#### Constrained superfields on metastable anti-D3-branes

Received: April
Constrained super elds on metastable anti-D3-branes
Lars Aalsma 0 1 3
Jan Pieter van der Schaar 0 1 3
Bert Vercnocke 0 1 2
Open Access 0 1
c The Authors. 0 1
0 Celestijnenlaan 200D , bus 2415, 3001 Leuven , Belgium
1 University of Amsterdam , Science Park 904, 1098 XH Amsterdam , The Netherlands
2 Institute for Theoretical Physics , KU Keuven
3 Institute for Theoretical Physics Amsterdam, Delta Institute for Theoretical Physics
We study the e ect of brane polarization on the supersymmetry transformations of probe anti-D3-branes at the tip of a Klebanov-Strassler throat geometry. As is well known, the probe branes can polarize into NS5-branes and decay to a supersymmetric state by brane- ux annihilation. The e ective potential has a metastable minimum as long as the number of anti-D3-branes is small compared to the number of ux quanta. We study the reduced four-dimensional e ective NS5-brane theory and show that in the metastable minimum supersymmetry is non-linearly realized to leading order, as expected for spontaneously broken supersymmetry. However, a strict decoupling limit of the higher order corrections in terms of a standard nilpotent super eld does not seem to exist. We comment on the possible implications of these results for more general low-energy e ective descriptions of in ation or de Sitter vacua.
D-branes; Superstring Vacua; Supersymmetry Breaking
1 Introduction
The bosonic KPV potential
The fermionic KPV potential
Supersymmetry transformations
At the south pole
At the north pole
At the metastable minimum
Comments and conclusions
A Details on fermions
A.1 Projection matrix
A.2 Fermionic action
The fermionic action up to second order
Reduction to four dimensions
Mass matrix in four dimensions
Mass matrix at the poles
Mass matrix at the metastable minimum
Fermionic action: orientifold compatible gauge choice
A.3 Supersymmetry transformations
Introduction
De Sitter vacua are at the heart of any cosmological model as both the early and late
universe are well-approximated by a de Sitter phase. It is therefore of great importance to
understand the construction of de Sitter vacua in string theory and supergravity. However,
such constructions have proven to be a tremendous challenge. Kachru, Kallosh, Linde and
Trivedi (KKLT) provided a generic mechanism of moduli stabilization in Anti-de Sitter
and an uplift to de Sitter vacua in ten-dimensional string theory already in 2003 [1] and
by now, many di erent approaches for de Sitter compacti cations have been uncovered.
In contrast, the equivalent mechanism for de Sitter vacua in an e ective
foursuper elds [2{6]. By imposing constraints on super elds it is not only possible to describe
elds transforming non-linearly under the broken supersymmetry, but also to eliminate
unwanted degrees of freedom. General prescriptions for constrained super elds from linearly
transforming ones in a supergravity context were given in [7, 8]. For some recent reviews
of constrained super elds and their applications to cosmology, see [9, 10].
Constrained super elds are often e ective descriptions of the low-energy excitations.
For example, in the context of four-dimensional spontaneous supersymmetry breaking the
massless goldstino can be packaged in a chiral super eld that satis es a nilpotency
constraint. This constraint arises after the bosonic superpartner of the goldstino (the
sgoldstino) becomes heavy enough to be integrated out [11{13]. As argued in [11] this can be
extended to multiple elds. In general, integrating out additional heavy degrees of freedom
results in extra constraints which describe the universal low-energy dynamics of the theory,
see also [7, 13].
It is of crucial importance to understand the embedding of constrained super elds in
a putative UV-complete description. Can we indeed realize large mass splittings such that
the constrained super elds correspond to a good approximation of the relevant low-energy
physics? This question is especially important when considering cosmological in ation.
As one typically accesses high energy scales during in ation it is necessary to ensure that
elds eliminated by the constraints have large enough masses to be integrated out.
Otherwise, a constrained super eld description will be invalid.
An important condition for obtaining universal (UV insensitive) couplings to the
goldstino, and standard constrained super eld descriptions, is that the masses of the heavy
superpartners should be large compared to the supersymmetry breaking scale. If that
condition is not ful lled, the constraints are higher-order and depend on the masses of the
heavy elds [14]. This issue was recently reconsidered featuring global supersymmetry [15]
and supergravity [16]. Those authors studied the emergence of the constraints by
integrating out massive elds, instead of imposing the constraints by hand. In [16] the corrections
to an in ationary model with two super elds were analyzed. One super eld was used to
describe spontaneous supersymmetry breaking and a second one contained the in aton
and its superpartner. The UV physics is described by a supergravity model with
additional heavy super elds and supersymmetry is broken by an O'Raifeartaigh-mechanism.
This particular UV model did not allow for an exact nilpotent super eld description,
because the strict in nite mass limit of the sgoldstino that would decouple its uctuations as
in [12] does not exist. Instead, corrections due to the
nite sgoldstino mass during in
ation signi cantly limit the range of parameters for which an e ective nilpotent description
is available. It is not clear whether more generic UV models have similar restrictions on
taking the large sgoldstino mass limit.
