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We review a series of developments concerning non-linear realizations of supersymmetry coupled to supergravity, with emphasis on some applications to inflationary Cosmology., Comment: 27 pages, 1 figure. Invited contribution to "Half a century of Supergravity'', eds. A. Ceresole and G. Dall'Agata (Cambridge Univ. Press, to appear)

We connect the indications of 2D CFT for branes in non-supersymmetric strings to actual spacetime profiles, taking the bulk tadpole potentials into account. We find exact solutions for the uncharged branes that spread in the internal intervals of Dudas-Mourad vacua for non-tachyonic orientifolds and heterotic strings. These solutions involve suitable dressings of the uncharged branes of the nine-dimensional tadpole-free theory. Similar exact results for uncharged branes that are transverse to the internal space, or for charged branes, appear more challenging. Nevertheless, we identify their large-distance behavior, which is determined by linearized equations, and show that it is compatible with the CFT analysis in all expected cases. We also provide some hints on the leading back-reaction of the form-fields., Comment: 47 pages, LaTeX; corrected some typos and a sign in eq.(4.10), which affects section 4.3. Some of the branes have different behaviors, but the main conclusions are unchanged. Final version to appear in JHEP, with two figures and minor additional text corrections added

After recasting the standard charged and uncharged brane profiles in the harmonic gauge, we explore solutions with the same isometries where the potentials $V = T \,e^{\gamma\,\phi}$ of ten-dimensional non-supersymmetric strings are taken into account. Combining a detailed catalog of the possible asymptotics with some numerical results suggests that these spherically symmetric backgrounds terminate at singularities within finite proper distances., Comment: 66 pages, Latex, 7 eps figures

We recently proposed a class of type IIB vacua that yield, at low energies, four--dimensional Minkowski spaces with broken supersymmetry and a constant string coupling. They are compactifications with an internal five-torus bearing a five--form flux $\Phi$ and warp factors depending on a single coordinate. The breaking of supersymmetry occurs when the internal space includes a finite interval. A probe-brane analysis revealed a gravitational repulsion and a charge attraction of equal magnitude from the left end of the interval, and a singularity at the other end. Here we complete the analysis revealing the presence, at one end, of an effective $O3$ of negative tension and positive five--form charge. We also determine the values of these quantities, and show that $T = -\, Q = \Phi$, and characterize the singularity present at the other end of the interval, which hosts an opposite charge. Finally, we discuss various forms of the gravity action in the presence of a boundary and identify a self--adjoint form for its fluctuations., Comment: 23 pages, LaTeX. Comments added. Final version to appear in the special issue of Journal of Physics A: Mathematical and Theoretical on "Fields, Gravity, Strings and Beyond: In Memory of Stanley Deser''

We recently constructed type-IIB compactifications to four dimensions depending on a single additional coordinate, where a five-form flux $\Phi$ on an internal torus leads to a constant string coupling. Supersymmetry is fully broken when the internal manifold includes a finite interval of length $\ell$, which is spanned by a conformal coordinate in a finite range $0 < z < z_m$. Here we examine the low-lying bosonic spectra and their classical stability, paying special attention to self-adjoint boundary conditions. Special boundary conditions result in the emergence of zero modes, which are determined exactly by first-order equations. The different sectors of the spectrum can be related to Schr\"odinger operators on a finite interval, characterized by pairs of real constants $\mu$ and $\tilde{\mu}$, with $\mu$ equal to ${1}/{3}$ or ${2}/{3}$ in all cases and different values of $\tilde{\mu}$. The potentials behave as $\frac{\mu^2-1/4}{z^2}$ and $\frac{\tilde{\mu}^2-1/4}{\left(z_m-z\right)^2}$ near the ends and can be closely approximated by exactly solvable trigonometric ones. With vanishing internal momenta, one can thus identify a wide range of boundary conditions granting perturbative stability, despite the intricacies that emerge in some sectors. For the Kaluza--Klein excitations of non-singlet vectors and scalars the Schr\"odinger systems couple pairs of fields, and the stability regions, which depend on the background, widen as the ratio ${\Phi}/{\ell^4}$ decreases., Comment: 119 pages, LaTeX, 24 eps figures

