#### Brane SUSY breaking and the gravitino mass

HJE
Brane SUSY breaking and the gravitino mass
so that the component @ 0 1
0 Hachioji , Tokyo 192-0397 , Japan
1 Department of Physics, Tokyo Metropolitan University
Supergravity models with spontaneously broken supersymmetry have been widely investigated over the years, together with some notable non-linear limits. Although in these models the gravitino becomes naturally massive absorbing the degrees of freedom of a Nambu-Goldstone fermion, there are cases in which the naive counting of degrees of freedom does not apply, in particular because of the absence of explicit gravitino mass terms in unitary gauge. The corresponding models require non-trivial de Sitter-like backgrounds, and it becomes of interest to clarify the fate of their Nambu-Goldstone modes. We elaborate on the fact that these non-trivial backgrounds can accommodate, consistently, gravitino elds carrying a number of degrees of freedom that is intermediate between those of massless and massive elds in a at spacetime. For instance, in a simple supergravity model of this type with de Sitter background, the overall degrees of freedom of gravitino are as many as for a massive spin{3=2 eld in at spacetime, while the gravitino remains massless in the sense that it undergoes null-cone propagation in the stereographic picture. On the other hand, in the ten-dimensional USp(32) Type I Sugimoto model with \brane SUSY breaking", which requires a more complicated background, the degrees of freedom of gravitino are half as many of those of a massive one, and yet it somehow behaves again as a massless one.
Supersymmetry Breaking; D-branes; Supergravity Models
1 Introduction
2
3
4
5
1
Massless propagation in de Sitter backgrounds
Massless gravitino in non-linear supersymmetry
Conclusions
Introduction
Akulov model [22], whose rst application to supergravity models was discussed in [23{25].
This framework remains of great interest, and its coupling to Supergravity was recently
reconsidered in [26{29], in the light of constrained super elds [30{33], and in the light of
non-BPS D-branes [34, 35]. As was explained in [25], one must introduce mass terms for
the gravitino and Nambu-Goldstone fermion and modify accordingly its supersymmetry
transformation in order to eliminate a cosmological constant term to arrive at at-space
models. The systematics of these constructions, examined in detail in [36{39], led
eventually to the no-scale models of [40], which also found a string realization in the presence of
internal uxes [41].
Typically, the gravitino becomes massive absorbing the degrees of freedom of a
NambuGoldstone fermion, a phenomenon that becomes manifest in a unitary gauge, but in the
ten-dimensional and six-dimensional orientifold models [42{51] with \brane supersymmetry
breaking" [52{57] supersymmetry is non-linearly realized and no explicit gravitino mass
term is allowed [58, 59], since the gravitino is a Majorana-Weyl fermion.1 On the other
1\Brane supersymmetry breaking" is a way to break supersymmetry by brane con gurations without
tachyon instabilities. For example, the simultaneous presence of branes and anti-branes in order to break
cosmological term [60, 61]. Since no gravitino mass term emerges in a unitary gauge, the
role of the Nambu-Goldstone fermion may appear confusing. Our aim here is to elaborate
on the fate of the degrees of freedom of Nambu-Goldstone fermions in this second class
of models. There are some interesting aspects in this story, since the curved spacetime is
crucial in a proper account of the related degrees of freedom.
Much e ort was devoted, over the years, to providing suitable de nitions of masses
and degrees of freedom in de Sitter spacetime, in particular in [62{66]. A highlight of
these works is that null-cone propagation takes naturally the place of masslessness in
at
space time [62], and the correspondence is illuminated by the special choice of \symmetric
density in a at slicing coordinate system for a de Sitter spacetime. In section 4 we
demonstrate that, in a simple model without gravitino mass term that will be introduced in the
next section, the gravitino remains massless in this sense, displaying its null-cone
propagation and also applying our new criterion. We then show that the gravitino in the Sugimoto
model remains massless, in the sense speci ed above, via our new criterion, which applies
insofar as the background can be understood as a cosmological evolution. The overall
number of degrees of freedom of the massless gravitino is the sum of those of a massless
gravitino in at spacetime and a Nambu-Goldstone fermion, which is half as many of those
of a massive spin{3=2 eld. The last section contains some concluding remarks.
2
Supergravity models with non-linear supersymmetry
Let us begin by considering pure supergravity
e
1
2
{ 2 {
1
4, which allows Majorana fermions, at the cost of
leaving aside some members of the supergravity multiplet. In this paper we follow the
conventions of [68]. This Lagrangian is invariant, up to a total divergence and higher
powers of gravitino, under the transformations
whose parameter is the Majorana spinor . The covariant derivative of the spinor eld
involves the spin connection ! and reads
HJEP04(218)
where local Lorentz indices are denoted by Latin letters. Let us now introduce a
Nambu
Goldstone fermion eld that provides a non-linear realization of supersymmetry
LNL =
e 2f 2
D
+ O ( )4 ;
whose lowest-order coupling to the gravitino
involves the supersymmetry current
e
m
=
=
1
D ;
m ;
D
=
Lcurrent = e
S =
f
S
:
= f :
(2.2)
(2.3)
(2.4)
(2.5)
(2.6)
(2.7)
(2.8)
(2.9)
1
2
1
4
e
1
2
The dimensionful quantity f de nes the scale of supersymmetry breaking and the
NambuGoldstone fermion eld transforms, to lowest order, according to
The total Lagrangian Lpure + LNL + Lcurrent
L =
1
that the gravitino is a simple massless spin{3=2 eld, which conforms to the fate of the
degrees of freedom introduced by Nambu-Goldstone fermion. The purpose of this paper is
to elaborate on what happens to the gravitino and the Nambu-Goldstone fermion in more
complicated models of broken Supersymmetry.
negative cosmological constant, so that
Lpure(massive) =
m ;
D
+
D
1
2
m
1
2
quires the introduction of a mass term for the Nambu-Goldstone fermion [70, 71],
Lmass =
e
1
The portion of the total Lagrangian that we are focussing on, Lpure(masive) +LNL +Lcurrent +
Lmass, is then invariant under the new gauge transformation, up to higher order terms or
contributions involving the other members of the gravity or matter multiplets. In
particular, a model without cosmological constant term would require tuning 2f 2 = C, and
thus linking the mass parameters of the gravitino and the Nambu-Goldstone fermion, m,
to the supersymmetry breaking scale f . In a model of this type admitting a at spacetime
background the gravitino has a mass term in the unitary gauge
= 0, and in fact it is a
conventional massive spin{3=2 eld.