In this paper we take a step back from in ation and study how universal the description
of de Sitter vacua with a nilpotent super eld is, in the context of string theory. We build on
supergravity and string theory. The uplift term of the KKLT mechanism is generated by
anti-D3 branes in a Giddings-Kachru-Polchinski (GKP) background [17]. This uplift is
an example of the generic string theory mechanism of supersymmetry breaking by branes
in backgrounds with
uxes. If the anti-D3-brane indeed breaks supersymmetry
spontaneously [18{20] it should be possible to package a worldvolume fermion into a nilpotent
super eld describing the goldstino. This expectation was con rmed explicitly by putting
The e ective description for the
rst constrained super eld models in the context
of KKLT arises by explicitly putting the anti-D3-brane on top of the orientifold plane.
To answer the question if a constrained super eld description of de Sitter vacua is still
appropriate in a more general background, we remove the orientifold projection. One of
us initiated this study with Kallosh and Wrase for a ten-dimensional at background [24]:
the non-linear transformations for all massless worldvolume elds (vector, scalars, fermion)
can indeed be described by constrained multiplets.
The full understanding of anti-D3-branes in ux backgrounds should introduce
corrections to the description in the at background of [24]. Anti-D3-branes at the bottom of a
warped throat can polarize into NS5-branes under the in uence of background
In this paper we show that one source of corrections is due to such polarization e ects.1
We write down the supersymmetric version of the action for the polarized brane and
consider small uctuations around the metastable minimum from the four-dimensional point
of view. This reveals that supersymmetry is indeed, to leading order in
uctuations,
nonlinearly realized at the minimum. The central question is at what scale the rst leading
corrections to the standard four-dimensional non-linear description appear. We nd that
this scale is not set by the mass of the scalar uctuations, but is instead smaller by a factor
background. Interestingly, the strict limit that would decouple these corrections does not
The rest of this paper is organized as follows. We review the potential for polarized
anti-D3 branes from the perspective of the NS5 worldvolume theory in section 2, with
special emphasis on the expected scale at which this description is valid. In section 3, we
construct the supersymmetric completion of the polarized NS5-brane action. We analyze
the four-dimensional supersymmetry transformations in section 4. Finally, in section 5 we
comment on our
ndings and the relation to the use of anti-branes in de Sitter uplifts.
Appendix A contains a technical derivation of the fermionic terms in the action and the
supersymmetry transformations, based on the S-dual D5-brane action in a ux background
The bosonic KPV potential
Let us start with a short review of some of the results of Kachru, Pearson and Verlinde
(KPV) [18]. KPV added p anti-D3-branes to the warped deformed conifold geometry of
Klebanov and Strassler [30]. The throat of this geometry is supported by M units of ux
through the A-cycle and K units through the B-cycle.
F3 = M
H3 =
The Klebanov-Strassler geometry is an example of a GKP background [17] that experiences
a high degree of warping near the bottom of the throat in the six-dimensional geometry.
1In recent years the literature has been divided on whether metastable anti-D3 probes are robust beyond
probe level, for recent work see [25{28] and references. We want to discuss the appearance of non-linear
supersymmetry and possible corrections rst at probe level and do not discuss back-reaction in this paper.
e4A0 p 3=gs. At the
north pole (
Q( ) at the metastable minimum ( min). This con guration can decay non-perturbatively to the
supersymmetric minimum at the south pole (
), where the p anti-D3-branes annihilated and
the nal con guration contains (M
p) D3-branes.
Since probe anti-D3 branes in the Klebanov-Strassler background feel a net force towards
the bottom of the throat we can describe the relevant physics by focusing on the region
near the tip of the throat, with topology R4
S3. The metric near the tip is [18]
ds2 = e2A0
dx dx + gsM b02(d 2 + sin2
rameter of the deformed conifold. Anti-branes carry opposite charge with respect to the
supersymmetric background, breaking all supersymmetry. By brane polarization [31], the
anti-branes can blow up to form an NS5-brane wrapping an S2 inside the S3. Depending on
the S2, or shrinks all the way to the opposite south pole of the S3, brane- ux annihilation
takes place and the
nal con guration becomes supersymmetric, see
gure 1. Since the
non-supersymmetric and supersymmetric states are continuously connected by moving the
NS5-brane from the north pole to the south pole on the S3, one expects the breaking of
supersymmetry in the metastable vacuum to be spontaneous and supersymmetry to be
realized non-linearly. We opt to describe the dynamics from the perspective of the e
ective NS5-brane worldvolume theory. The bosonic action describing the NS5 worldvolume
theory is given by2
SNS5 =
We wrote the DBI term in terms of the metric components Gk spanned by the (anti-)D3
brane coordinates (Minkowski coordinates plus possibly motion in ) and G?, spanned by
2The e ective action on the NS5-brane is obtained by S-duality of the D5-brane DBI theory. Strictly
speaking this description is therefore only valid for large gs, but some (supersymmetric) properties and
structures are expected to be invariant.
