these terms and compute them directly.
However, we will instead derive this property by computing the potential for a probe
antiD3 brane in the KS solution to polarize into a D5 brane wrapping the contracting S2 of the
warped deformed conifold. This potential does not allow for brane polarization, but one
nds that the quadratic term is twice larger than it would be if the polarization potential
were a perfect square. This is described in appendix B. This veri es that the three scalars
corresponding to the motion of the brane away from the bottom of the warped deformed
conifold have a masssquared that is twice the wouldbe supersymmetric value.
These two calculations indicate that the sum of the squares of the treelevel scalar
masses and the sum of the squares of the treelevel fermion masses are the same, which
agrees with the more general result recently found in [32] that this is a property of all
D3 branes at equilibrium. Note however that the traceless part of the scalar mass matrix
3When antiD3 branes are placed in an imaginary antiselfdual (IASD) background, the fermions are
massless. In addition, if the background is of IASD Gran~aPolchinskiGKPtype [36, 37], the antibranes
feel no potential when moving in the transverse directions, and hence the six scalars are also massless.
{ 3 {
depends on the features of the geometry near the location of the branes and hence is not
determined by the fermion masses.
Having obtained the treelevel brane action, we next compute the eldtheory loop
corrections. The easiest way to compute these corrections is to observe that the antibrane
worldvolume theory has the following structure. Consider N = 4 SuperYangMills (SYM)
theory, broken to N = 1 by giving equal masses to the three chiral multiplets. As we will
show, the antibrane worldvolume theory is a particular N = 0 theory originating from this
equalmass N = 1 theory by the addition of a traceless scalar bilinear term (a Bterm) that
breaks the remaining supersymmetry but preserves the SO(3)
SO(3) symmetry. One can
then apply a combination of certain general results on
niteness obtained by Parkes and
West [38{41] to nd that this theory is nite to all orders in perturbation theory. Thus the
masses of the chiral multiplets and the Bterms receive no perturbative corrections, which
implies that the interbrane potential along the S3 at the bottom of the deformed conifold
is exactly zero. This is the main result of our paper.
This result is all the more striking because it agrees exactly with the result obtained
1 by studying backreacted antibranes in the KS solution [24]. Indeed, one
can compute the PolchinskiStrassler [42] polarization potential of fullybackreacted
antiD3 branes localized in the KS solution and one nds, after a pair of surprising cancellations,
that the quadratic piece in the polarization potential along the S3 at the bottom of the
warped deformed conifold is exactly zero.4 This in turn indicates that the three scalars that
describe the motion of the antiD3 branes at the bottom of the KS solution are massless
4It is important to emphasize that the underlying reason for this cancellation is not understood, and
moving at the bottom of the throat is repulsive [14].
As this expression suggests, the metrics which have a tilde are unwarped. The C4 potential
takes the form5
where the function
is related to the warp factor e4A by6
2.1
The KlebanovStrassler background
The purpose of this subsection is to review some properties of the KS background [1], to
extract certain relations that we need for our analysis, and to introduce notation. We will
not give a full review of KS; for more details we refer the reader to ref. [43], which agrees
with most of our conventions.
The KS background is a supersymmetric, noncompact,
Gran~aPolchinskiGKPtype [36, 37] solution. By this we mean the following. The G3
F3
ie
H3
ux has
(2,1) complexity, the dilaton e
= gs is constant, and the tendimensional (stringframe)
HJEP07(216)3
metric is a warped product of R3;1 with a CalabiYau base:
(2.1)
(2.2)
(2.3)
(2.4)
(2.5)
where
0 is a constant that we gauge x by requiring that ( = 0) = 0.
In the KS solution, the CalabiYau base is a deformed conifold with topology R+
S2
S3, and all the functions that determine the solution, such as
and A above, depend
only on the radial direction of the deformed conifold, which is commonly denoted as
and
parameterizes the R+.
the throat,
This solution has four supercharges, which are compatible with those of D3 branes.
Hence, probe antiD3 branes experience a potential that forces them to the bottom of
ds120 = e2Ads~42 + e 2Ads~62 :
C4 = ?~4
= e
e4A
0 ;
In this paper we consider antiD3 branes localized at the bottom of the deformed
conifold, and hence we employ a local R6 coordinate system [24] parameterizing the
neighborhood of the branes,
ds120 = e2A
dx dx + e 2Ag~mndxmdxn ;
where the internal metric is given by g~mn = b mn for some constant b that depends on
conventions.
5We will use the notation that F5 = ?10dC4, or equivalently that F5 is the internal part of the selfdual
veform
eld strength F5 = (1 + ?10)F5.
6Note that we are using the stringframe metric throughout this paper. This means that there is an
extra factor of the (constant) dilaton in this expression compared to the corresponding formula in [37],
written using the Einsteinframe metric.
For later convenience we introduce complex coordinates
with i = 1; 2; 3, in terms of which we have
2
/ (x72 + x82 + x92) ;
1
2
1
2
z
i
p (xi+3 + ixi+6) ;
In such a local coordinate system, the local expansion in terms of is now an expansion
in x7;8;9  the local coordinates of the R+
S2. This means that the 2 term of the antiD3
brane potential will have the form
which, as mentioned in the Introduction, can be decomposed as the sum of a quadratic
term transforming in the 1 (singlet) of the SO(6) Rsymmetry and a term transforming in
the 200 traceless symmetric representation,
2
/ (x72 + x82 + x92)
1
2
=
(x42 + x52 + x62 + x72 + x82 + x92)
To derive the supersymmetric term on the antiD3 brane worldvolume, we will make
use of the F5 Bianchi identity
dF5 = H3 ^ F3 ;
written here for a sourceless background, such as the KS solution. Expanding around the
bottom of the deformed conifold, and using the fact that e 4A has no linear term in this
expansion for KS, we can write the l.h.s. of eq. (2.11) as
using the metric and the C4 potential. Hence
in real coordinates, or in the complex coordinates introduced before
dF5 =
e 8A0 d~?6d j0 ;
dF5 = 2e 8A0 b 1e
where we also used the relation (2.3).