Internal intervals spanned by finite ranges of a conformal coordinate $z$ and terminating at a pair of singularities are a common feature of many string compactifications with broken supersymmetry. The squared masses emerging in lower-dimensional Minkowski spaces are then eigenvalues of Schr\"odinger-like operators, whose potentials have double poles at the ends of the intervals. For one-component systems, the possible self-adjoint extensions of Schr\"odinger operators are described by points in $AdS_3 \times S^1$, and those corresponding to independent boundary conditions at the ends of the intervals by points on the boundary of $AdS_3$. The perturbative stability of compactifications to Minkowski space time depends, in general, on these choices of self-adjoint extensions. We apply this setup to the orientifold vacua driven by the ``tadpole potential'' $V=T \ e^{\,\frac{3}{2}\,\phi}$ and find, in nine dimensions, a massive scalar spectrum, a unique choice of boundary conditions with stable tensor modes and a massless graviton, and a wide range of choices leading to massless and/or massive vector modes., Comment: 49 pages, LaTeX, 13 figures. References added, misprints corrected in eqs. (3.71) and (3.72). Final version to appear in JHEP (with some additional misprints removed while correcting the proofs)

We consider the type-IIB supergravity vacua that include an internal $T^5$, depend on a single coordinate $r$ and respect a four-dimensional Poincar\'e symmetry, with the aim of highlighting low-energy spectra with broken supersymmetry and a bounded string coupling. These vacua are characterized by the flux $\Phi$ of the self-dual five form in the internal torus, the length $\ell$ of the interval described by the coordinate $r$, a dilaton profile that is inevitably constant and a strictly positive dimensionless parameter $h$. As $\ell\rightarrow\infty$ while retaining finite values for $\Phi$ and $\ell \,h^{-\,\frac{5}{4}}$, half of the original ten-dimensional supersymmetry is recovered, while finite values of $\ell$ break it completely. In the large-$\ell$ limit one boundary disappears but the other is still present, and is felt as a BPS orientifold by a probe brane. In this paper we focus on the fermionic zero modes and show that, although supersymmetry is broken for finite values of $\ell$, they are surprisingly those of four-dimensional $N=4$ supergravity coupled to five vector multiplets. The gravitini can acquire masses via radiative corrections, absorbing four of the massless spin-$\frac{1}{2}$ modes., Comment: 63 pages, LaTeX, 3 eps figures. Minor clarifications added. Final version to appear in JHEP

We investigate the effects of the leading tadpole potentials of 10D tachyon-free non-supersymmetric strings in warped products of flat geometries of the type M(p+1) x R x T(10-p-2) depending on a single coordinate. In the absence of fluxes and for p<8, there are two families of these vacua for the orientifold disk-level potential, both involving a finite internal interval. Their asymptotics are surprisingly captured by tadpole-free solutions, isotropic for one family and anisotropic at one end for the other. In contrast, for the heterotic torus-level potential there are four types of vacua. Their asymptotics are always tadpole-dependent and isotropic at one end lying at a finite distance, while at the other end, which can lie at a finite or infinite distance, they can be tadpole-dependent isotropic or tadpole-free anisotropic. We then elaborate on the general setup for including symmetric fluxes, and present the three families of exact solutions that emerge when the orientifold potential and a seven-form flux are both present. These solutions include a pair of boundaries, which are always separated by a finite distance. In the neighborhood of one, they all approach a common supersymmetric limit, while the asymptotics at the other boundary can be tadpole-free isotropic, tadpole-free anisotropic or again supersymmetric. We also discuss corresponding cosmologies, with emphasis on their climbing or descending behavior at the initial singularity. In some cases the toroidal dimensions can contract during the cosmological expansion., Comment: 67 pages, LaTeX, 6 tables, 9 eps figures