the coordinates on the S2. The form elds in the action are
2 F2 = 2 F2
dB6 =
charge p carried by the NS5-brane:
F2 = 2 p ;
min =
and F3 = dC2.
becomes3
over the S2:
3Where we added a constant to the action such that the potential is zero at the supersymmetric minimum.
where for later convenience we introduced the position-dependent angle ( ), which takes
1 at the poles of the S3:
cos( ( ))
We plot the potential in gure 1. It has a metastable minimum for relatively small values
We are speci cally interested in the e ective dynamics in the angular direction
the S3, which is transverse to the NS5-brane wrapped on an S2 inside the S3. The action
S =
F2 =
? = b20M
Q describes the e ective D3-charge at position .
From the action one can nd the potential (the Hamiltonian at zero momentum):
V ( ) =
3 e4A0 pQ2 + P 2 (1 + cos( )) ;
By expanding for small values of
up to fourth order we nd
Around the north pole of the S3 (
In a moment we will expand the action in
around the minimum
general expansion of an arbitrary potential in
up to cubic order can be
) = V ( min) +
Using the standard normalization in four dimensions, a scalar eld
and the cubic
coupling 3 have mass dimension 1. For
< m2 = 3 the quadratic term is a
good approximation of the relevant physics, but for larger
uctuations the cubic term
dominates, signaling a breakdown of the quadratic approximation. For natural couplings
is some (high-energy) cut-o
scale, this would just restrict the
uctuations to values below m
, but for `unnaturally' large cubic couplings the
quadratic approximation would only be valid for uctuations signi cantly smaller than the
mass scale m .
The second situation is exactly what we observe in the Klebanov-Strassler throat.
Expanding (2.11) up to cubic order around
min we obtain
) =
From this expression, we see that 3=m
larger than the quadratic coupling. In the remainder of this article, we are interested
in the quadratic approximation. We are then forced to restrict to
uctuations that are
not only small compared to the dimensionless mass parameter m
= 2 2=b20, which is of
order one, but small compared to a dimensionless parameter set by the eld value in the
metastable minimum:
The importance of this basic observation will become clear when we discuss the corrected
supersymmetry transformations in the metastable vacuum.
In the rest of this paper, we continue with the discussion of the fermions on the NS5
worldvolume. We will be concerned with the leading behaviour at a xed but small value of
p=M . Then we consider the small
expansion, and discuss small eld
uctuations around
a xed background position for small . We consider up to quadratic order in the scale
From the action it is straightforward to obtain the potential for the canonically
normalized eld. For small uctuations around a minimum at
1, the kinetic term gets a
with the potential
V~ ( ) = p 1 pQ( )2 + P ( )2
We will arrange the kinetic terms of the fermions to have the same constant prefactor (for
small uctuations at least), such that we can consistently compare mass scales.
SNS5 =
Since we expect the metastable minimum to break supersymmetry spontaneously, there
should exist an associated massless goldstino. For a single anti-D3-brane on top of an
orientifold plane, the goldstino was identi ed as the 4d fermion on the worldvolume of the
anti-brane, which is a singlet under the SU(3) holonomy of the 6d internal space [21, 22].
Removing the orientifold plane, we now want to revisit the situation for the polarized
NS5brane. Based on the physical picture of the previous section, we expect the e ective 4d
worldvolume description to reduce to the known results for p anti-D3-branes at the north
pole and M
p D3-branes at the south pole, both probing the GKP background.
The fermionic action up to second order
Just as for the bosonic action, we formally obtain the fermionic NS5-brane worldvolume
action from S-duality of a D5-brane. The action up to quadratic order in fermions is given
by [29] (notice that we have a background with a constant dilaton)
det(g + 2 gsF ) (1
= g
= r
We only included terms in the action that are non-zero at the tip of the throat, because we
are not interested in dynamics taking us away from the tip (we dropped terms with
form and one-form
eld strengths). The indices m; n are ten-dimensional curved indices,
; indicate worldvolume indices. To avoid confusion with the equations below, we wrote
the pullbacks of gamma matrices on the worldvolume with hats: ^
we underline tangent space indices (m; n : : :). The fermion is a doublet of Majorana-Weyl
spinors with positive chirality.
We now use the speci c embedding of the NS5-brane of the previous section and use
the leg structure of the three-forms to simplify the expressions. The F3
ux is fully along
the S3 spanned by ( ; ; ) while H3 is orthogonal to F3 in the internal space. This means
we can drop H3 terms with legs along the worldvolume of the NS5-brane. Also we will
drop the terms with @
coming from the pullbacks of gamma matrices, as those do not
contribute to the mass matrix. We only highlight the main points of the calculation here.
For more general expressions and more detailed information, see appendix A.
The combination in right brackets of (3.1) gives:
= (M~ 1
cos(2 )Fmnp 3 + (1 + sin2 )gs 1Hmnp 1
with the position-dependent angle
de ned in (2.10).