To evaluate the r.h.s. of eq. (2.11) we rely on the imaginary self duality of the KS G3
ux, which implies that the NSNS and RR
uxes are related by H3 = e ?6 F3. Together
with F3 = 12 (G3 + G3), this gives
{ 6 {
where (G3)ijk = jG3j ijk and jG3j is real. Since A a function of
only, we can now write
HJEP07(216)3
(G3)mnp(G3)mnp = 3(G3)ijk(G3)ijk :
3
2
4 s
1 g2b 2e8A0 jG3j2 i :
In due course we shall also make use of the other quadratic terms in the Taylor
expansion of e4A. Again using the fact that A is a function of
only, we can use eqs. (2.7){(2.10)
to nd that these terms are given by
where indices are contracted using the warped metric. Transforming to complex coordinates
and using the fact that G3 is purely (2,1), only one type of contraction is nonvanishing,
(2.16)
(2.17)
(2.18)
(2.19)
(2.20)
These terms transform in the 200, unlike the singlet term in (2.18). As one can see from
eqs. (2.7){(2.10), the contributions to the potential along the S3 directions coming from
the terms in the 1 and the 200 are equal and opposite, while along the R+
S2 directions
they add.
2.2
The bosonic terms in the worldvolume theory
We now derive the worldvolume gauge theory of antiD3 branes at the bottom of the KS
throat. We start with the bosonic terms in the action, and we compute the fermionic terms
in the next subsection.
The worldvolume gauge eld will not play an important role, so for ease of presentation
we shall suppress it in what follows, with the understanding that the full action contains
the usual gauge kinetic terms and covariant derivative couplings. In addition, since the
U(1) sector of the gauge theory is free, we focus on the SU(N3) sector. With these points
understood, the bosonic part of the theory is given by the DBI and WZ parts of the
brane Lagrangian7
p
3 Tr nP [ei {2' C ^ eB2 ]o
det(P [Mab]) det(Qmn) ;
o
0123
:
We work with the stringframe metric. The indices a; b; : : : are tendimensional, the indices
; ; : : : are fourdimensional worldvolume indices parallel to the brane, and m; n; : : : are
sixdimensional indices transverse to the brane. The tensors in the above expression are
7The Lagrangian is written with explicit signs corresponding to an antibrane in our conventions. We
use the generic term brane to refer to both antiDp and Dp branes, where explicit signs and dimensions
determine the details.
{ 7 {
de ned as follows:
2 `s2 and the D3 brane charge is 3 = 2 =(2 `s)4, where `s is the string length.8
The Hermitian scalars 'm transform in the adjoint of SU(N3) and parameterize the brane
positions. We also choose a gauge for B2 such that B2j0 = 0.
Expanding in powers of , the DBI Lagrangian becomes
{ 8 {
where we have used the fact mentioned below eq. (2.3) that in our gauge we have ( =
0) = 0 and, hence, at the location of the branes dC6 = F7 =
?10 F3.
The two trilinear terms can be combined into an expression involving a particular
combination of G3 and its complex conjugate,
L tri =
G3)mnpTrf'm'n'pg :
3
2 3 e4A0 (G3
i = p
1
2
'i+3 + i'i+6 :
It is convenient to introduce complex coordinates that we shall use from now on,
In these complex coordinates the full Lagrangian, L = LDBI + LWZ, becomes
where a bar on the scalars indicates Hermitian conjugation.
8Our conventions are described in appendix A.
3
i e4A0 e
1
2
e
1
2
e
o
1
2
e
(2.22)
HmnpTrf'm'n'pg +
g~mqg~npTr f['m; 'n]['q; 'p]g
where as usual the H3 coupling arises from the Taylor expansion of B2, and where we have
dropped the constant term proportional to e4A0 .
Similarly, expanding the WZ Lagrangian gives
3
i e4A0 Trf'm'n'pg(?6F )mnp
i
+ e4A0 GijkTrf
i j kg + h.c.
i j g + h.c.
g~i{g~jTrf[ i; j ][ {; ] [ i; ][ j ; {]
g + : : :
(2.23)
(2.24)
(2.25)
(2.26)
In order to proceed we make a constant rescaling of the scalars to obtain a
canonicallynormalized kinetic term. This is achieved by de ning
^i
i
Having done this, we immediately drop the hat from the rescaled expressions and
exclusively use the canonicallynormalized scalars from now on.
After applying the relation (2.18) derived from the Bianchi identity in the previous
subsection, the Lagrangian becomes
L =
i

g
i
2
+
+
1
2
2
p
pgs e4A0 jG3jb 3=2
p
gs1=2 !
i j g + h.c.
2
pgs e4A0 jG3jb 3=2
i{ jTr [ i; j ][ {; ]
ijkTrf
i j kg + h:c:
(2.28)
will shortly write part of the Lagrangian in terms of an N = 1 superpotential and, as is
wellknown, the Fterm part of the quadrilinear interaction of N = 4 SYM is contained in
the superpotential, while the Dterm part is not (see for example [44]).
We can identify the bilinear and trilinear scalar interactions in the language of soft
supersymmetrybreaking. To this end we introduce complexscalar masses (m2B)i, Bterms
bij , and the (2,1) trilinear interaction rijk via
1
2 bij Trf
Lsoft =
(m2B)iTrf
By matching these terms with eq. (2.28), we nd the following bosonic soft
supersymmetrybreaking terms:
Complexscalar masses: (m2B)i = m2B i ;
mB
2
pgs e4A0 jG3jb 3=2
;
Bterms: bij =
(2,1) trilinear: rijk =
m2B ij ;
p
gs1=2 !
We note that the treelevel Bterms are real. A priori these terms could have had
an imaginary part, which would correspond to o diagonal elements in the real matrix
2.3
The fermionic terms in the worldvolume theory
The fermionic part of the action for D3 branes in transverse RR and NSNS threeform
uxes was rst explicitly written down in ref. [33] and further studied in refs. [45{47].
{ 9 {
where indices have been raised with i{. Then the fermion mass matrix is proportional to
the complex conjugate of S [33],
By expanding the G3 ux at the location of the branes, one nds a diagonal fermion mass
matrix with equal entries,
(2.31)
(2.32)
(2.33)
(2.34)
We now compute the terms in this fermionic action and their precise normalization for
antibranes in KS.
The calculation deriving the fermion bilinear [33] and that determining the scalar
trilinear [34] were performed using di erent conventions, so it is necessary to x the relative
normalizations of the bosonic and fermionic actions. To x this overall factor, we now
examine the form of the fermionic bilinear terms.