Models that are similar to some extent to those in eq. (2.9) are realized in orientifolds
with brane supersymmetry breaking, and the ten-dimensional Sugimoto model is the
simplest example in this class. At tree level the supersymmetry present in the original type
IIB closed string sector is halved by the orientifold projection, as in the type-I superstring,
while in the open sector supersymmetry is completely broken, or non-linearly realized. The
low-energy e ective Lagrangian of the Sugimoto model was discussed in detail in [58, 59].
In Einstein frame and in unitary gauge it combines open and closed contributions,
where
Lclosed =
L = Lclosed + Lopen;
1
1
p
D
]
p
2
1
2
D
i
F
{ 4 {
describes the contribution from the closed sector, while
Lopen =
1
describes the contribution from open sector. Beginning from the closed sector,
model thus reduces to
L =
(
1
2 210 e R
1
2
Now the eld equation of the Nambu-Goldstone fermion yields
D
1
2
=
3
p
2
D
;
which identi es the dilatino and the gamma trace of the gravitino. We see that there is no
mass term for gravitino, and therefore this model belongs to the same class as the model
of eq. (2.9). There is an additional reason for the absence of a gravitino mass term: in
this ten-dimensional setting this eld is a Majorana-Weyl fermion eld, for which the mass
term simply does not exist. The goal of this paper is to elaborate on what happens to the
gravitino in this more complicated model, in the presence of non-trivial background elds.
3
Massless propagation in de Sitter backgrounds
It is well known that one can de ne a D-dimensional de Sitter spacetime starting from the
quadratic constraint
=
curvature tensors are described by the metric tensor g , independently of the coordinate
system, as
R
= H2 ( g
g ) ;
R
= R
= (D
1)H2g ;
(3.2)
with ; ; ;
1, and the scalar curvature is a constant, with R = D(D
1)H2. In this paper use two concrete coordinate systems, the symmetric coordinates and
the at slicing coordinates.
The symmetric coordinates [67], the result from a stereographic projection of the
generalized hyperboloid of eq. (3.1) to D-dimensional at spacetime, de ned via
In this fashion the metric tensor takes the form
with
= (1 + s) 1, while
=
x
1 + s
;
D =
1
is a corresponding vielbein. The Christo el symbols and the spin connection are then
ds2 = g dx dx =
dx0 2
+ e2Hx0 ij dxidxj ;
and a corresponding vielbein is
e m = diag 1; eHx0 ; eHx0 ;
:
{ 6 {
(3.3)
(3.4)
(3.5)
(3.6)
(3.7)
(3.8)
(3.9)
2
H r2eHx0 ;
(3.10)
(3.11)
(3.12)
1
4
! mn
mn =
=
mem
=
H2
2
H2
4
1
x
;
=
x
x
;
x ;
respectively, where
Since this coordinate system yields a conformally at metric, it is a convenient choice to
investigate null-cone propagation and conformal covariance of the eld equations.
The more conventional at slicing coordinates of de Sitter spacetime are de ned as
1
H
0 =
sinh Hx0 +
2
H r2eHx0 ;
i = xieHx0 ;
D =
cosh Hx0
1
H
where r2 = ij xixj with i; j = 1;
1. As is well known, this coordinate system covers
half of the whole de Sitter spacetime, the portion where space expands in time, which
a ords a natural cosmological interpretation.
In a Minkowski spacetime the Casimir operator
p^ p^ of Poincare group, where p^
is the generator of spacetime translation, is used to de ne the mass of the eld. Since
there is no such Casimir operator in the de Sitter SO(D; 1) group, there is no natural way
to de ne the mass of elds. Still, null-cone propagation remains a convincing criterion for
masslessness.
Let us now brie y recall the massless criterion via null-cone propagation in the
symmetric coordinate system [62]. To this end, let us consider a simple spin{1=2 spinor eld
in de Sitter background, for which
The eld equation can be rewritten as where is the rescaled eld de ned by
Lspin 1=2 =
e
D
w1=2 ;
{ 7 {
(3.13)
(3.14)
(3.15)
(3.16)
(3.17)
(3.18)
(3.20)
(3.21)
with the conformal weight of the spinor eld w1=2 = (1
D)=2. One is thus led to null-cone
propagation, and in this respect the spinor eld can be considered a massless eld in de
Sitter background. A similar conclusion can be reached starting from the second-order
eld equation.
D
D
=
g
D D
D(D
4
1) H2
=
3+D
2
(3.19)
which leads consistently to null-cone propagation and thus points again to the notion of a
massless spinor eld.
The action of this model is invariant under the de Sitter SO(D; 2) conformal
transformation. Since local Weyl invariance is enough for de Sitter conformal invariance [66], we
show local Weyl invariance of this model. The Weyl transformation is de ned as
e m
!
W e m
;
em
!
1
W e m
;
to be combined with
T
=
T00 =
1
2
/ e
D2 1 Hx0 ;
D
D
2
:
o
where
W is the local Weyl scaling factor. Using the de nition of the gamma matrices in
a non-trivial background,
=
mem , and the transformation of spin connection
! mn
! ! mn + (em e
n
en e
W ;
it is straightforward to see the invariance of the Lagrangian of eq. (3.15) provided one
chooses the conformal weight w1=2 = (1
D)=2. Since conformal invariance re ects the
absence of a speci c scale, it is natural to regard the spinor eld as a massless eld.