We investigate in detail solutions of supergravity that involve warped products of flat geometries of the type M(p+1) x R x T(D-p-2) depending on a single coordinate. In the absence of fluxes, the solutions include flat space and Kasner-like vacua that break all supersymmetries. In the presence of a symmetric flux, there are three families of solutions that are characterized by a pair of boundaries and have a singularity at one of them, the origin. The first family comprises supersymmetric vacua, which capture a universal limiting behavior at the origin. The first and second families also contain non--supersymmetric solutions whose behavior at the other boundary, which can lie at a finite or infinite distance, is captured by the no--flux solutions. The solutions of the third family have a second boundary at a finite distance where they approach again the supersymmetric backgrounds. These vacua exhibit a variety of interesting scenarios, which include compactifications on finite intervals and (p+1)-dimensional effective theories where the string coupling has an upper bound. We also build corresponding cosmologies, and in some of them the string coupling can be finite throughout the evolution., Comment: 63 pages, LaTeX

We review the main results of our investigations motivated by the tadpole potentials of ten-dimensional strings with broken supersymmetry. While these are at best partial indications, it is hard to resist the feeling that they do capture some lessons of String Theory. For example, these very tadpole potentials lead to weak-string-coupling cosmologies that appear to provide clues on the onset of the inflation from an initial fast roll. The transition, if accessible to us, would offer a natural explanation for the lack of power manifested by the CMB at large angular scales. In addition, the same tadpole potentials can drive spontaneous compactifications to lower-dimensional Minkowski spaces at corresponding length scales. Furthermore, the cosmological solutions exhibit an intriguing "instability of isotropy" that, if taken at face value, would point to an accidental origin of compactification. Finally, symmetric static AdS x S solutions driven by the tadpole potentials also exist, but they are unstable due to mixings induced by their internal fluxes. On the other hand, the original Dudas-Mourad solution is perturbatively stable, and we have gathered some detailed evidence that instabilities induced by internal fluxes can be held under control in a similar class of weak-coupling type-IIB compactifications to Minkowski space., Comment: Invited contribution to the special issue of Letters in High Energy Physics (LHEP) on "Swampland and String Theory Landscape", edited by I. Antoniadis, K. Benakli and E. Dudas

We elaborate on integrable dynamical systems from scalar-gravity Lagrangians that include the leading dilaton tadpole potentials of broken supersymmetry. In the static Dudas-Mourad compactifications from ten to nine dimensions, which rest on these leading potentials, the string coupling and the space-time curvature become unbounded in some regions of the internal space. On the other hand, the string coupling remains bounded in several corresponding solutions of these integrable models. One can thus identify corrected potential shapes that could grant these features generically when supersymmetry is absent or non-linearly realized. On the other hand, large scalar curvatures remain present in all our examples. However, as in other contexts, the combined effects of the higher-derivative corrections of String Theory could tame them., Comment: 66 pages, latex, 34 figures

We address some general issues related to torsion and Noether currents for Fermi fields in the presence of boundaries, with emphasis on the conditions that guarantee charge conservation. We also describe exact solutions of these boundary conditions and some implications for string vacua with broken supersymmetry., Comment: 12 pages, LaTeX

We study the perturbative stability of four settings that arise in String Theory, when dilaton potentials accompany the breaking of Supersymmetry, in the USp(32) and U(32) orientifold models, and also in the heterotic SO(16)xSO(16) model. The first two settings are a family of AdS3xS7 orientifold vacua and a family of AdS7xS3 heterotic vacua, supported by form fluxes, with small world-sheet and string-loop corrections within wide ranges of parameters. In both cases we find some unstable scalar perturbations, as a result of mixings induced by fluxes, confirming for the first class of vacua a previous result. However, in the second class they only affect the l=1 modes, so that a Z2 projection induced by an overall internal parity suffices to eliminate them, leading to perturbative stability. Moreover, the constant dilaton profiles of these vacua allow one to extend the analysis to generic potentials, thus exploring the possible effects of higher-order corrections, and we exhibit wide nearby regions of perturbative stability. The solutions in the third setting have nine-dimensional Poincare' symmetry. They include regions with large world-sheet or string-loop corrections, but we show that these vacua have no perturbative instabilities. Finally, the last setting concerns cosmological solutions in ten dimensions where the "climbing" phenomenon takes place: they have bounded string-loop corrections but large world-sheet ones close to the initial singularity. We find that perturbations generally decay, but homogeneous tensor modes exhibit an interesting logarithmic growth that signals a breakdown of isotropy. If the Universe then proceeds to lower dimensions, milder potentials from other branes force all perturbations to remain bounded., Comment: 59 pages, latex, 7 pdf figures. One reference added. Some comments also added in the Introduction and in Sections 2, 3, 4. Final version to appear in JHEP