Recall that upon writing N = 4 SYM in an N = 1 super eld formalism, one obtains
three chiral multiplets. The KS background has only (2; 1) primitive threeform
ux, so
the only additional term in the fermion action is a mass for the three Weyl fermions in
the chiral multiplets, and the gaugino remains massless [33]. We write the (2,1) primitive
This determines the correct normalization of the fermionic terms in the Lagrangian, and
implies that the sum of the squares of the treelevel scalar masses and the sum of the squares
of the treelevel fermion masses are the same. This fact agrees with the more general result
recently found in ref. [32] that this is a property of all D3 branes at equilibrium in warped
compacti cations.
Having established this relation between fermion and boson masses, we now observe
that the (2; 1) scalar trilinear coupling is proportional to the mass of the fermions. This
S{ = S {
)
miFj = mF ij :
Next, we observe that the three massive Hermitian scalars have a masssquared which
is twice the one that they would have if supersymmetry had been preserved. This can
be seen from computing the polarization potential of a D5 brane carrying antiD3 brane
charge and wrapping the shrinking S2 of the KS background. As outlined in appendix B,
the D5 brane polarization potential does not have a niteradius minimum. However, when
the term originating from the mass is halved, the polarization potential becomes a perfect
square and has a supersymmetric polarization minimum. Hence the supersymmetric
masssquared is half the masssquared of the three scalars corresponding to motion away from
the tip. From the argument made around eq. (2.7) we see that the supersymmetric mass is
the mass of the complex scalars, given in eq. (2.30). Thus the mass of the Weyl fermions
and the complex scalars are equal,
mF = mB =
2
pgs e4A0 jG3jb 3=2 :
Lsusy = Tr
i j
+
c gYM
3
ijkTr n i j ko
where c is a numerical parameter that depends on conventions. The supersymmetry is
broken from N = 4 ! N = 1? by the three equal masses of the chiral multiplets. The
i
are the chiral multiplet super elds, written in component elds as
fact, combined with the equality of fermion and boson masses in the absence of the Bterm,
allows us to temporarily put aside the Bterm (and the Dterm quadrilinear interaction),
and write the remainder of the Lagrangian in terms of N = 1 super elds. Similar
observations have been made in refs. [
46, 47
].
The Lagrangian for these terms can then be written as
Upon eliminating the auxiliary elds F i, the Lagrangian becomes
Lsusy =
i
i
)
ig
i = i + p
2
i + 2 i
F :
(mF )(mF ) i{ Trf
i {
g
miFj Trf
jc gYMj2 i{ jTr [ i; j ][ {; ] ;
where again indices are contracted with i or its inverse. We can now read o 9
(2.35)
(2.36)
rijk = mF c gYM ijk :
Combining eqs. (2.30), (2.34) and (2.39), we observe that the relation between fermion
masses and scalar trilinear couplings takes the explicit form:
This simple relation, which is also present in more general theories on Dbrane worldvolume,
will be a crucial ingredient in our analysis of perturbative corrections, as in general it leads
to signi cant simpli cations in beta functions [48].
The supersymmetric Lagrangian (2.38) reproduces the fermionic terms and all terms in
the bosonic Lagrangian (2.28) except for the Bterms and the Dterm quadrilinear. Thus we
see that the Bterms are the only terms responsible for breaking the N = 1 supersymmetry,
and making the theory an N = 0? theory. As mentioned earlier, the gauge elds have been
suppressed, but can easily be reintroduced. This concludes the calculation of the antiD3
brane worldvolume treelevel Lagrangian.
9There is a redundancy in conventions in how one exactly chooses the value of the constant c, and how
one relates gYM to gs; for our purposes we will not need to x this redundancy.
=
+
+
+
+
+
+
+
+
: : :
HJEP07(216)3
scalar couplings and the Yukawa couplings.
for planar diagrams, and blue for nonplanar diagrams.
3
Loop corrections and nonrenormalization theorems
Having derived the treelevel action of the antiD3 worldvolume gauge theory, we now
proceed to investigate quantum corrections. These corrections would generically cause the
masses them to run logarithmically with the energy, and this running can be thought of
as coming from the backreaction of the antiD3 branes on the corresponding
supergravity elds.
The worldvolume gauge theory of a stack of N3 coincident antiD3 branes is a U(N3) =
SU(N3)
U(1) theory. All of the interaction terms derived above, except the mass terms
and the Bterms, are antisymmetrized and hence, as usual, the U(1) sector is free and
decouples. In the SU(N3) sector, the diagrams that provide the corrections to the masses
of the scalars are summarized schematically in gure 1. These diagrams are the usual
eldtheory limit of openstring diagrams (see gure 2). These diagrams come with a factor of
gsN3 for each additional boundary, and hence we are in a regime of perturbative control
1. Thus, naively, one would expect a oneloop correction to the scalar mass
(1)(m2B)
(m2B)tree
/ gsN3 :
In the previous section we derived the structure of how supersymmetry is broken at
tree level on antiD3 branes in KS,
W4 =

c gYM
3
ijkTr
N{=z4
i j k
! W1 = W4 +
}

1
2
N{=z1?
miFj Tr
i j
}

N{=z0?
}
! W1 & Bterms :
(3.2)
The purpose of this section is to describe how this type of supersymmetry breaking a ects
the running of the masses and couplings. Since along the S3 directions there is a
perfect cancellation between the supersymmetric scalar mass terms and the real part of the
supersymmetrybreaking Bterms, this direction is at at tree level. At loop level there
(3.1)
are a priori three possibilities. Either the real part of the Bterms and the
supersymmetric masses run di erently, and then the spectrum along the S3 becomes either gapped or
tachyonic, or they run in the same way, preserving the masslessness of the S3 scalars.
Perturbative corrections to supersymmetric gauge theories of the kind we are interested
in were investigated by Parkes and West [38{41]. They considered the addition of mass
terms that preserve some supersymmetry, and they also applied the spurion method [49]
to study theories in which supersymmetry is completely broken. They derived several
powerful allloop results, a subset of which we now combine for our analysis.
The rst step in the breaking of supersymmetry is the addition of a mass term to the
N = 4 superpotential, resulting in an equalmass N = 1? theory. This theory was shown
to remain nite to all loops [38].
niteness to all orders in perturbation theory [39].10
The second step is to add to the Lagrangian the particular Bterms induced by the KS
background. These Bterms have the \X2 Y 2" form (2.10). It was shown that adding such
terms, in conjunction with supersymmetric masses, to N = 4 Super YangMills, preserves
Combining these two results implies that neither the supersymmetric masses nor the
Bterms receive any higherloop correction, and hence the masslessness of the three scalars
along the S3 direction is preserved to all orders in the loop expansion. This absence of
these perturbative corrections applies not only to the bosonic masses, but to all terms in
the Lagrangian of the antiD3 brane gauge theory.