We now turn to propose a massless criterion for fermion elds in the at slicing
coordinate system, referring again to the above model of a spinor eld. The explicit form of
the eld equation in the at slicing coordinate is
D
2
1
H
= 0:
Here, we have also assumed that the eld depends only on time, since we anticipate the use
of some physical arguments related to the homogeneous expansion of the Universe. The
important property of the solution of this equation is that it is not oscillatory,
and consequently does not contribute to the energy density T00, where
The explicit form of T00 in our present setting
(3.22)
(3.23)
(3.24)
(3.25)
(3.26)
(3.27)
(3.28)
(3.29)
clearly indicates that it could be non-vanishing for an oscillatory solution
\rest mass" m. Hence, the non-oscillatory behavior of the Fermi eld can regarded as an
/ e imx0 with
indication of masslessness.
We can now propose a similar argument for the second-order eld equation,
anticipating some steps that will prove useful for the application to the gravitino in the next section.
The explicit form of the second-order eld equation is
+ (D
D
2
Let us now perform the rescaling of the eld as in eq. (3.18) with
for this choice of
is that the metric can be described, in conformal time , as
= eHx0 . The reason
ds2 =
2
d 2 + ij dxidxj
with
= 1=jH j = eHx0 , so that this setting compares naturally with eq. (3.5). The eld
equation now takes the simple form
If we assume that a solution should be nite in the limit of x0
which is consistent with eq. (3.24). In this respect, a constant solution for the rescaled eld
in the at slicing coordinate system can be a criterion of masslessness.
! 1,
should be a constant,
Consider the model of eq. (2.9) in the background de Sitter spacetime required by the
cosmological constant 2f 2. The eld equations of the gravitino and the Nambu-Goldstone
fermion are
In the unitary gauge = 0, these equations reduce to one eld equation with one constraint:
the combination that was discussed as a model of null-cone propagation without de Sitter
conformal invariance in [62]. Applying
to the eld equation gives another constraint
Eliminating and x using the explicit forms of the two constraints yields nally the eld equation
D 1
2
D
=
and if D 6= 2 the eld equation becomes
rescaled eld
with w3=2 = (3
D)=2, is now
In the symmetric coordinate system the explicit form of this eld equation for the
D
D
+ f
f
This is a well-known conformally covariant eld equation. It was described in [62], and
was derived in [72] in D = 4. The di erence with respect to the massless Rarita-Schwinger
equation lies in the special value of the coe cient of the second term, which would be one
for the massless Rarita-Schwinger equation rather than 2=D. Note that one can not take
to obtain it we had to perform a division by H2 in order to eliminate the x
a naive at limit H ! 0 in this conformally covariant eld equation, since in the process
term. As
a result, the number of degrees of freedom of this spin{3=2 eld undergoes a discontinuity
in moving between at and de Sitter spacetimes.
{ 9 {
(4.1)
(4.2)
(4.3)
(4.4)
(4.5)
(4.6)
(4.7)
(4.8)
(4.9)
carrying spin{1=2 degrees of freedom propagates on the null
cone. As a result, the other components of the eld ~ , with @ ~ = 0 (and also
~ = 0)
satisfy the eld equation
which means again null-cone propagation. Therefore, the gravitino of this model can be
regarded as massless in de Sitter spacetime, although its degrees of freedom combine those
of a massless spin{3=2 eld and of a massless spin{1=2 eld in a at Minkowski background.
Let us stress that this is a di erent state of a airs from the \partially massless eld" in [73].
In the at slicing coordinate system, under the assumption that the eld depends only
on time, the second order eld equation before rescaling the eld
D
D
= g
D D
1
2
R
1
4
R
(4.10)
(4.11)
(4.12)
(4.13)
(4.14)
(4.15)
(4.16)
(4.17)
(4.18)
(4.19)
takes the form
The constraint D
= 0 now reads
(D
5)(D
4
1
2
(D
5)(D
4
1) H2 0 = 0;
1) H2 i + H2 0i 0
H2 i
= 0 :
and a simple solution of the above constraint and
= 0 is
0 = 0;
i
i = 0:
In this fashion the second-order eld equation becomes
D
2
This equation is very similar to eq. (3.27) for the spinor eld. Rescaling the eld
according to
where
x
0
i
!
w3=2 i;
= eHx0 and w3=2 = (3
D)=2 is the conformal weight of vector-spinor eld, the
eld equation takes nally the very simple form
If we add the reasonable assumption that the solution should be
nite in the limit of
! 1, i can only be a constant. This means that gravitino is massless, exactly as was
the case for the spinor eld in previous section. The energy-momentum tensor of the eld
before rescaling can be recast in the form
T
=
1
4
g
D
D
D
D
1
4
and the explicit form of T00 in our present setting is
T00 =
1
We see that this energy density function vanishes for the non-oscillatory solutions, as
HJEP04(218)
pertains to elds with a vanishing \rest mass".
We are now ready to investigate the behavior of the gravitino in the more complicated
Sugimoto model. The relevant starting point was already introduced at the end of section 2,
in eqs. (2.18) and (2.19). A time-dependent background, which can be interpreted as a
cosmological evolution, was rst obtained in [60, 61] starting from the ansatz
ds2 =
e2B dx0 2
+ e2A ij dxidxj ;
for the metric, where A and B are functions only of x0. The gauge condition B =
which corresponds to a convenient choice for the time coordinate reduces the equations for
the background to
(4.20)
(4.21)
(4.22)
3 =4
(4.23)
(4.24)
(4.25)
This result of [60, 61] is a special instance of a class of solutions for Einstein gravity
minimally coupled to a scalar eld in the presence of an exponential potential proportional
to exp(
), which were studied in detail in [74]. The exponent selected by String Theory
for \brane supersymmetry breaking",
= 3=2 in the Einstein frame, has the key property
of separating two regions of solutions with widely di erent behavior. As pointed out in [75{
77], this value marks the onset of the \climbing phenomenon", according to which the scalar
eld has no other option, when emerging from the initial singularity, than climbing up the
potential before reaching a turning point and starting its descent. This is important, since
>>:>36 A
_ 2
8 + 9A_
B
_ _ =
8A
8A_ B_ + 36 A
3;
+
where dots indicate derivatives with respect to the dimensionless time
solution then reads
= p
E x0. The
Notice that
= eA can be regarded as the scale factor of cosmological expansion, while
de nes implicitly the cosmic time via
this dynamics sets naturally an upper bound on the string coupling (although there are
no indications of a similar bound on 0 corrections [78]), and suggests that the resulting
descent could help one model the onset of in ation. A climbing phase could provide the
impulse to start in ation [79{91], and in general an early fast-roll would introduce a
lowfrequency cut in the primordial power spectrum of scalar perturbations. This option was
explored over the years in di erent contexts [92{111], but it arguably obtains, in \brane
supersymmetry breaking", an enticing input from String Theory. There are also some
signs, away from the Galactic plane, of an encouraging comparison with the lack of power
apparently present in the low{` CMB [112{115].