The lack of power of large--angle CMB anisotropies is known to increase its statistical significance at higher Galactic latitudes, where a string--inspired pre--inflationary scale $\Delta$ can also be detected. Considering the Planck 2015 data, and relying largely on a Bayesian approach, we show that the effect is mostly driven by the \emph{even}--$\ell$ harmonic multipoles with $\ell \lesssim 20$, which appear sizably suppressed in a way that is robust with respect to Galactic masking, along with the corresponding detections of $\Delta$. On the other hand, the first \emph{odd}--$\ell$ multipoles are only suppressed at high Galactic latitudes. We investigate this behavior in different sky masks, constraining $\Delta$ through even and odd multipoles, and we elaborate on possible implications. We include low--$\ell$ polarization data which, despite being noise--limited, help in attaining confidence levels of about 3 $\sigma$ in the detection of $\Delta$. We also show by direct forecasts that a future all--sky $E$--mode cosmic--variance--limited polarization survey may push the constraining power for $\Delta$ beyond 5 $\sigma$., Comment: 49 pages, 19 figures. Figures and final discussion simplified, references added. Final version to appear in Physics of the Dark Universe

"Brane supersymmetry breaking" is a peculiar phenomenon that can occur in perturbative orientifold vacua. It results from the simultaneous presence, in the vacuum, of non-mutually BPS sets of BPS branes and orientifolds, which leave behind a net tension and thus a runaway potential, but no tachyons. In the simplest ten-dimensional realization, the low-lying modes combine the closed sector of type-I supergravity with an open sector including USp(32) gauge bosons, fermions in the antisymmetric 495 and an additional singlet playing the role of a goldstino. We review some properties of this system and of other non-tachyonic models in ten dimensions with broken supersymmetry, and we illustrate some puzzles that their very existence raises, together with some applications that they have stimulated., Comment: Dedicated to the memory of Yassen S. Stanev

We identify a cubic holomorphic constraint that subtends the total breaking of N=2 supersymmetry in a vector multiplet and exhibit its microscopic origin. The new constraint leaves behind, at low energies, a vector and the two goldstini, in a non-linear Lagrangian that generalizes the N=2 Volkov-Akulov model., Comment: 26 pages, LaTeX

The fortieth anniversary of the original construction of Supergravity provides an opportunity to combine some reminiscences of its early days with an assessment of its impact on the quest for a quantum theory of gravity., Comment: 27 pages, LaTeX. Based in part on the talk delivered by S. F. at the "Infeld Colloquium and Discrete", in Warsaw, on December 1 2016, and on a joint CERN Courier article. Dedicated to John H. Schwarz on the occasion of his 75-th birthday. Comment and references added

We describe how unbounded three--form fluxes can lead to families of $AdS_3 \times S_7$ vacua, with constant dilaton profiles, in the $USp(32)$ model with "brane supersymmetry breaking" and in the $U(32)$ 0'B model, if their (projective--)disk dilaton tadpoles are taken into account. We also describe how, in the $SO(16) \times SO(16)$ heterotic model, if the torus vacuum energy $\Lambda$ is taken into account, unbounded seven--form fluxes can support similar $AdS_7 \times S_3$ vacua, while unbounded three--form fluxes, when combined with internal gauge fields, can support $AdS_3 \times S_7$ vacua, which continue to be available even if $\Lambda$ is neglected. In addition, special gauge field fluxes can support, in the $SO(16) \times SO(16)$ heterotic model, a set of $AdS_{n}\times S_{10-n}$ vacua, for all $n=2,..,8$. String loop and $\alpha'$ corrections appear under control when large form fluxes are allowed., Comment: 13 pages, LaTeX. References added. Final version to appear in Phys. Lett. B