4
Physical interpretation
In this section we discuss the physical signi cance of the allloop result obtained above.
To do this, we rst described the region of parameter space in which we work, and then
compare it to previous results. The open string loop expansion is valid for gsN3
1 and,
since the U(1) sector is free, we focus on the SU(N3) sector and hence work at N3 > 1.
This is the opposite regime to the one used to analyze fullybackreacted antibranes in
supergravity (gsN3
1) and we believe that the striking agreement between our results
and those of [24] strongly suggests that the scalars corresponding to motion along the S3
at the bottom of the KS solution remain massless for all values of gsN3.
We work in the usual lowcurvature supergravity limit, where the length scale, L,
associated to the curvature of the background is much larger than the string length, `s=L
1. This suppresses higherderivative terms in the brane action, coming from the Taylor
expansion of the supergravity elds, such as
1
X
=0
!
gmn(') =
(4.1)
where p
`s=L.
Thirdly, we work in the AdS/CFT decoupling regime, in which the lowenergy gauge
theory on the brane decouples from the supergravity
elds. This regime corresponds to
10In addition, the \X2
Y 2" terms preserve
niteness to all orders when added to a nite N = 2
theory [40], and preserve twoloop [41] and oneloop [50] niteness in N = 1 gauge theories.
r
AdS/CFT
HJEP07(216)3
out in the di erent limits include: A. [29], B. [23], C. [24], D. The present work. The AdS/CFTlike
decoupling limit is shaded in blue, and the vertical (green) dashed line is gsN3 = 1.
+
+
: : :
elds [29]. Crosses represent external supergravity elds.
+
+
+
: : :
represent closed string vertex operators corresponding to the external legs in gure 4.
sending the distance r? from the branes to zero with r?= 0 xed [51, 52], which means
that one has
r
4
?
gsN3( 0)2
Since we work in the weaklycoupled gauge theory, the conjectured bulk dual is
stronglycurved and the corresponding sigmamodel is stronglycoupled. We depict our regime and
compare it to the regimes considered in other works in gure 3.
The di erence between our approach and that of [29] is that the latter considers a
lowenergy EFT involving both brane and supergravity
elds, that is valid for r
The diagrams that enter in the calculation of this EFT, depicted in
gure 4, are the
masslessclosedstring limit of the string diagrams in
gure 5. Our diagrams are simply
the lightopenstring limit of the same diagrams. For example, if we insert external
openstring vertex operators in
gure 5, we see that this diagram and the oneloop openstring
diagrams in
gure 2 are the same. Furthermore, as mentioned above, the wouldbe eld
?
`s.
theory corrections to the fermion and boson mass terms can be thought of as representing
the correction to the corresponding supergravity
elds caused by the backreaction of the
brane. Therefore, the diagrams in gure 4 and those in gure 1 compute the same quantity
in two di erent regimes.
The advantage of our regime is that it allows one to do precise calculations, which,
given the current technology of string loop calculations in RamondRamond backgrounds,
does not appear possible in the approach of [29] in the near future. Furthermore, it is
entirely possible that if such calculations were done, the exact cancellations leading to the
at directions along the S3 may not survive in the regime of parameters of [29], and thus
the corresponding EFT might have tachyonic terms. A priori, such terms could arise both
for multiple branes and for a single brane.11
If such tachyonic terms are present, the dynamics becomes complicated and falls
beyond what can be computed with current technology, either analytically or numerically.
One can ask whether there may be a nearby metastable minimum that could be used
for uplifting to de Sitter (as discussed in [29]); such a minimum would
rst need to be
found, and then a stability analysis would need to be performed. An analytic approach
would be to use the brane e ective action, which involves calculating string diagrams in a
RamondRamond background. A numerical approach would be to build a fullybackreacted
solution and analyze its stability; this is a cohomogeneitythree problem. Both approaches
are currently computationally out of reach. Taken together with the existing results on
antibrane instabilities, we believe that the burden of proof for claiming the existence of a new
metastable minimum rests on those who would make such a claim. Hence, our reluctance
to share the optimism expressed in [29] regarding antiD3 brane uplift [22].
In the analysis of fully backreacted antibranes in the KS solution [24] it was argued that
the cancellation of the bosonic potential along the S3 is not the full story. More speci cally,
this cancellation comes from a nontrivial relation between the real part of the Bterms and
the trace of the boson mass matrix, but in addition the Bterms could also have imaginary
parts that are not prohibited by the symmetry of the problem, and one therefore expects to
nd them generically. Such a term would give rise to tachyonic instabilities in o diagonal
directions [24].
In our analysis, the imaginary parts of the Bterms are not present at tree level, and
are also not generated by loops. However, since there are no symmetries protecting against
such terms, it is entirely possible that they will arise nonperturbatively in gsN3 or at
subleading order in the expansions discussed above that take us away from the regime of
parameters in which we work (the blue region in gure 3).
If such corrections preserve the balance between the real part of the Bterms and
the scalar mass terms, then nonzero imaginary parts of the Bterms would give rise to
tachyons. While one expects this balance to be preserved in the decoupling limit when
interpolating from weak to strong coupling, there is a priori no reason why it should be
preserved away from the decoupling limit. This in turn could generate a gap or could lead
11Although, of course, for a single brane, such a tachyon cannot be interpreted as indicating branebrane
repulsion.
to tachyons even without nonzero imaginary parts of the Bterms. Thus, while our result
does not prove that the potential has a tachyonic direction, it shows that the potential is
vulnerable to tachyonic corrections of the types discussed above, which may be expected
to be generically present by standard EFT reasoning.
Another question which one can ask is whether in our regime of parameters one can see
a nonperturbative brane ux annihilation e ect of the type proposed in [25]. If one rst
considers antiD3 branes in the Sdual of KS geometry, their worldvolume theory has Higgs
vacua. These vacua correspond to the polarization of the antiD3 branes into D5 branes
wrapping an S2 inside the S3 at the bottom of the throat. However, in the limit in which
we work, `s=L ! 0, the height of the energy barrier that these D5 brane have to traverse
in order to trigger brane ux annihilation is in nite, and hence the tunneling probability
of the antiD3 branes is zero. It is almost certain that this height is also in nite in the KS
geometry in our limit, because the size of the S3 that the NS5 brane with antiD3 brane
charge has to sweep out diverges.