There is an important link with the preceding case, since close to the turning point
for the climbing scalar eld, where _ = 0 and A = 0, the Universe in this case bears some
similarities with de Sitter spacetime with temporally constant A_ in
= eA. The Christo el
symbols and the spin connection for this background are
0
ij = p
E A_ e2A 2B
ij ;
0
respectively. The curvature tensors are described as
where in the application to the original model one should set D = 10.
The eld equations of the gravitino and the dilatino that follow from the Lagrangian
of eq. (2.18) are
R
=
R
=
h
E 0
and are to be combined with the constraint of eq. (2.19).
equation of the gravitino becomes
Let us now try to nd a solution with vanishing dilatino
= 0. In this case, the eld
(4.26)
(4.27)
(4.28)
(4.29)
(4.30)
(4.31)
(4.32)
(4.33)
0
0
1)
h
D
D
+
+
1
p
with the constraints
= 0, which follow from the eld
equation of the dilatino. Applying
to the eld equation gives the additional constraint
= 0, while the covariant divergence of the eld equation gives further additional
constraint R
It is simple to see that all of these four constraints are solved by 0 = 0 and i i = 0,
using the explicit forms of the Christo el symbols, the spin connection and the Ricci tensor.
Therefore, the problem is reduces to solve the eld equation
D
i = 0;
where the eld is subject to the constraint i i = 0. The explicit form of the second order
equation
D
D
i = 0 is
HJEP04(218)
i +
7A_ +
but rescaling the eld according to
it takes a very simple form:
i
!
Using the explicit expression of the background in eq. (4.24), the eld equation reduces to
whose solution can be expressed in terms of the incomplete -function, according to
dd2 2i +
3
4
2
3
3
2
d
d i = 0;
i = C(
1
) + C(
2
)
i i
1
4
;
16
where C(
1
) and C(
2
) are integration constants that satisfy the conditions iC(
1
) = 0 and
iC(
2
) = 0. They key point is that these solutions are not oscillatory. Moreover, if we
i
require a nite i at
! 1, the only option is
with iC(
1
) = 0, which points again to a massless gravitino in view of our new criterion,
namely no \rest mass" or no contribution to the energy density T00. In this respect, we
can conclude that the gravitino is massless in this cosmological vacuum, which somehow
replaces at spacetime in the Sugimoto model.
The next issue concerns the actual number of degrees of freedom that are carried by
the gravitino in the unitary gauge. To begin with, a Majorana vector-spinor eld
has
2[D=2] real degrees of freedom, or 320 in D = 10. Since
0 = 0 and i i = 0, this
number is readily cut to 256 in D = 10. Moreover,
is also a Weyl eld, so that number
i = C(
1
)
i
is again reduced by a factor of two, or to 128 degrees of freedom in D = 10. As usual, the
rst order Dirac equation that is left in unitary gauge
D
amounts, in our background, to a condition 0
= 0, and considering projection operators
P
= (1
i 0)=2, one is
nally left with 64 degrees of freedom. This number results
precisely from the 56 degrees of freedom carried by the original gravitino and the 8 degrees
of freedom carried by the Nambu-Goldstone fermion.
To reiterate, the gravitino in the Sugimoto model behaves as a massless eld in its
cosmological background of [60, 61], although it combines the degrees of freedom that a
massless gravitino would describe in a
at spacetime with those of the absorbed
NambuGoldstone fermion. The two elds recover separate lives, consistently with this
interpretation and known facts, if one turns o
the tadpole potential.
Moreover, this number is the half of the degrees of freedom that a massive spin{3=2 eld would have in a tendimensional Minkowski spacetime, as demanded by the orientifold projection that underlies the Sugimoto model.
5
Conclusions
We have investigated the behavior of the gravitino in a class of orientifold models [42{51]
with \brane supersymmetry breaking" [52{57], where supersymmetry is non-linearly
realized while a gravitino mass term is not allowed. These models require non-trivial spacetime
backgrounds, so that the notion of mass entails some subtleties and is di erent from the
more familiar case of a at spacetime. In a de Sitter spacetime null-cone propagation in
the symmetric coordinate system provides an accepted criterion of masslessness, and along
these lines we have proposed a new criterion for fermions that applies in the
at slicing
coordinate system. The new criterion rests on a cosmological interpretation, and applies
to more general background spacetimes. These include the spatially at geometries where
\brane supersymmetry breaking" brings along the climbing mechanism [75{77]. This could
provide the initial impulse to start an in ationary phase [79{91], and with a short in ation
this type of dynamics could have had some bearing on the low{` CMB anomalies [112{115].
We have investigated the gravitino mass in the Sugimoto model, with reference to the
background eld con guration of [75{77], which is more complicated than de Sitter
spacetime, arriving at conclusions that are reasonable with a massless gravitino. The overall
lesson is consistent with massless gravitinos in non-trivial backgrounds that entail di erent
numbers of degrees of freedom than in
at spacetime. Let us also stress that the
\cosmological" criterion of masslessness in de Sitter-like backgrounds also applies to a conformal
scalar
eld theory with non-minimal coupling to the scalar curvature, consistently with
our view. Even in this case, the eld equation for a suitably rescaled scalar eld allows
non-oscillatory solutions for which the energy density vanishes, although there is also a
di erent option, which is proportional to a D, where a = exp(Hx0) is the scale factor.