Abdus Salam was a true master of 20th Century Theoretical Physics. Not only was he a pioneer of the Standard Model (for which he shared the Nobel Prize with S. Glashow and S.Weinberg), but he also (co)authored many other outstanding contributions to the field of Fundamental Interactions and their unification. In particular, he was a major contributor to the development of supersymmetric theories, where he also coined the word "Supersymmetry" (replacing the earlier "Supergauges" drawn from String Theory). He also introduced the basic concept of "Superspace" and the notion of "Goldstone Fermion"(Goldstino). These concepts proved instrumental for the exploration of the ultraviolet properties and for the study of spontaneously broken phases of super Yang-Mills theories and Supergravity. They continue to play a key role in current developments in Early-Universe Cosmology. In this contribution we review models of inflation based on Supergravity with spontaneously broken local supersymmetry, with emphasis on the role of nilpotent superfields to describe a de Sitter phase of our Universe., Comment: 20 pages, Latex. Contribution to the Proceedings of the "Memorial Meeting for Nobel Laureate Prof. Abdus Salam's 90th Birthday", Nanyang Technological University, Singapore, 25-28 January 2016. arXiv admin note: text overlap with arXiv:1509.01500

We elaborate on how to build, in a systematic fashion, two-field Abelian extensions of the Born-Infeld Lagrangian. These models realize the non-trivial duality groups that are allowed in this case, namely U(2), SU(2) and U(1)xU(1). For each class, we also construct an explicit example. They all involve an overall square root and reduce to the Born-Infeld model if the two fields are identified, but differ in quartic and higher interactions. The U(1)xU(1) and SU(2) examples recover some recent results obtained with different techniques, and we show that the U(1)xU(1) model admits an N=1 supersymmetric completion. The U(2) example includes some unusual terms that are not analytic at the origin of field space., Comment: 30 pages. Final version with reference added, typo corrected and comment added in the Conclusions. To appear in a special Nucl. Phys. B issue dedicated to the memory of Raymond Stora

Models of inflation are instructive playgrounds for supersymmetry breaking in Supergravity and String Theory. In particular, combinations of branes and orientifolds that are not mutually BPS can lead to \emph{brane supersymmetry breaking}, a phenomenon where non--linear realizations are accompanied, in tachyon--free vacua, by the emergence of steep exponential potentials. When combined with milder terms, these exponentials can lead to slow--roll after a fast ascent and a turning point. This leaves behind distinctive patterns of scalar perturbations, where pre--inflationary peaks can lie well apart from an almost scale invariant profile. I review recent attempts to connect these power spectra to the low--$\ell$ CMB, and a corresponding one--parameter extension of $\Lambda$CDM with a low--frequency cut $\Delta$. A detailed likelihood analysis led to $\Delta = (0.351 \pm 0.114) \times 10^{-3} \, \mbox{Mpc}^{-1}$, at $99.4\%$ confidence level, in an extended Galactic mask with $f_{sky}=39\%$, to be compared with a nearby value at $88.5\%$ in the standard Planck 2015 mask with $f_{sky}=94\%$. In these scenarios one would be confronted, in the CMB, with relics of an epoch of deceleration that preceded the onset of slow--roll., Comment: 16 pages, LaTeX, 22 eps figures. Based on the presentations at the International School for Subnuclear Physics, 53rd Course, "The Future of our Physics Including New Frontiers," Erice, June 24 -- July 3 2015, and at "Physics on the Riviera 2015," Sestri Levante, September 16--18 2015

Theories with elementary scalar degrees of freedom seem nowadays required for simple descriptions of the Standard Model and of the Early Universe. It is then natural to embed theories of inflation in supergravity, also in view of their possible ultraviolet completion in String Theory. After some general remarks on inflation in supergravity, we describe examples of minimal inflaton dynamics which are compatible with recent observations, including higher-curvature ones inspired by the Starobinsky model. We also discuss different scenarios for supersymmetry breaking during and after inflation, which include a revived role for non-linear realizations. In this spirit, we conclude with a discussion of the link, in four dimensions, between "brane supersymmetry breaking" and the super--Higgs effect in supergravity., Comment: 27 pages, LaTeX. Based in part on the Plenary and Parallel Session talks given by S.F. at the "Fourteenth Marcel Grossmann Meeting - MG14," Rome, July 12-18 2015, on the talk given by S.F. at "The String Theory Universe," 21st European String Workshop, Leuven, September 7-11, 2015, and on the plenary talk given by A.S. at "Planck 2015," Ioannina, May 25-29 2015. Misprints fixed, references added