We note in passing that it has been suggested that antiD3 brane singularities may
possibly be resolved by polarization into NS5branes [23]. We do not study such
polarization in this work, since it requires one to work in the regime of parameters gsN
1, as
discussed in the Introduction. We recall however that it was argued in [53] that a tachyonic
term in the polarization potential renders this NS5 con guration unstable to developing
shape modes that break spherical symmetry. The branebrane repulsion results in the
tachyonic accumulation of antiD3 density (encoded by the worldvolume ux on the NS5)
near the endpoints of the major axis of an NS5 ellipsoid inside the S3. Another possible
manifestation of branebrane repulsion is the expulsion of antiD3 branes from the NS5
(analogous to the antiM2 expulsion discussed in [14]).
5
Discussion
In this paper we have computed the potential of antiD3 branes placed at the bottom of
the KS throat, in the regime 0 < gsN
1 and in the AdS/CFT limit, to all orders in
perturbation theory.
We rst computed the treelevel Lagrangian, and determined the
pattern of (soft) supersymmetry breaking. We then applied certain wellestablished results
on
niteness to show that this Lagrangian does not receive corrections to all loops in
perturbation theory, and hence three of the scalars on the worldvolume of the antiD3
branes in KS remain massless to all orders in the loop expansion. The fact that this
result matches the one obtained in the fullybackreacted regime (gsN3
1) in ref. [24] is
strong evidence that the spectrum of antiD3 branes in KS does not become gapped in any
regime of parameters where exact calculations can be made. Furthermore, since there is
no symmetry prohibiting an imaginary Bterm in the e ective action on the branes, and
since such a term will always introduce tachyons, the optimism about branebranerepelling
tachyons disappearing when gsN3
1 appears premature.
Although the explicit analysis performed in this paper is for the KS background, we
expect the masslessness of some of the worldvolume scalars to be a generic feature of
antiD3 branes in conical highlywarped geometries: for a conical geometry to be regular at its
HJEP07(216)3
bottom it is necessary to have some nite cycle, such as the S3 of the deformed conifold
or the S2 of the resolved conifold. From the perspective of the worldvolume theory of an
antibrane at the bottom of this geometry, this would imply that some of the scalars are
massless at tree level and hence the Bterm will be nonzero.
In a generic conical geometry with ISD
uxes the theory on the antiD3 branes will
contain fermion bilinears and scalar trilinears, that will consist of both \supersymmetric"
terms (of the kind we found in KS) coming from the primitive (2,1) components of G3,
and also nonsupersymmetric terms coming from the (0,3) and the nonprimitive (1,2) G3
components. For example, a (0,3) component would introduce a hijkTrf
scalar trilinear term and a gaugino mass M
. The relation between hijk and M is exactly
of the same form as that between the supersymmetric fermion masses and scalar trilinear
i j k
g + h.c.
couplings that we analyzed in detail in section 2.3, and this, combined with the vanishing
of the mass supertrace at tree level [32], implies that the betafunctions of the theory will
vanish both at one and two loops [48, 54{56].12 Hence, the scalars that are massless at
tree level will remain massless at least to two loops.
A similar argument can be made for a more generic background that contains a
combination of primitive (2,1), (0,3), and nonprimitive (1,2) ux: these uxes would give rise
to a symmetric 4
4 fermion mass matrix that can be diagonalized (this corresponds to
changing the complex structure), and the resulting theory would be the one with (2,1) and
(0,3) uxes discussed above.
Our result appears to be in tension with the argument that the spectrum of
antiD3 branes in the KS geometry is gapped [57], and also with the related argument that
separated antiD3 branes at the bottom of the KS solution should be screened by ux and
therefore should attract each other [58]. To explicitly compute the e ect of this screening
on the potential between two antiD3 branes, one must perform a full string calculation,
which cannot be done with current technology. However, in the limit where the branes are
close to each other, this string calculation reduces to our eldtheory calculation, which
nds that the treelevel masslessness of three of the six scalars that describe the antiD3
branes is preserved to all loops. Therefore, the spectrum of the antiD3 branes remains
ungapped in our regime of parameters. Hence, the intuition that antibranes at the bottom
of KS are screened and therefore attract each other does not appear to give the correct
physics in either of the two regimes of parameters where precise calculations can be done:
the regime we have considered and the largebackreaction regime [24].
The absence of a gap may also be problematic for phenomenological applications.
When the KS throat is glued to a compact manifold, the gluing introduces perturbations
to the KS throat, which in turn can give very small masses to the scalars that are massless
in the in nite KS solution. These masses were estimated in ref. [59], and were found to be
exponentially smaller than the typical mass scale of the light elds at the bottom of the
throat, and hence phenomenologically problematic. Had our calculations found instead
that the interbrane degrees of freedom were gapped, there would have only been three
such light elds, corresponding to the centerofmass degrees of freedom, and these elds
12In particular, this implies that the mass supertrace will remain zero at one and two loops [32].
could presumably have been uplifted in some other way. However, uplifting 3N3 massless
modes appears a more and more onerous task as one increases N3. It would be interesting
to compare these corrections to those discussed in section 4, in particular to see whether
any possible tachyonic term could overwhelm these very light masses.
We have also discussed the regime of parameters in which our calculation is done and its
relation to the brane e ective action approach of [29]: the Feynman diagrams that enter
in our
eldtheory calculation arise from the lightopenstring limit of string diagrams,
which in the opposite masslessclosedstring limit, reduce to the supergravity amplitudes
considered in [29].
There is another di erence between these approaches, which has to do with the number
of antiD3 branes we consider. The theory on N3 antiD3 branes has a U(N3) gauge group,
and its dynamics can be split in an SU(N3) sector and a U(1) sector. The U(1) sector
describes the centerofmass motion of the branes, and is a free theory, re ecting the fact
that there is no potential to the stack of antibranes (or a single antibrane) moving together
on the S3, as mentioned in [29]. It is important to note that this fact is not a result
of a calculation done using the brane e ective action, but simply a result of symmetry
considerations.13
As we have noted, when going away from our eldtheory limit, both the U(1) and
the SU(N3) sectors may receive corrections, which are capable of introducing a gap or a
tachyon. There is no symmetry that prohibits these corrections, even for the U(1) sector,
although the interpretation of the possible existence of a tachyon for a single antibrane
is unclear.