It is important for our calculations to note that
NS5 is o -diagonal. As explained in
appendix A, at the tip of the deformed conifold, this projector takes on a fairly simple form:
NS5 =
45 sin( )) :
We still need to gauge x the kappa-symmetry on the brane. We do this by taking the
gauge xing condition on the doublet
= ( 1; 2)
3 =
1 = 0 :
Now we can express the action in terms of the spinor 2 only. This gauge xing condition
is convenient due to its simplicity, but it is not suitable when one also wants to perform
an orientifold projection. The calculation for the mass matrix can also be done in a gauge
where we set (1 +
that this choice of gauge does not change the mass matrix.
We introduce the notation for the remaining spinor components
Taking care of the o -diagonal matrix NS5 and using that for a 10d Majorana-Weyl spinor
the only fermion bilinears that are non-zero have three or seven gamma matrices, we nd
SNS5 =
with d the volume element on the unit two-sphere.
The only terms that contribute to the mass matrix M are
M =
cos(2 )Fmnp
This is the mass matrix on the six-dimensional world volume. The reduction to four
dimensions could also pick up extra mass terms coming from the reduction of the kinetic
term [32]. To determine if these extra mass terms still allow for a massless fermion, we
have to make sure the internal piece of the modi ed Dirac operator together with the mass
matrix [(M~ 1
+ M] has a zero mode.
In the remainder of this section we show that this is indeed the case and the
lowest Kaluza-Klein modes reveal the existence of a massless fermionic mode, which we will
identify as the massless goldstino.
In the previous section we obtained the action for the worldvolume fermions from the
sixdimensional point of view. We now discuss the four-dimensional interpretation.
we perform the reduction to four dimensions, we will write
in terms of four fermions:
a singlet 0 and a triplet i under the SU(3) holonomy of the six-dimensional transverse
internal space. This decomposition can for instance be found in [22].
Let us rst focus on the reduction of the mass matrix. We observe that, up to angles
that parameterize the position of the NS5 on the S3, it is completely determined by the
ux of the background, which can be written in terms of the complexi ed three-form
= mij i+ j+ + m{| { | ;
G3 = F3
This immediately implies that the only relevant structure for the fermionic mass matrix
we have to reduce to four dimensions is the real part of the complex three-form:
M =
(cos(2 ) + cos( )) (G3 + G3)mnp
Up to the coordinate-dependent prefactor (cos(2 ) + cos( )), this is the known mass term
for anti-D3 branes in a supersymmetric background with
uxes that carry only D3-brane
charges, as reviewed in [22]. The general discussion of our mass terms also carries through
directly as in [22]. The background three-form is (2,1) and primitive, and therefore we nd
that the only non-zero contributions to the mass matrix come from the triplet:
where the mij are linear in the components of the background ux and
subscripts denote
4d Weyl spinors
massless, similar to a single anti-D3-brane that does not polarize [22].
The kinetic term of the fermions still contains a `modi ed Dirac operator'
= ((g + 2 gs 3F ) 1)
that could contribute to the mass matrix in four dimensions. We can ask whether there is a
fermion that remains massless and signals the spontaneous breaking of supersymmetry. The
ux is crucial. If we would reduce the Dirac operator on an S2 without worldvolume ux
F , this would leave no fermion massless, as the 2-sphere admits no covariantly constant
spinors. However, we have a non-zero worldvolume
ux F on the S2 that induces the
(anti-)D3 brane charge. This allows for the possibility that the gauge eld twists the Dirac
operator on the S2 such that the modi ed Dirac operator can have a zero mode on the
2-sphere, along the lines of [33]. If that zero mode agrees with the 0 direction, we can
0 as the four-dimensional goldstino, as was suggested in [19].
Instead of explicitly solving (M~ 1
mode from the dual perspective of the non-abelian gauge theory on the anti-D3 branes.
From this point of view the situation is more transparent because a reduction to four
dimensions is not needed. In the non-abelian theory all elds become matrix-valued. The
transverse scalars i have a potential that describes the brane polarization. One nds that
the local minimum of this potential occurs when the scalars take an irreducible
representation of SU(2) [18], which agrees with the metastable minimum of the wrapped NS5-brane.
In the non-abelian theory, the analogue of the abelian 2-sphere with coordinates ; is a
non-commutative fuzzy 2-sphere.
The non-abelian theory is studied in detail in [19], with a decomposition of the 10d
worldvolume fermion
(which is promoted to a matrix) to the 4d
0 (`gaugino')
and i (`modulini'), analogous to the abelian theory. For supersymmetry preserving ISD
couplings between i
G3- ux, the gaugino mass terms vanish and only the modulini are massive, in agreement
with our NS5 mass matrix M. An additional mass contribution might come from Yukawa
The scalars i have a vacuum expectation value in the metastable minimum such that the
Yukawa coupling can be viewed as an o -diagonal contribution to the mass matrix. To
nd the massless goldstino, we expand the fermions in terms of eigenfunctions on the fuzzy
sphere (the non-commutative analogue of spherical harmonics [34]). One
nds that the
leaving 0 massless. Higher (` > 0) modes correspond to a Kaluza-Klein tower [33] and can
be ignored when the radius of the fuzzy sphere is su ciently small. Clearly, ignoring ` > 0
modes we are left with an abelian truncation of the non-abelian fermionic action where
we can identify
0 as a goldstino. This veri es the idea that in this setup spontaneous
supersymmetry breaking should come with a massless fermion.