A scale invariant Goldstino theory coupled to Supergravity is obtained as a standard supergravity dual of a rigidly scale invariant higher--curvature Supergravity with a nilpotent chiral scalar curvature. The bosonic part of this theory describes a massless scalaron and a massive axion in a de Sitter Universe., Comment: CERN-PH-TH-2015-193. Misprints corrected. Final version to appear in Phys. Lett. B

String Theory and Supergravity allow, in principle, to follow the transition of the inflaton from pre-inflationary fast roll to slow roll. This introduces an infrared depression in the primordial power spectrum that might have left an imprint in the CMB anisotropy, if it occurred at accessible wavelengths. We model the effect extending $\Lambda$CDM with a scale $\Delta$ related to the infrared depression and explore the constraints allowed by {\sc Planck} data, employing also more conservative, wider Galactic masks in the low resolution CMB likelihood. In an extended mask with $f_{sky}=39\%$, we thus find $\Delta = (0.351 \pm 0.114) \times 10^{-3} \, \mbox{Mpc}^{-1}$, at $99.4\%$ confidence level, to be compared with a nearby value at $88.5\%$ with the standard $f_{sky}=94\%$ mask. With about 64 $e$--folds of inflation, these values for $\Delta$ would translate into primordial energy scales ${\cal O}(10^{14})$ GeV., Comment: 16 pages, 6 figures, 1 table. Final version to appear in Physics of the Dark Universe. Contains: more detailed discussion of galactic masking, improved estimate

We construct Supergravity models where the goldstino multiplet has a gravitational origin, being dual to the chiral curvature superfield. Supersymmetry is nonlinearly realized due to a nilpotent constraint, while the goldstino arises from $\gamma$-traces of the gauge-invariant gravitino field strength. After duality transformations one recovers, as expected, the standard Volkov-Akulov Lagrangian coupled to Supergravity, but the gravitational origin of the goldstino multiplet restricts the available types of matter couplings. We also construct explicitly some inflationary models of this type, which contain both the inflaton and the nilpotent superfield., Comment: 19 pages, LaTeX. Dedicated to the memory of Raymond Stora. References and comment on the duality added. Final version to appear in JHEP

We argue that the lack of power exhibited by cosmic microwave background (CMB) anisotropies at large angular scales might be linked to the onset of inflation. We highlight observational features and theoretical hints that support this view, and present a preliminary estimate of the physical scale that would underlie the phenomenon., Comment: 7 pages, LaTeX, 2 eps figures. Essay Written for the 2015 Gravity Research Foundation Awards for Essays on Gravitation. Selected for Honorable Mention

This review is devoted to some aspects of non-linear Supersymmetry in four dimensions that can be efficiently described via nilpotent superfields, in both rigid and curved Superspace. Our focus is mainly on the partial breaking of rigid $N=2$ Supersymmetry and on a class of generalized Born-Infeld systems that originate from Special Geometry and on some prototype cosmological models, starting from the Supergravity embedding of Starobinsky inflation. However, as an aside we also review briefly some interesting two-field extensions of the Born-Infeld Lagrangian whose field equations enjoy extended duality symmetries., Comment: Contribution to the Proceedings of "Group Theory, Probability, and the Structure of Spacetime", A Conference on the Occasion of Professor V.S. Varadarajan's Retirement, UCLA Mathematics Department, November 7-9, 2014. To appear in a special issue of "p-Adic Numbers, Ultrametric Analysis and Applications". 35 pages LaTeX, 1 eps figure. Typos corrected, reference added

The self-duality of the N=1 supersymmetric Born--Infeld action implies a double self-duality of the tensor multiplet square-root action when the scalar and the antisymmetric tensor are interchanged via Poincare' duality. We show how this phenomenon extends to D space-time dimensions for non-linear actions involving pairs of forms of rank p and D-p-2. As a byproduct, we construct a new two-field generalization of the Born-Infeld action whose equations of motion are invariant under a U(1) duality. In these systems, the introduction of Green-Schwarz terms results in explicit non-linear mass-like terms for dual massive pairs., Comment: 17 pages, LaTeX. References added, duality group of eq. (4.16) changed to SU(2). Final version to appear in JHEP