Our result that the branebrane potential along the S3 at the bottom of the KS solution
remains
at agrees exactly with the strongcoupling calculation of [24]. In [24] it was
furthermore argued that the symmetries of the problem do not prohibit the existence of
another term in the branebrane potential (an imaginary part of the Bterms) and thus,
following the usual EFT reasoning, one expects that such a term will generically be present.
This term does not a ect the value of this potential along the S3 but on the other hand
gives rise to a branebrane repelling tachyon along a direction misaligned with the S3.
It is important to ask whether a similar term could possibly also appear and give rise to
a branebranerepelling tachyon in the weakcoupling regime in which we work. In the eld
theory on the branes this term is zero at tree level, and one can also show that the
betafunctions associated to its running are exactly zero. Hence one possibility is that this term
is exactly zero in the weakcoupling regime, and only appears in the largebackreaction
regime. However, if one examines the problem a bit deeper, and excludes mathematical
oddities, this possibility appears quite unlikely. Rather, if the branebranepotential on the
S3 remains at all the way from weak to strong coupling, and the tachyoninducing term
is considerable at large gsN3, one expects that a leftover of this term, however small, will
be visible at weak coupling, perhaps as a nonperturbative e ect. Whenever this term is
13There has also been interest in studying bound states of an antiD3 brane and an O3
plane [60{64],
which entirely removes the six antiD3 scalar degrees of freedom, and hence any potential branebrane
repelling tachyons. Besides the fact that these objects have
nite charge but zero mass and so strongly
violate the BPS bound, these constructions so far lack the explicitness available in KS.
not exactly zero, a branebranerepelling tachyon is present [24], and thus we expect that
this tachyon will be present at all nite values of gsN3.
Finally, let us comment on the implications of our result for the possibility of using
antiD3 branes in long warped KSlike throats to uplift the cosmological constant and obtain
a landscape of metastable de Sitter solutions in String Theory. Our computation found a
at direction in the branebrane potential which is preserved at all loops, indicating that
the system is ungapped and is vulnerable to tachyons which, from an EFT perspective, are
likely to be present. Therefore antiD3 brane uplifting mechanisms remain questionable
even when backreaction is small.
Our paper thus contains yet another calculation that a priori could have either agreed
or disagreed with the viability of antiD3 brane uplifting constructions. Taken together with
the negative results obtained in the large backreaction regime [4{18], our result further adds
to the evidence against the existence of a de Sitter multiverse obtained using antibranes.14
Acknowledgments
We thank Daniel Baumann, Micha Berkooz, Ulf Danielsson, Emilian Dudas, Anatoly
Dymarsky, Mariana Gran~a, Stanislav Kuperstein, Emil Martinec, Stefano Massai, Praxitelis
Ntokos, Giulio Pasini, Joe Polchinski, Andrea Puhm, Rodolfo Russo, and Thomas Van
Riet for useful discussions. This work was supported by the John Templeton Foundation
Grant 48222. The work of I.B. was also supported by the ERC Starting Grant 240210
StringQCDBH, and by a grant from the Foundational Questions Institute (FQXi) Fund,
a donor advised fund of the Silicon Valley Community Foundation on the basis of proposal
FQXiRFP31321 (this grant was administered by Theiss Research). The work of J.B. and
D.T. was also supported by the CEA Eurotalents program.
A
Conventions
In this appendix we record our conventions and their relation to those used in a selection
of related literature [33, 34, 45{47, 62]. Our conventions are:
G3
F3 e
H3 and EMN
BMN , which implies that the RR elds strengths
are F = dC + H ^ C;
Our Hodgestar conventions are those described in appendix A of [66];
The antiD3 brane worldvolume theory has interaction terms induced by ISD
uxes,
?6G3 = iG3.
Our H3 = dB2 has the opposite sign compared to that of refs. [
34, 46, 47, 62
]. Ref. [
45
]
does not follow the same Hodgestar conventions as us, and ref. [46] switches to a
mostlyminus signature for their fourdimensional theory while we keep strictly to a mostlyplus
signature. Note that we also start from the stringframe brane action, which is a choice
that becomes irrelevant after the constant rescaling done in eq. (2.27).
14If one desires, one can even quantify this evidence in a Bayesian approach [
65
].
B
In addition, we di er from ref. [34] in our conventions for the RR
elds. We have
gs 1H3 while ref. [34] has Fn
H3. Finally, in our conventions we have `s2 = 0.
Fermion masses from D5 polarization
As discussed in section 2.3, the relative normalizations of the bosonic and fermionic actions
can be directly derived from the D5 polarization potential. In the KS background a D5
brane carrying antiD3 brane charge and wrapping the shrinking S2 at a nite distance, ,
from the bottom of the deformed conifold, has the action
VD5 = 2 N3c2 2
c3 3 +
1
where c2;3;4 are constants. Details of the derivation of this potential can be found for
example in ref. [13]. This potential has no minimum away from
= 0, and hence antiD3
branes in KS cannot polarize into D5 branes.
We are interested in the coe cient c2: this is proportional to @2(e4A + )j0 and is hence
proportional to the masssquared of the scalars that correspond to motion away from the
bottom of the warped deformed conifold.
Now, if we deform the potential by taking c2 ! c2=2, the potential can now be written
as a perfect square. Explicitly, the deformed potential is
V~D5 =
b
2
gs
128 N3
2
p
2b22 e4A0 jG3j gs
N3
!2
:
(B.2)
This expression has been translated into our conventions using the local R6 coordinates of
ref. [24] (modi ed to be consistent with our complex coordinates de ned in eq. (2.8)) and
the KS conventions found in ref. [13].
This deformed potential obtained by c2 ! c2=2 has a supersymmetric minimum at
nonzero . Thus, the masssquared of the three massive Hermitian scalars in the undeformed
potential is twice its wouldbe supersymmetric value, i.e. the masssquared of the fermions.
Open Access.
This article is distributed under the terms of the Creative Commons
Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
[1] I.R. Klebanov and M.J. Strassler, Supergravity and a con ning gauge theory: duality cascades
and SB resolution of naked singularities, JHEP 08 (2000) 052 [hepth/0007191] [INSPIRE].
[2] M. Cvetic, G.W. Gibbons, H. Lu and C.N. Pope, Ricci at metrics, harmonic forms and
brane resolutions, Commun. Math. Phys. 232 (2003) 457 [hepth/0012011] [INSPIRE].
[3] B. Janssen, P. Meessen and T. Ort n, The D8brane tied up: string and brane solutions in
massive type IIA supergravity, Phys. Lett. B 453 (1999) 229 [hepth/9901078] [INSPIRE].