The latter contribution is typical of massless radiation.
There is a host of evidence that our Universe has never been exactly a at Minkowski
spacetime, and that now it is close to a de Sitter spacetime. Fields with unusual
numbers of degrees of freedom can thus be of interest, in principle, for Cosmology. A recent
investigation along these lines can be found in [116{119], for example.
Acknowledgments
The author would like to thank S. Ferrara for helpful discussions and A. Sagnotti for helpful
discussions, suggestions and a careful reading of the manuscript. The author would also
like to thank Scuola Normale Superiore for the kind hospitality and for partial support
while this work was in progress.
Open Access.
This article is distributed under the terms of the Creative Commons
Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
[4] J. Wess and B. Zumino, A Lagrangian Model Invariant Under Supergauge Transformations,
(1971) 452]. [INSPIRE].
Nucl. Phys. B 34 (1971) 632 [INSPIRE].
Phys. Lett. B 49 (1974) 52 [INSPIRE].
70 (1974) 39 [INSPIRE].
[5] J. Wess and B. Zumino, Supergauge Transformations in Four-Dimensions, Nucl. Phys. B
[6] A. Salam and J.A. Strathdee, Supergauge Transformations, Nucl. Phys. B 76 (1974) 477
[7] S. Ferrara and B. Zumino, Supergauge Invariant Yang-Mills Theories, Nucl. Phys. B 79
[8] A. Salam and J.A. Strathdee, Supersymmetry and Nonabelian Gauges, Phys. Lett. B 51
[9] P. Fayet and J. Iliopoulos, Spontaneously Broken Supergauge Symmetries and Goldstone
Spinors, Phys. Lett. B 51 (1974) 461 [INSPIRE].
[10] D.Z. Freedman, P. van Nieuwenhuizen and S. Ferrara, Progress Toward a Theory of
Supergravity, Phys. Rev. D 13 (1976) 3214 [INSPIRE].
[11] S. Deser and B. Zumino, Consistent Supergravity, Phys. Lett. B 62 (1976) 335 [INSPIRE].
[12] D.Z. Freedman and A. Van Proeyen, Supergravity, Cambridge University Press, Cambridge
U.K. (2012).
[13] P. Van Nieuwenhuizen, Supergravity, Phys. Rept. 68 (1981) 189 [INSPIRE].
Cim. 40 (2017) 1 [J. Phys. Conf. Ser. 873 (2017) 012014] [arXiv:1702.00743] [INSPIRE].
(2012).
(1973) 109 [INSPIRE].
[22] D.V. Volkov and V.P. Akulov, Is the Neutrino a Goldstone Particle?, Phys. Lett. B 46
[23] D.V. Volkov and V.A. Soroka, Higgs E ect for Goldstone Particles with Spin 1/2, JETP
Lett. 18 (1973) 312 [Pisma Zh. Eksp. Teor. Fiz. 18 (1973) 529] [INSPIRE].
[24] D.V. Volkov and V.A. Soroka, Gauge elds for symmetry group with spinor parameters,
Theor. Math. Phys. 20 (1974) 829 [Teor. Mat. Fiz. 20 (1974) 291] [INSPIRE].
[25] S. Deser and B. Zumino, Broken Supersymmetry and Supergravity, Phys. Rev. Lett. 38
(1977) 1433 [INSPIRE].
JHEP 09 (2015) 217 [arXiv:1507.07842] [INSPIRE].
[26] E. Dudas, S. Ferrara, A. Kehagias and A. Sagnotti, Properties of Nilpotent Supergravity,
[27] F. Hasegawa and Y. Yamada, Component action of nilpotent multiplet coupled to matter in
4 dimensional N = 1 supergravity, JHEP 10 (2015) 106 [arXiv:1507.08619] [INSPIRE].
[28] E.A. Bergshoe , D.Z. Freedman, R. Kallosh and A. Van Proeyen, Pure de Sitter
Supergravity, Phys. Rev. D 92 (2015) 085040 [Erratum ibid. D 93 (2016) 069901]
[arXiv:1507.08264] [INSPIRE].
[29] N. Cribiori, G. Dall'Agata and F. Farakos, From Linear to Non-linear SUSY and Back
Again, JHEP 08 (2017) 117 [arXiv:1704.07387] [INSPIRE].
[30] R. Casalbuoni, S. De Curtis, D. Dominici, F. Feruglio and R. Gatto, Nonlinear Realization
of Supersymmetry Algebra From Supersymmetric Constraint, Phys. Lett. B 220 (1989) 569
[31] A. Brignole, F. Feruglio and F. Zwirner, On the e ective interactions of a light gravitino
with matter fermions, JHEP 11 (1997) 001 [hep-th/9709111] [INSPIRE].
[32] Z. Komargodski and N. Seiberg, From Linear SUSY to Constrained Super elds, JHEP 09
(2009) 066 [arXiv:0907.2441] [INSPIRE].
[33] S.M. Kuzenko and S.J. Tyler, Relating the Komargodski-Seiberg and Akulov-Volkov actions:
Exact nonlinear eld rede nition, Phys. Lett. B 698 (2011) 319 [arXiv:1009.3298]
and de Sitter supergravity, JHEP 02 (2016) 080 [arXiv:1511.03024] [INSPIRE].
constrained super elds and matter in N = 1 supergravity, JHEP 11 (2016) 109
Spontaneous Symmetry Breaking and Higgs E ect in Supergravity Without Cosmological
Constant, Nucl. Phys. B 147 (1979) 105 [INSPIRE].
[37] A.H. Chamseddine, R.L. Arnowitt and P. Nath, Locally Supersymmetric Grand Uni cation,
HJEP04(218)
Phys. Rev. Lett. 49 (1982) 970 [INSPIRE].
Lett. B 118 (1982) 103 [INSPIRE].
212 (1983) 413 [INSPIRE].
[38] J. Bagger and E. Witten, The Gauge Invariant Supersymmetric Nonlinear -model, Phys.