M =
Mass matrix in four dimensions
To facilitate comparison with similar treatments in the literature, we will now explicitly
compute the mass matrix M at the three relevant positions: the two poles of the S3 and
the metastable minimum at
min. As mentioned before, we can rewrite the mass
matrix in terms of the complexi ed three-form G3. The general form of the mass matrix
then becomes
(cos(2 ) + cos( )) (G3 + G3)mnp
We now give the four-dimensional reduction and discuss the fermionic mass matrix on the
positions of interest.
Mass matrix at the poles
The mass matrix M becomes
At the North pole,
we have M
= 0) =
= ) = 0 :
From the small
expansion of
In terms of G3 ux we have the mass matrix
( min) =
M =
These match earlier results for anti-D3 branes or D3 branes on GKP backgrounds derived
in [22, 35] (note that we are working in an S-dual frame compared to those references, so
one should take G3 !
igs 1G3 for comparison to those references.)
Mass matrix at the metastable minimum
To obtain the mass matrix in the metastable minimum, we expand cos( ) to lowest
nontrivial order and we evaluate this expression at the minimum:
M =
Supersymmetry transformations
In the previous section we argued that in the metastable minimum supersymmetry is
spontaneously broken by identifying the corresponding massless goldstino. This also suggests
that the e ective low-energy dynamics can be described in terms of a nilpotent
supereld [11]. In this section we analyze the supersymmetry transformations to verify this
picture and identify the leading corrections.
To begin we need the expressions for the supersymmetry transformations in non-trivial
ux backgrounds, which can be found in short in appendix A, adapted from [29].
Supersymmetry of the background requires that
(1 + i 2 0123) = 0
2 =
With a slight abuse of notation, we will write the 32-component Majorana-Weyl spinor
2 2. We have the following supersymmetry transformations:
(1 + ) and the operator
de ned as
0123 +, see eq. (A.12):
We do not write fermion terms in , as those result in transformations that take use beyond
the quadratic fermion order in the action.
More details on these transformations can be found in appendix A. From here, we can
already see the general form of the transformations around the poles, since
= 0 :
= 1 + : : : ;
= +1 + : : : ;
where the ellipses denote terms with eld uctuations. So around
= 0 we nd non-linear
transformations and at
linear ones.
To obtain four-dimensional supersymmetry transformations, in the end we always
decompose the spinor into the singlet 0 and the triplet i under the SU(3) holonomy.
Moreorientation of the S2 inside the transverse S3, corresponding to the superpartner of the
at the south pole where supersymmetry is restored. The other directions come
along for the ride and we will ignore them throughout.
We are also interested in the
supersymmetry transformations with parameter 0, the SU(3) singlet component of the
32-component Majorana-Weyl spinor , as this is the supersymmetry preserved by the
With all the relevant information in place, we present a summary of the
fourdimensional fermionic, scalar and gauge eld supersymmetry transformations at the di
erent locations of interest: both poles and most importantly the metastable minimum.
At the south pole
Let us rst analyze the south pole
= 1 +
and the reduction of the supersymmetry transformations to four dimensions gives
0 =
3 = p
~ = p
where we rede ned the scalar as follows.
and rescaled spinors as
~ =
(gsM b02)1=2 ;
! p12 . We conclude that, as expected, at
and ( 3; ) correspond to a chiral multiplet. If we would have included the other two
directions on the S2 that we now have ignored, they would form two additional chiral
At the north pole
At the (unstable) north pole we expect the e ective description to formally reduce to
the results for a supersymmetry breaking anti-D3-brane in a GKP background. We will
write the transformations to at most quadratic order in
uctuations. Since sin( ) =
O( 2); cos( ) = 1 + O( 4), we set cos
the results of [24].
order in the supersymmetry transformations. Then we indeed reproduce to quadratic order
We will expand the supersymmetry transformations up to the rst non-trivial order in
the elds. Then we only have to expand the operator
to rst order:
The supersymmetry transformations around
= 0 are
= 1
~ =
~ =
~ =
~ =
~ =
~ = A
the collection of all elds
= f ; ; A g
. We recognize the rst terms as the
standard non-linear transformations. By requiring the elds to transform non-linearly
under the supersymmetry we can perform appropriate
eld rede nitions of the spinors,
scalar and gauge eld, that x the transformations uniquely:
and we have the standard-looking transformations
With an additional rescaling of the spinors ~
2 , we then
nd the
following supersymmetry transformations in terms of the appropriate four-dimensional elds
~0 = 0 + O( 2)
~3 = 0 + O( 2)
~ = (~0
~ = (~0
We conclude that indeed, as anticipated by the physical interpretation in terms of
braneux decay, this seems to describe an exact non-linear realization of (broken) supersymmetry
when adding anti-D3-branes to the GKP background and ignoring the (higher order)
dynamics describing the polarization in the transverse S3 directions. This matches the results
for anti-D3 branes in supersymmetric backgrounds of [21{24]. Note that this (direct)
expansion of the theory around the north pole is only a formal result: since the scalar eld
sits at the maximum of its potential, this is an expansion around an unstable con guration.