[4] I. Bena, M. Gran~a and N. Halmagyi, On the existence of metastable vacua in
KlebanovStrassler, JHEP 09 (2010) 087 [arXiv:0912.3519] [INSPIRE].
05 (2011) 053 [arXiv:1102.1734] [INSPIRE].
[INSPIRE].
(Anti)brane backreaction beyond perturbation theory, JHEP 02 (2012) 025
[arXiv:1111.2605] [INSPIRE].
and singularities, JHEP 09 (2013) 123 [arXiv:1301.5647] [INSPIRE].
[arXiv:1303.1809] [INSPIRE].
antibrane singularities, JHEP 10 (2012) 078 [arXiv:1205.1798] [INSPIRE].
KlebanovStrassler, JHEP 09 (2013) 142 [arXiv:1212.4828] [INSPIRE].
(2014) 173 [arXiv:1402.2294] [INSPIRE].
(2013) 063012 [arXiv:1212.5162] [INSPIRE].
[arXiv:1409.0534] [INSPIRE].
JHEP 04 (2014) 064 [arXiv:1309.2949] [INSPIRE].
[16] I. Bena, J. Blaback, U.H. Danielsson and T. Van Riet, Antibranes cannot become black, Phys.
Rev. D 87 (2013) 104023 [arXiv:1301.7071] [INSPIRE].
Nucl. Phys. B 883 (2014) 107 [arXiv:1310.1372] [INSPIRE].
[17] A. Buchel and D.A. Galante, Cascading gauge theory on dS4 and string theory landscape,
[18] J. Blaback, U.H. Danielsson, D. Junghans, T. Van Riet and S.C. Vargas, Localised
antibranes in noncompact throats at zero and nite T , JHEP 02 (2015) 018
[19] F. Apruzzi, M. Fazzi, D. Rosa and A. Tomasiello, All AdS7 solutions of typeII supergravity,
[20] D. Junghans, D. Schmidt and M. Zagermann, Curvatureinduced resolution of antibrane
singularities, JHEP 10 (2014) 034 [arXiv:1402.6040] [INSPIRE].
[21] G.S. Hartnett, Localised antibranes in ux backgrounds, JHEP 06 (2015) 007
[arXiv:1501.06568] [INSPIRE].
Rev. D 68 (2003) 046005 [hepth/0301240] [INSPIRE].
[22] S. Kachru, R. Kallosh, A.D. Linde and S.P. Trivedi, De Sitter vacua in string theory, Phys.
[23] D. CohenMaldonado, J. Diaz, T. van Riet and B. Vercnocke, Observations on uxes near
antibranes, JHEP 01 (2016) 126 [arXiv:1507.01022] [INSPIRE].
(2015) 146 [arXiv:1410.7776] [INSPIRE].
[25] S. Kachru, J. Pearson and H.L. Verlinde, Brane/ ux annihilation and the string dual of a
nonsupersymmetric eld theory, JHEP 06 (2002) 021 [hepth/0112197] [INSPIRE].
con gurations in MQCD, JHEP 11 (2006) 088 [hepth/0608157] [INSPIRE].
HJEP07(216)3
realized gauge theories, Nucl. Phys. B 755 (2006) 239 [hepth/0606061] [INSPIRE].
dynamics, JHEP 09 (2015) 021 [arXiv:1412.5702] [INSPIRE].
(1975) 157 [INSPIRE].
Rev. D 65 (2002) 025011 [hepth/0104170] [INSPIRE].
[31] W.D. Goldberger and M.B. Wise, Renormalization group ows for brane couplings, Phys.
[32] I. Bena, M. Gran~a, S. Kuperstein, P. Ntokos and M. Petrini, D3brane model building and
the supertrace rule, Phys. Rev. Lett. 116 (2016) 141601 [arXiv:1510.07039] [INSPIRE].
[33] M. Gran~a, D3brane action in a supergravity background: the fermionic story, Phys. Rev. D
66 (2002) 045014 [hepth/0202118] [INSPIRE].
[34] R.C. Myers, Dielectric branes, JHEP 12 (1999) 022 [hepth/9910053] [INSPIRE].
[35] W. Taylor and M. Van Raamsdonk, Multiple D0branes in weakly curved backgrounds, Nucl.
Phys. B 558 (1999) 63 [hepth/9904095] [INSPIRE].
Rev. D 63 (2001) 026001 [hepth/0009211] [INSPIRE].
[36] M. Gran~a and J. Polchinski, Supersymmetric three form ux perturbations on AdS5, Phys.
[37] S.B. Giddings, S. Kachru and J. Polchinski, Hierarchies from
uxes in string
compacti cations, Phys. Rev. D 66 (2002) 106006 [hepth/0105097] [INSPIRE].
[38] A.J. Parkes and P.C. West, N = 1 supersymmetric mass terms in the N = 4 supersymmetric
YangMills theory, Phys. Lett. B 122 (1983) 365 [INSPIRE].
[39] A. Parkes and P.C. West, Finiteness and explicit supersymmetry breaking of the N = 4
supersymmetric YangMills theory, Nucl. Phys. B 222 (1983) 269 [INSPIRE].
[40] A. Parkes and P.C. West, Explicit supersymmetry breaking can preserve niteness in rigid
N = 2 supersymmetric theories, Phys. Lett. B 127 (1983) 353 [INSPIRE].
[41] A. Parkes and P.C. West, Finiteness in rigid supersymmetric theories, Phys. Lett. B 138
(1984) 99 [INSPIRE].
theory, hepth/0003136 [INSPIRE].
[42] J. Polchinski and M.J. Strassler, The string dual of a con ning fourdimensional gauge
[43] C.P. Herzog, I.R. Klebanov and P. Ouyang, Remarks on the warped deformed conifold, in the
proceeding of Modern trends in string theory: 2nd Lisbon school superstrings on g theory
superstrings, July 13{17, Lisbon, Portugal (2001), hepth/0108101 [INSPIRE].
[hepth/0312232] [INSPIRE].
Phys. B 689 (2004) 195 [hepth/0311241] [INSPIRE].
07 (2012) 188 [arXiv:1206.0754] [INSPIRE].
(1999) 101 [hepph/9903365] [INSPIRE].
65 [INSPIRE].
[INSPIRE].
nite N = 1
supersymmetric gauge theories, Phys. Lett. B 148 (1984) 317 [INSPIRE].
[51] J.M. Maldacena, The largeN limit of superconformal eld theories and supergravity, Int. J.
