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.