At the metastable minimum
Now let us include the polarization dynamics and determine the transformations at the
true metastable minimum
the S3. We rst expand in
min, which should include corrections due to the dynamics on
and then in the uctuations around the metastable minimum.
The expansion for
around the metastable minimum is then
) =
captured by expanding
in powers of :
j =0 ;
where j =0 is given by (4.13).
We nd that after the eld rede nition (4.15) and the spinor rescalings the
transformations (4.16) are corrected by the -expansion (or equivalently
~ =
~ =
~ =
j =0
j =0
A j =0 +
The transformations in the metastable minimum become
~0 = 0
~3 = 0
~ = (~0
~ = (~0
The rst terms correspond to the standard non-linear transformations. Remember that the
expansion of
min is given by (4.21). We identify two types of corrections. First
terms are just proportional to (the square of)
p=M and re ect the shift towards
the metastable minimum. In fact, if we could ignore the eld
(as well as the spinor 3),
the probe limit would consistently reproduce a subset of the non-linear supersymmetry
transformations at the north pole. In other words, if the
elds were in nitely
massive, the probe limit takes you to the north pole and a constrained super eld description
of the goldstino and the gauge eld would be adequate.
However, it can be seen from (2.14) that the mass of the scalar
is always of the same
order of the potential energy scale in the metastable vacuum, so uctuations in
be decoupled. Interestingly the corrections that are proportional to
2 are all, except for
large and one should include (all) higher order terms. This is in line with the discussion of
section 2: at order
are forced to conclude that a strict decoupling limit in which the e ective description in
terms of non-linearly realized supersymmetry becomes UV independent does not exist. As
a consequence the validity of a constrained super eld description is restricted. Just how
restricted can be estimated by observing that the corrections become comparable to the
shift term when the uctuation
is of order p=M or equivalently
not come as a complete surprise, since this is where the expansion in
min. This should
breaks down. We
can translate this into a corresponding mass scale using the potential, giving a scale that is
In other words, the description in terms of non-linearly realized supersymmetry seems to
break down at scales far below the mass scale of relevance in the metastable vacuum.
Closing this section, we would like to make a nal comment. It is important to realize
that one should not perform an additional eld rede nition at
min that would remove the
leading corrections. For instance, an additional eld rede nition of 3 that removes the
corrections at the same time modi es the form of the transformations at the north pole and
also changes the fermionic mass matrix for 0. In this case, the rede ned spinor cannot be
identi ed with the massless goldstino.
Comments and conclusions
Constrained super elds provide a powerful technique in the context of a universal (UV
insensitive) low-energy description of spontaneously broken supersymmetry. A crucial
requirement is a stable and large enough hierarchy between the scale of the elds that are
projected out by the constraints and the relevant scale of the low energy e ective theory. In
some cases such a hierarchy might not be achievable, precluding the existence of a standard
constrained super eld description. In general however the appropriate constrained
supereld description is valid up to some energy scale that should be identi ed and compared
to the supersymmetry breaking scale. In this work we studied the leading corrections to
the nilpotent goldstino super eld description of anti-D3-branes in the GKP background
from polarization e ects. Our main observation is that the (non-linear) supersymmetry
transformations in the metastable vacuum receive corrections that cannot be `decoupled'
and actually become large in the probe limit p ! 0.
To arrive at that result we constructed, to leading order in the elds, the
supersymmetric completion of the e ective theory on an NS5-brane wrapped on an S2 inside the
transverse S3 at the tip of the KS throat geometry of [18]. We identi ed the massless
goldstino of spontaneous supersymmetry breaking as well as the gauge eld and transverse
that describes the position of the S2 inside the S3. In the absence of an
orientifold plane that projects out the bosonic degrees of freedom, they should also transform
non-linearly. In the metastable state we again identi ed
0, the singlet under the SU(3)
holonomy of the `internal' space, as the massless goldstino associated with the
spontaneously broken supersymmetry. We argued this from the non-abelian point of view, which
1). From the abelian
perspective this should correspond to twisting the Dirac operator with a gauge
the 2-sphere, as was done in [36]. A full treatment of the modi ed Dirac operator on the
4d reduced abelian NS5-brane should also reveal this zero mode at the position of the
metastable minimum. We hope to come back to this question in future work.