We present a semi--analytic exploration of some low--$\ell$ angular power spectra inspired by "Brane Supersymmetry Breaking". This mechanism splits Bose and Fermi excitations in String Theory, leaving behind an exponential potential that is just too steep for the inflaton to emerge from the initial singularity while descending it. As a result, the scalar generically bounces against the exponential wall, which typically introduces an infrared depression and a pre--inflationary peak in the power spectrum of scalar perturbations. We elaborate on a possible link between this phenomenon and the low--$\ell$ CMB. For the first 32 multipoles, combining the hard exponential with a milder one leading to $n_s\simeq 0.96$ and with a small gaussian bump we have attained a reduction of $\chi^{\,2}$ to about 46% of the standard $\Lambda$CDM setting, with both WMAP9 and PLANCK 2013 data. This result corresponds to a $\chi^{\,2}/DOF$ of about 0.45, to be compared with a $\Lambda$CDM value of about 0.85. The preferred choices combine naturally quadrupole depression, a first peak around $\ell=5$ and a wide minimum around $\ell=20$. We have also gathered some evidence that similar spectra emerge if the hard exponential is combined with more realistic models of inflation. A problem of the preferred examples is their slow convergence to an almost scale--invariant profile., Comment: 12 pages, latex, 7 figures. Power and limitations of the models and of the semi-analytic approach stressed, misprints corrected. Final version to appear in Mod. Phys. Lett. A

We consider massive dual pairs of p-forms and (D-p-1)-forms described by non-linear Lagrangians, where non-linear curvature terms in one theory translate into non-linear mass-like terms in the dual theory. In particular, for D=2p and p even the two non-linear structures coincide when the non-linear massless theory is self-dual. This state of affairs finds a natural realization in the four-dimensional massive N=1 supersymmetric Born-Infeld action, which describes either a massive vector multiplet or a massive linear (tensor) multiplet with a Born-Infeld mass-like term. These systems should play a role for the massive gravitino multiplet obtained from a partial super-Higgs in N=2 Supergravity., Comment: 17 pages, LaTeX. Misprints corrected. Final version to appear in JHEP

We investigate $U(1)^{\,n}$ supersymmetric Born-Infeld Lagrangians with a second non-linearly realized supersymmetry. The resulting non-linear structure is more complex than the square root present in the standard Born-Infeld action, and nonetheless the quadratic constraints determining these models can be solved exactly in all cases containing three vector multiplets. The corresponding models are classified by cubic holomorphic prepotentials. Their symmetry structures are associated to projective cubic varieties., Comment: 17 pages, LaTeX, 1 eps figure. Comments added and misprints corrected. Final version to appear in Fortschritte der Physik - Progress of Physics

"Brane Supersymmetry Breaking" is a peculiar string-scale mechanism that can unpair Bose and Fermi excitations in orientifold models. It results from the simultaneous presence, in the vacuum, of collections of D-branes and orientifolds that are not mutually BPS, and is closely tied to the scale of string excitations. It also leaves behind, for a mixing of dilaton and internal breathing mode, an exponential potential that is just too steep for a scalar to emerge from the initial singularity while descending it. As a result, in this class of models the scalar can generically bounce off the exponential wall, and this dynamics brings along, in the power spectrum, an infrared depression typically followed by a pre-inflationary peak. We elaborate on a possible link between this type of bounce and the low-$\ell$ end of the CMB angular power spectrum. For the first 32 multipoles, one can reach a 50 % reduction in $\chi^{\,2}$ with respect to the standard $\Lambda$CDM setting., Comment: 15 pages, LaTeX, 30 eps figures. To appear in the Proceedings of ICNFP 2014. References added

We derive new types of $U(1)^n$ Born-Infeld actions based on N=2 special geometry in four dimensions. As in the single vector multiplet (n=1) case, the non--linear actions originate, in a particular limit, from quadratic expressions in the Maxwell fields. The dynamics is encoded in a set of coefficients $d_{ABC}$ related to the third derivative of the holomorphic prepotential and in an SU(2) triplet of N=2 Fayet-Iliopoulos charges, which must be suitably chosen to preserve a residual N=1 supersymmetry., Comment: 16 pages, LaTeX. Misprints corrected, comment added. Final version to appear in JHEP