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 t
he 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.
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 fi
elds. 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.
We construct a supergravity model whose scalar degrees of freedom arise from a chiral superfield and are solely a scalaron and an axion that is very heavy during the inflationary phase. The model includes a second chiral superfield $X$, which is subj
ect however to the constraint $X^2=0$ so that it describes only a Volkov - Akulov goldstino and an auxiliary field. We also construct the dual higher - derivative model, which rests on a chiral scalar curvature superfield ${cal R}$ subject to the constraint ${cal R}^2=0$, where the goldstino dual arises from the gauge - invariant gravitino field strength as $gamma^{mn} {cal D}_m psi_n$. The final bosonic action is an $R+R^2$ theory involving an axial vector $A_m$ that only propagates a physical pseudoscalar mode.
I elaborate on a link between the string--scale breaking of supersymmetry that occurs in a class of superstring models and the onset of inflation. The link rests on spatially flat cosmologies supported by a scalar field driven by an exponential poten
tial. If, as in String Theory, this potential is steep enough, under some assumptions that are spelled out in the text the scalar can only climb up as it emerges from an initial singularity. In the presence of another mild exponential, slow--roll inflation is thus injected during the ensuing descent and definite imprints are left in the CMB power spectrum: the quadrupole is systematically reduced and, depending on the choice of two parameters, an oscillatory behavior can also emerge for low multipoles l < 50, in qualitative agreement with WMAP9 and PLANCK data. The experimentally favored value of the spectral index, n_s ~ 0.96, points to a potentially important role for the NS fivebrane, which is unstable in this class of models, in the Early Universe.
In string models with brane supersymmetry breaking exponential potentials emerge at (closed-string) tree level but are not accompanied by tachyons. Potentials of this type have long been a source of embarrassment in flat space, but can have interesti
ng implications for Cosmology. For instance, in ten dimensions the logarithmic slope |V/V| lies precisely at a critical value where the Lucchin--Matarrese attractor disappears while the scalar field is emph{forced} to climb up the potential when it emerges from the Big Bang. This type of behavior is in principle perturbative in the string coupling, persists after compactification, could have trapped scalar fields inside potential wells as a result of the cosmological evolution and could have also injected the inflationary phase of our Universe.
The simplest higher-spin interactions involve classical external currents and symmetric tensors $phi_{m_1 ... m_s}$, and convey three instructive lessons. The first is a general form of the van Dam-Veltman-Zakharov discontinuity in flat space for thi
s class of fields. The second is the rationale for its disappearance in (A)dS spaces. Finally, the third is a glimpse into an option which is commonly overlooked in Field Theory, and which both higher spins and String Theory are confronting us with: one can well allow in the Lagrangians non-local terms that do not spoil the local nature of physical quantities.
We determine the current exchange amplitudes for free totally symmetric tensor fields $vf_{mu_1 ... mu_s}$ of mass $M$ in a $d$-dimensional $dS$ space, extending the results previously obtained for $s=2$ by other authors. Our construction is based on
an unconstrained formulation where both the higher-spin gauge fields and the corresponding gauge parameters $Lambda_{mu_1 >... mu_{s-1}}$ are not subject to Fronsdals trace constraints, but compensator fields $alpha_{mu_1 ... mu_{s-3}}$ are introduced for $s > 2$. The free massive $dS$ equations can be fully determined by a radial dimensional reduction from a $(d+1)$-dimensional Minkowski space time, and lead for all spins to relatively handy closed-form expressions for the exchange amplitudes, where the external currents are conserved, both in $d$ and in $(d+1)$ dimensions, but are otherwise arbitrary. As for $s=2$, these amplitudes are rational functions of $(ML)^2$, where $L$ is the $dS$ radius. In general they are related to the hypergeometric functions $_3F_2(a,b,c;d,e;z)$, and their poles identify a subset of the partially-massless discrete states, selected by the condition that the gauge transformations of the corresponding fields contain some non-derivative terms. Corresponding results for $AdS$ spaces can be obtained from these by a formal analytic continuation, while the massless limit is smooth, with no van Dam-Veltman-Zakharov discontinuity.