We found that uctuations of the
scalar eld around the metastable minimum cannot
be decoupled.
Moreover, corrections to the non-linear supersymmetry transformations
become large at a scale far below the mass scale set by the scalar
uctuations in the
brane number and the background
ux. This limits a
nite parameter window where
an e ective low-energy description of the metastable vacuum in terms of a constrained
super eld is appropriate.
This might not come as a total surprise.
When the source of spontaneous
supersymmetry breaking is intrinsically higher-dimensional, it might not admit any low-energy
description in terms of (simple) constrained super elds. This is clearest for more energetic
uctuations around the metastable minimum, with
p=M . Those uctuations are not
localized around the metastable minimum, as they exceed the energy di erence between
the metastable state and the north pole (left maximum in gure 1). However, they are still
localized on the northern hemisphere of the S3, as they have less energy than the
absolute maximum of the potential. Those uctuations describe full 6-dimensional uctuations
around the nilpotent super eld description of anti-D3 branes, governing the non-linear
transformations around the north pole
further will invalidate the non-linear description altogether, and will lead to a restoration
of the linear transformations by higher-dimensional excitations.
The uctuations we study in this paper are of a di erent nature. They capture
excitations very close to the metastable minimum and obey
p=M . They can be captured
in a four-dimensional language (albeit not with standard constrained super elds).
Determining the relevant uctuations in the KK reduction to four dimensions is subtle, since we
discussed two di erent descriptions with opposite regimes of validity. The polarized
NS5brane point of view is only valid for a large S2 and is hence intrinsically 6-dimensional.
In section 3.2, we argued however from the dual non-abelian anti-D3 point of view that
the set of lowest mass states of the KK spectrum in four-dimensions contains the
massless goldstino.
It is straightforward to check that the requirement
p=M is a direct consequence of
the relevance of higher order terms in the DBI action around the metastable vacuum. The
expansion of the polarization potential around the metastable minimum (2.14) shows that
the higher order terms become important when
p=M , as we also concluded
from the supersymmetry transformations. From the low-energy e ective eld theory point
of view the theory becomes strongly coupled as soon as
Our observations appear to be in line with the discussion of [14]. The mass of the
around the minimum of the potential is in fact of the same order as the
supersymmetry breaking scale, as can easily be seen from (2.14)
m2 =
As explained in [14], integrating out massive elds with masses of the order of the
supersymmetry breaking scale does not lead to universal couplings of the goldstino and instead
give rise to generalized holomorphic constraints on super elds. The UV dependence in our
setup becomes apparent at scales
correct the supersymmetry transformations. Whether and how this can be described in
terms of generalized (higher order) constrained super elds, or in another approach such as
the `goldstino brane' [37, 38], is a question we hope to come back to in the future.
Let us nally brie y elaborate on what the general consequences of our ndings might
be in the context of string cosmology. Following the arguments of [14], to allow for a
standard universal nilpotent super eld description one would require a stable hierarchy
between the scale of supersymmetry breaking and the mass of the transverse scalar .
In the original KKLT scenario, the scale of supersymmetry breaking is set by the uplift
energy of the metastable anti-D3 brane and hence seems to remain of the order of the mass
uctuations around the metastable vacuum. As a consequence the uplift with
p metastable polarized branes might lead to a similar breakdown of a putative universal
constrained super eld description at energies far below the supersymmetry breaking scale.
An e ective description of the metastable minimum by nilpotent super elds all the way up
to the supersymmetry breaking scale with polarized anti-branes would require a version
of the KKLT mechanism where the supersymmetry breaking scale and the uplift energy
can be decoupled. Broad classes of such models are available: for instance in [39], or
antibrane uplifts of an AdS minimum where supersymmetry is already broken, as in the Large
Volume Scenario and related work [40{42]. We hope to address some of these questions in
future work.
Acknowledgments
We thank Daniel Baumann, Eric Bergshoe , Nikolay Bobev, Ben Freivogel, Thomas
Hertog, Dan Roberts, Gary Shiu, Hagen Triendl, Thomas Van Riet for discussions; Riccardo
Argurio, Luca Martucci and Timm Wrase for feedback on a draft version of the paper;
and Renata Kallosh and Timm Wrase for collaboration on related work. BV thanks the
Galileo Galilei Institute for Theoretical Physics for hospitality and the INFN for partial
support during the completion of this work. BV was supported during the initial stages
of work by: the European Commission through the Marie Curie Intra-European fellowship
328652-QM-sing and Starting Grant of the European Research Council (ERC-2011-SrG
279617 TOI). Currently, BV is supported in part by the Interuniversity Attraction Poles
Programme initiated by the Belgian Science Policy (P7/37), by the European Research
Council grant no. ERC-2013-CoG 616732 HoloQosmos and the KU Leuven C1 grant
Horizons in High-Energy Physics. This work is part of the Delta ITP consortium, a program of
the Netherlands Organisation for Scienti c Research (NWO) that is funded by the Dutch
Ministry of Education, Culture and Science (OCW). The work of LA and JPvdS is also
supported by the research program of the Foundation for Fundamental Research on Matter
(FOM), which is part of the Netherlands Organization for Scienti c Research (NWO).