Theor. Phys. 38 (1999) 1113 [Adv. Theor. Math. Phys. 2 (1998) 231] [hepth/9711200]
[44] J.M. Maldacena, Strings in at space and plane waves from N = 4 super YangMills, Ann.
CalabiYau orientifolds with Dbranes and uxes, Nucl. Phys. B 690 (2004) 21
[52] O. Aharony, S.S. Gubser, J.M. Maldacena, H. Ooguri and Y. Oz, LargeN
eld theories,
string theory and gravity, Phys. Rept. 323 (2000) 183 [hepth/9905111] [INSPIRE].
[53] I. Bena and S. Kuperstein, Brane polarization is no cure for tachyons, JHEP 09 (2015) 112
[arXiv:1504.00656] [INSPIRE].
[54] S.P. Martin and M.T. Vaughn, Two loop renormalization group equations for soft
supersymmetry breaking couplings, Phys. Rev. D 50 (1994) 2282 [Erratum ibid. D 78 (2008)
039903] [hepph/9311340] [INSPIRE].
[55] Y. Yamada, Two loop renormalization group equations for soft SUSY breaking scalar
interactions: supergraph method, Phys. Rev. D 50 (1994) 3537 [hepph/9401241] [INSPIRE].
[56] I. Jack and D.R.T. Jones, Soft supersymmetry breaking and niteness, Phys. Lett. B 333
(1994) 372 [hepph/9405233] [INSPIRE].
[57] S. Kachru, Fluxes and moduli stabilization, website, video and lecture notes, (2012).
[58] O. DeWolfe, S. Kachru and H.L. Verlinde, The giant in aton, JHEP 05 (2004) 017
[hepth/0403123] [INSPIRE].
[59] O. Aharony, Y.E. Antebi and M. Berkooz, Open string moduli in KKLT compacti cations,
Phys. Rev. D 72 (2005) 106009 [hepth/0508080] [INSPIRE].
[60] A.M. Uranga, Comments on nonsupersymmetric orientifolds at strong coupling, JHEP 02
(2000) 041 [hepth/9912145] [INSPIRE].
[61] R. Kallosh and T. Wrase, Emergence of spontaneously broken supersymmetry on an
antiD3brane in KKLT dS vacua, JHEP 12 (2014) 117 [arXiv:1411.1121] [INSPIRE].
[62] E.A. Bergshoe , K. Dasgupta, R. Kallosh, A. Van Proeyen and T. Wrase, D3 and dS, JHEP
05 (2015) 058 [arXiv:1502.07627] [INSPIRE].
[5] I. Bena , G. Giecold and N. Halmagyi , The backreaction of antiM 2 branes on a warped Stenzel space , JHEP 04 ( 2011 ) 120 [arXiv: 1011 .2195] [INSPIRE].
[6] A. Dymarsky , On gravity dual of a metastable vacuum in KlebanovStrassler theory , JHEP [7] I. Bena , G. Giecold, M. Gran~a, N. Halmagyi and S. Massai, The backreaction of antiD3 branes on the KlebanovStrassler geometry , JHEP 06 ( 2013 ) 060 [arXiv: 1106 .6165] [8] J. Blaback , U.H. Danielsson , D. Junghans , T. Van Riet , T. Wrase and M. Zagermann , [9] I. Bena , M. Gran~a, S. Kuperstein and S. Massai, AntiD3 branes: singular to the bitter end , Phys. Rev. D 87 ( 2013 ) 106010 [arXiv: 1206 .6369] [INSPIRE].
[10] F.F. Gautason , D. Junghans and M. Zagermann , Cosmological constant, near brane behavior [11] G. Giecold , F. Orsi and A. Puhm , Insane antimembranes?, JHEP 03 ( 2014 ) 041 [12] I. Bena , D. Junghans , S. Kuperstein , T. Van Riet , T. Wrase and M. Zagermann , Persistent [13] I. Bena , M. Gran~a, S. Kuperstein and S. Massai , PolchinskiStrassler does not uplift [14] I. Bena , M. Gran~a, S. Kuperstein and S. Massai, Tachyonic antiM 2 branes , JHEP 06 [15] I. Bena , A. Buchel and O.J.C. Dias , Horizons cannot save the landscape , Phys. Rev. D 87 [26] I. Bena , E. Gorbatov , S. Hellerman , N. Seiberg and D. Shih , A note on (meta)stable brane [27] S. Franco and A.M. . Uranga, Dynamical SUSY breaking at metastable minima from Dbranes at obstructed geometries , JHEP 06 ( 2006 ) 031 [ hep th/0604136] [INSPIRE].
[28] H. Ooguri and Y. Ookouchi , Landscape of supersymmetry breaking vacua in geometrically [29] B. Michel , E. Mintun , J. Polchinski , A. Puhm and P. Saad , Remarks on brane and antibrane [30] T. Damour , A new and consistent method for classical renormalization , Nuovo Cim. B 26 [46] P.G. Camara , L.E. Iban ~ez and A.M. Uranga , Flux induced SUSY breaking soft terms , Nucl.
[47] P. McGuirk , G. Shiu and F. Ye , Soft branes in supersymmetrybreaking backgrounds , JHEP [48] I. Jack and D.R.T. Jones , Nonstandard soft supersymmetry breaking , Phys. Lett. B 457 [49] L. Girardello and M.T. Grisaru , Soft breaking of supersymmetry, Nucl. Phys. B 194 ( 1982 ) [50] D.R.T. Jones , L. Mezincescu and Y.P. Yao , Soft breaking of two loop Henri Poincare 4 ( 2003 ) S111 [Braz . J. Phys . 34 ( 2004 ) 151].
[45] M. Gran ~a, T.W. Grimm, H. Jockers and J. Louis , Soft supersymmetry breaking in [63] R. Kallosh , F. Quevedo and A.M. Uranga , String theory realizations of the nilpotent goldstino , JHEP 12 ( 2015 ) 039 [arXiv: 1507 .07556] [INSPIRE].
[64] I. Garc aEtxebarria, F. Quevedo and R. Valandro , Global string embeddings for the nilpotent goldstino , JHEP 02 ( 2016 ) 148 [arXiv: 1512 .06926] [INSPIRE].
[65] J. Polchinski , Why trust a theory? Some further remarks (part 1) , arXiv: 1601 .06145 [66] U.H. Danielsson , S.S. Haque , G. Shiu and T. Van Riet , Towards classical de Sitter solutions