[39] E. Cremmer, S. Ferrara, L. Girardello and A. Van Proeyen, Yang-Mills Theories with Local
Supersymmetry: Lagrangian, Transformation Laws and SuperHiggs E ect, Nucl. Phys. B
[40] E. Cremmer, S. Ferrara, C. Kounnas and D.V. Nanopoulos, Naturally Vanishing
Cosmological Constant in N = 1 Supergravity, Phys. Lett. B 133 (1983) 61 [INSPIRE].
[41] S.B. Giddings, S. Kachru and J. Polchinski, Hierarchies from
uxes in string
compacti cations, Phys. Rev. D 66 (2002) 106006 [hep-th/0105097] [INSPIRE].
[42] A. Sagnotti, Open Strings and their Symmetry Groups, in Cargese '87, Non-Perturbative
Quantum Field Theory, G. Mack et al. eds., Pergamon Press (1988), p. 521
[hep-th/0208020] [INSPIRE].
B 294 (1992) 196 [hep-th/9210127] [INSPIRE].
17 (2000) R41 [hep-ph/0006190] [INSPIRE].
(2003) 339] [hep-th/0204089] [INSPIRE].
[47] M. Bianchi and A. Sagnotti, Twist symmetry and open string Wilson lines, Nucl. Phys. B
[48] M. Bianchi, G. Pradisi and A. Sagnotti, Toroidal compacti cation and symmetry breaking
in open string theories, Nucl. Phys. B 376 (1992) 365 [INSPIRE].
[49] A. Sagnotti, A Note on the Green-Schwarz mechanism in open string theories, Phys. Lett.
[50] E. Dudas, Theory and phenomenology of type-I strings and M-theory, Class. Quant. Grav.
(1999) 38 [hep-th/9908023] [INSPIRE].
B 566 (2000) 126 [hep-th/9908064] [INSPIRE].
[54] C. Angelantonj, Comments on open string orbifolds with a nonvanishing B(ab), Nucl. Phys.
[INSPIRE].
arXiv:1711.11494 [INSPIRE].
[55] G. Aldazabal and A.M. Uranga, Tachyon free nonsupersymmetric type IIB orientifolds via
Brane - anti-brane systems, JHEP 10 (1999) 024 [hep-th/9908072] [INSPIRE].
Phys. B 615 (2001) 33 [hep-th/0107090] [INSPIRE].
[59] G. Pradisi and F. Riccioni, Geometric couplings and brane supersymmetry breaking, Nucl.
[60] E. Dudas and J. Mourad, Brane solutions in strings with broken supersymmetry and dilaton
tadpoles, Phys. Lett. B 486 (2000) 172 [hep-th/0004165] [INSPIRE].
[61] R. Blumenhagen and A. Font, Dilaton tadpoles, warped geometries and large extra
dimensions for nonsupersymmetric strings, Nucl. Phys. B 599 (2001) 241
[hep-th/0011269] [INSPIRE].
[62] S. Deser and R.I. Nepomechie, Gauge Invariance Versus Masslessness in de Sitter Space,
Annals Phys. 154 (1984) 396 [INSPIRE].
[63] S. Deser and A. Waldron, Null propagation of partially massless higher spins in (A)dS and
cosmological constant speculations, Phys. Lett. B 513 (2001) 137 [hep-th/0105181]
[INSPIRE].
[64] S. Deser and A. Waldron, Gauge invariances and phases of massive higher spins in (A)dS,
Phys. Rev. Lett. 87 (2001) 031601 [hep-th/0102166] [INSPIRE].
[65] T. Garidi, What is mass in de Sitterian physics?, hep-th/0309104 [INSPIRE].
[66] S. Deser and A. Waldron, Conformal invariance of partially massless higher spins, Phys.
Lett. B 603 (2004) 30 [hep-th/0408155] [INSPIRE].
[67] F. Gursey and T.D. Lee, Spin 1=2 Wave Equation in de Sitter Space, Proc. Nat. Acad. Sci.
49 (1963) 179 [INSPIRE].
[INSPIRE].
(1977) 189 [INSPIRE].
[68] Y. Tanii, Introduction to supergravity, SpringerBriefs in Mathematical Physics (2014).
[69] D.Z. Freedman, Supergravity with Axial Gauge Invariance, Phys. Rev. D 15 (1977) 1173
[70] B. Zumino, Nonlinear Realization of Supersymmetry in de Sitter Space, Nucl. Phys. B 127
[71] S. Ferrara and A. Van Proeyen, Mass Formulae for Broken Supersymmetry in Curved
Space-Time, Fortsch. Phys. 64 (2016) 896 [arXiv:1609.08480] [INSPIRE].
[72] A.O. Barut and B.-W. Xu, Conformal Covariance and the Probability Interpretation of
Wave Equations, Phys. Lett. A 82 (1981) 218 [INSPIRE].
(2001) 577 [hep-th/0103198] [INSPIRE].
[74] J.G. Russo, Exact solution of scalar tensor cosmology with exponential potentials and
transient acceleration, Phys. Lett. B 600 (2004) 185 [hep-th/0403010] [INSPIRE].
B 694 (2011) 80 [arXiv:1009.0874] [INSPIRE].
[76] A. Sagnotti, Brane SUSY breaking and in ation: implications for scalar elds and CMB
distortion, Phys. Part. Nucl. Lett. 11 (2014) 836 [arXiv:1303.6685] [INSPIRE].
[77] P. Fre, A. Sagnotti and A.S. Sorin, Integrable Scalar Cosmologies I. Foundations and links
with String Theory, Nucl. Phys. B 877 (2013) 1028 [arXiv:1307.1910] [INSPIRE].
[78] C. Condeescu and E. Dudas, Kasner solutions, climbing scalars and big-bang singularity,
JCAP 08 (2013) 013 [arXiv:1306.0911] [INSPIRE].
[79] A.A. Starobinsky, A New Type of Isotropic Cosmological Models Without Singularity, Phys.
Lett. B 91 (1980) 99 [INSPIRE].