Details on fermions
In this appendix we review and apply the relevant details of the fermionic action of a
Dp-brane of [29, 43, 44], its supersymmetry transformations and gauge
xing. We take
the results for a D5 brane with worldvolume
ux in the S-dual background to
KlebanovStrassler. We follow the conventions of [29]. For easy comparison with the literature on
gauge- xed fermionic D-brane actions, we keep this appendix wholly in that `D5-frame'
and we adapt notation slightly to match as much as possible the related work for Dp-branes
at space [45] used in the recent literature on non-linear supersymmetries on anti-D3
branes [22{24, 46].
To transform the results of this appendix (`app') to the expressions used in the text,
one has to apply the following S-duality rules to the NS5-frame:
H3app =
= (gs 1)text ;
Projection matrix
We obtain the matrix
D5 from [29]:
We have +
= 1 and the relation
denote pull-backs on the worldvolume ^
= @ XM
(F ) =
+( F ). Note that hats on gamma matrices
D5 =
(D05) =
We will split the eld and the metric in a four-dimensional part (along the D3 worldvolume)
and a transverse part along the two-sphere as:
F = F
It is not hard to see that the matrix in the projector splits as:
+ =
2 F ?
8 F 1 2 F 3 4
The ellipses indicates terms higher order in elds and indices have been raised and lowered
with the metric G
k and XI are the transverse coordinates. This is the straightforward
covariantization of the kappa-symmetry matrix for a D3-brane.
+? = cos( )
+k =
where Greek letters still refer to worldvolume indices, but we make a split: the middle of
The calculation of the term
+? follows straightforwardly from the discussion of
secF ? =
Q( )volS2 :
The four-dimensional part of the projector parallels that of the projector dubbed
appendix of [24]. Note that we only consider the bosonic terms, as fermionic terms in
would take us beyond the quadratic fermionic order in the action. The result for +k is
+k =
+? =
+k =
det(G? + F ?)
det(Gk + F k)
( ; : : : = 4; 5).
tion 2, with
Fermionic action
and writing
We split the terms not involving a covariant derivative along the four-dimensions and the
two-sphere as
k = [G
M? = [(G? + 2 gs 3F ) 1]
= M
We nd (using ; for directions on the two sphere, and ; for four-dimensions)
k =
M? = sin2( )
Hmnp 3 + e Fmnp 1
cos( ) =
det(G? + F )
sin( ) =
det(G? + F )
The signs in these last two equations are chosen for later convenience.
Now we use that the ux H3 is fully along S3 and F is along S2, while F3 is orthogonal.
So the non-zero terms in Mk; M? are
k =
M? = sin2( )
Hmnp 3 + e Fmnp 1
which gives the result (3.3).
(as they are higher order in the action), we get
From (A.2) and (A.12) we nd that for vanishing F and neglecting the derivative terms
D5 =
Now we use that for Majorana-Weyl bilinears only terms with three or seven gamma
matrices are non-zero.
= 0
for n 2= f3; 7g
We now see that the last term in M? will not contribute at all and we nd
cos(2 )Hmnp +
cos(2 )Hmnp + cos( )e Fmnp 0123
mnp = 0
mnp 3 =
mnp 1 =
2 Hmnp
2 cos( ) Fmnp
Hmnp + e cos( )Fmnp
We then nd after some algebra that
Where we again used (A.23) to eliminate some terms. The total mass matrix is then
cos(2 )Hmnp + e cos( )Fmnp 0123
= (?6F )mnp ;
with ?6 the Hodge star operator on the six-dimensional internal manifold. This yields the
nal result (3.8):
cos(2 )Hmnp + e cos( )Fmnp 0123
Fermionic action: orientifold compatible gauge choice
For completeness, we show that taking the alternative gauge choice
D5) = 0
1 =
0123(cos( ) + sin( ) 45) 2 ;
to x the kappa-symmetry we obtain the same mass matrix. This gauge choice is useful
when one also wants to perform an orientifold projection, which has to be compatible with
the gauge xing condition. Using this condition, we can write the terms appearing in Mk
and M? completely in terms of
With the identity (F
matrices, we nd
M =
M =
The elds on the brane enjoy a combination of supersymmetry transformations,
kappasymmetry with spinorial parameter
and di eomorphisms (we leave out the possibility of
gauge transformations of the gauge eld). To linear order in the fermions , these are:
Xm =
= 0 :
2 =
XI =
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As explained in [29, 45], we can x the gauge redundancy in the following way. We x
this remains valid under the combined transformation
The di eomorphism invariance can be xed by requiring static gauge, such that X
= 0.
The background spinor obeys
2 = 0 ;
We will denote the transverse scalars by XI and with slight abuse of notation
Then the SUSY transformations after xing the kappa gauge that leave the quadratic
action (3.7) invariant are (see also [29])
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