241 (1980) L59 [INSPIRE].
[80] D. Kazanas, Dynamics of the Universe and Spontaneous Symmetry Breaking, Astrophys. J.
[81] K. Sato, Cosmological Baryon Number Domain Structure and the First Order Phase
Transition of a Vacuum, Phys. Lett. B 99 (1981) 66 [INSPIRE].
[82] A.H. Guth, The In ationary Universe: A Possible Solution to the Horizon and Flatness
Problems, Phys. Rev. D 23 (1981) 347 [INSPIRE].
[83] A.D. Linde, A New In ationary Universe Scenario: A Possible Solution of the Horizon,
Flatness, Homogeneity, Isotropy and Primordial Monopole Problems, Phys. Lett. B 108
(1982) 389 [INSPIRE].
[84] A. Albrecht and P.J. Steinhardt, Cosmology for Grand Uni ed Theories with Radiatively
Induced Symmetry Breaking, Phys. Rev. Lett. 48 (1982) 1220 [INSPIRE].
U.K. (2005).
(2011).
[88] S. Weinberg, Cosmology, Cambridge University Press, Cambridge U.K. (2008).
[89] D.H. Lyth and A.R. Liddle, The primordial density perturbation: Cosmology, in ation and
the origin of structure, Cambridge University Press, Cambridge U.K. (2009).
[90] D.S. Gorbunov and V.A. Rubakov, Introduction to the theory of the early universe:
Cosmological perturbations and in ationary theory, World Scienti c, Hackensack U.S.A.
[91] J. Martin, C. Ringeval and V. Vennin, Encyclop dia In ationaris, Phys. Dark Univ. 5-6
(2014) 75 [arXiv:1303.3787] [INSPIRE].
[hep-ph/9807493] [INSPIRE].
[92] A.D. Linde, A Toy model for open in ation, Phys. Rev. D 59 (1999) 023503
the CMB anisotropies, JCAP 07 (2003) 002 [astro-ph/0303636] [INSPIRE].
[astro-ph/0502343] [INSPIRE].
HJEP04(218)
early fast-roll in ation: MCMC analysis of WMAP and SDSS data, Phys. Rev. D 78
(2008) 023013 [arXiv:0804.2387] [INSPIRE].
[98] F.J. Cao, H.J. de Vega and N.G. Sanchez, Quantum slow-roll and quantum fast-roll
in ationary initial conditions: CMB quadrupole suppression and further e ects on the low
CMB multipoles, Phys. Rev. D 78 (2008) 083508 [arXiv:0809.0623] [INSPIRE].
in ation and the low CMB multipoles, JCAP 01 (2009) 009 [arXiv:0809.3915] [INSPIRE].
[arXiv:0903.3543] [INSPIRE].
103516 [arXiv:1111.7131] [INSPIRE].
[arXiv:0912.2994] [INSPIRE].
(2012) 103517 [arXiv:1202.0698] [INSPIRE].
[104] E. Ramirez, Low power on large scales in just enough in ation models, Phys. Rev. D 85
[105] Z.-G. Liu, Z.-K. Guo and Y.-S. Piao, Obtaining the CMB anomalies with a bounce from the
contracting phase to in ation, Phys. Rev. D 88 (2013) 063539 [arXiv:1304.6527]
[106] F.G. Pedro and A. Westphal, Low-` CMB power loss in string in ation, JHEP 04 (2014)
[107] M. Cicoli, S. Downes, B. Dutta, F.G. Pedro and A. Westphal, Just enough in ation: power
spectrum modi cations at large scales, JCAP 12 (2014) 030 [arXiv:1407.1048] [INSPIRE].
[108] R. Bousso, D. Harlow and L. Senatore, In ation after False Vacuum Decay, Phys. Rev. D
91 (2015) 083527 [arXiv:1309.4060] [INSPIRE].
[109] Z.-G. Liu, Z.-K. Guo and Y.-S. Piao, CMB anomalies from an in ationary model in string
theory, Eur. Phys. J. C 74 (2014) 3006 [arXiv:1311.1599] [INSPIRE].
[110] A.Y. Kamenshchik, A. Tronconi and G. Venturi, Quantum Gravity and the Large Scale
Anomaly, JCAP 04 (2015) 046 [arXiv:1501.06404] [INSPIRE].
higher spin
elds through statistical anisotropy in the CMB and galaxy power spectra, Phys.
Rev. D 97 (2018) 023503 [arXiv:1709.05695] [INSPIRE].
[1] P. Ramond , Dual Theory for Free Fermions , Phys. Rev. D 3 ( 1971 ) 2415 [INSPIRE].
[2] Yu . A. Golfand and E.P. Likhtman , Extension of the Algebra of Poincare Group Generators and Violation of p Invariance , JETP Lett . 13 ( 1971 ) 323 [Pisma Zh . Eksp. Teor. Fiz . 13 [3] J.-L. Gervais and B. Sakita , Field Theory Interpretation of Supergauges in Dual Models, [15] M.B. Green , J.H. Schwarz and E. Witten , Superstring Theory , 2 volumes , Cambridge University Press, Cambridge U.K. ( 1987 ).
[16] J. Polchinski , String theory, 2 volumes , Cambridge University Press, Cambridge U.K.
[17] C.V. Johnson , D-branes, Cambridge University Press, Cambridge U.K. ( 2003 ).
[18] B. Zwiebach , A rst course in string theory , Cambridge University Press, Cambridge U.K.
[19] K. Becker , M. Becker and J.H. Schwarz , String theory and M-theory: A modern introduction , Cambridge University Press, Cambridge U.K. ( 2007 ).
[20] E. Kiritsis , String theory in a nutshell , Cambridge University Press, Cambridge U.K. ( 2007 ).
[21] P. West , Introduction to strings and branes, Cambridge University Press, Cambridge U.K.
[35] I. Bandos , M. Heller , S.M. Kuzenko , L. Martucci and D. Sorokin , The Goldstino brane, the [36] E. Cremmer , B. Julia , J. Scherk , S. Ferrara , L. Girardello and P. van Nieuwenhuizen , [43] G. Pradisi and A. Sagnotti , Open String Orbifolds, Phys. Lett. B 216 ( 1989 ) 59 [INSPIRE].
[44] P. Horava , Strings on World Sheet Orbifolds, Nucl. Phys. B 327 ( 1989 ) 461 [INSPIRE].
[45] P. Horava , Background Duality of Open String Models, Phys. Lett. B 231 ( 1989 ) 251 [46] M. Bianchi and A. Sagnotti , On the systematics of open string theories , Phys. Lett. B 247 [51] C. Angelantonj and A. Sagnotti , Open strings, Phys. Rept . 371 ( 2002 ) 1 [Erratum ibid . 376 [52] S. Sugimoto , Anomaly cancellations in type-I D-9-D-9 system and the USp(32) string theory , Prog. Theor. Phys . 102 ( 1999 ) 685 [ hep -th/9905159] [INSPIRE].
[53] I. Antoniadis , E. Dudas and A. Sagnotti , Brane supersymmetry breaking , Phys. Lett. B 464 [56] C. Angelantonj , I. Antoniadis, G. D'Appollonio , E. Dudas and A. Sagnotti , Type I vacua with brane supersymmetry breaking , Nucl. Phys. B 572 ( 2000 ) 36 [ hep -th/9911081] [57] J. Mourad and A. Sagnotti , An Update on Brane Supersymmetry Breaking , [58] E. Dudas and J. Mourad , Consistent gravitino couplings in nonsupersymmetric strings , Phys. Lett. B 514 ( 2001 ) 173 [ hep -th/0012071] [INSPIRE]. [75] E. Dudas , N. Kitazawa and A. Sagnotti , On Climbing Scalars in String Theory, Phys . Lett.
[85] A.D. Linde , Chaotic In ation, Phys. Lett. B 129 ( 1983 ) 177 [INSPIRE].
[86] N. Bartolo , E. Komatsu , S. Matarrese and A. Riotto , Non-Gaussianity from in ation: Theory and observations , Phys. Rept . 402 ( 2004 ) 103 [ astro -ph/0406398] [INSPIRE].
[87] V. Mukhanov , Physical foundations of cosmology, Cambridge University Press, Cambridge [93] C.R. Contaldi , M. Peloso , L. Kofman and A.D. Linde , Suppressing the lower multipoles in [94] Y.-S. Piao , B. Feng and X. -m. Zhang, Suppressing CMB quadrupole with a bounce from contracting phase to in ation , Phys. Rev. D 69 ( 2004 ) 103520 [ hep -th/0310206] [INSPIRE].
[95] Y.-S. Piao , A Possible explanation to low CMB quadrupole , Phys. Rev. D 71 ( 2005 ) 087301 [96] D. Boyanovsky , H.J. de Vega and N.G. Sanchez, CMB quadrupole suppression. 2. The early fast roll stage , Phys. Rev. D 74 ( 2006 ) 123007 [ astro -ph/0607487] [INSPIRE].
[97] C. Destri , H.J. de Vega and N.G. Sanchez , The CMB Quadrupole depression produced by [99] R.K. Jain , P. Chingangbam , J.-O. Gong , L. Sriramkumar and T. Souradeep , Punctuated [100] E. Ramirez and D.J. Schwarz , 4 in ation is not excluded , Phys. Rev. D 80 ( 2009 ) 023525 [101] E. Ramirez and D.J. Schwarz , Predictions of just-enough in ation , Phys. Rev. D 85 ( 2012 ) [102] R.K. Jain , P. Chingangbam , L. Sriramkumar and T. Souradeep , The tensor-to-scalar ratio in punctuated in ation , Phys. Rev. D 82 ( 2010 ) 023509 [arXiv: 0904 .2518] [INSPIRE].
[103] C. Destri , H.J. de Vega and N.G. Sanchez , The pre-in ationary and in ationary fast-roll eras and their signatures in the low CMB multipoles , Phys. Rev. D 81 ( 2010 ) 063520 [111] Y.-F. Cai , E.G.M. Ferreira , B. Hu and J. Quintin , Searching for features of a string-inspired in ationary model with cosmological observations , Phys. Rev. D 92 ( 2015 ) 121303 [112] E. Dudas , N. Kitazawa , S.P. Patil and A. Sagnotti , CMB Imprints of a Pre-In ationary Climbing Phase , JCAP 05 ( 2012 ) 012 [arXiv: 1202 .6630] [INSPIRE].
[113] A. Gruppuso and A. Sagnotti , Observational Hints of a Pre{In ationary Scale? , Int. J. Mod. Phys. D 24 ( 2015 ) 1544008 [arXiv: 1506 .08093] [INSPIRE].
[114] A. Gruppuso , N. Kitazawa , N. Mandolesi , P. Natoli and A. Sagnotti , Pre-In ationary Relics in the CMB? , Phys. Dark Univ . 11 ( 2016 ) 68 [arXiv: 1508 .00411] [INSPIRE].
[115] A. Gruppuso , N. Kitazawa , M. Lattanzi , N. Mandolesi , P. Natoli and A. Sagnotti , The Evens and Odds of CMB Anomalies, Phys . Dark Univ. 20 ( 2018 ) 49 [arXiv: 1712 .03288] [116] A. Kehagias and A. Riotto , On the In ationary Perturbations of Massive Higher-Spin Fields , JCAP 07 ( 2017 ) 046 [arXiv: 1705 .05834] [INSPIRE].
[117] N. Bartolo , A. Kehagias , M. Liguori , A. Riotto , M. Shiraishi and V. Tansella , Detecting [118] D. Baumann , G. Goon, H. Lee and G.L. Pimentel , Partially Massless Fields During In ation, arXiv: 1712 .06624 [INSPIRE].
[119] G. Franciolini , A. Kehagias and A. Riotto , Imprints of Spinning Particles on Primordial Cosmological Perturbations, JCAP 02 ( 2018 ) 023 [arXiv: 1712 .06626] [INSPIRE].