We analyze the exact perturbative solution of N=2 Born-Infeld theory which is believed to be defined by Ketovs equation. This equation can be considered as a truncation of an infinite system of coupled differential equations defining Born-Infeld acti
on with one manifest N=2 and one hidden N=2 supersymmetries. We explicitly demonstrate that infinitely many new structures appear in the higher orders of the perturbative solution to Ketovs equation. Thus, the full solution cannot be represented as a function depending on {it a finite number} of its arguments. We propose a mechanism for generating the new structures in the solution and show how it works up to 18-th order. Finally, we discuss two new superfield actions containing an infinite number of terms and sharing some common features with N=2 supersymmetric Born-Infeld action.
We consider the Attractor Equations of particular $mathcal{N}=2$, d=4 supergravity models whose vector multiplets scalar manifold is endowed with homogeneous symmetric cubic special K{a}hler geometry, namely of the so-called $st^{2}$ and $stu$ models
. In this framework, we derive explicit expressions for the critical moduli corresponding to non-BPS attractors with vanishing $mathcal{N}=2$ central charge. Such formulae hold for a generic black hole charge configuration, and they are obtained without formulating any textit{ad hoc} simplifying assumption. We find that such attractors are related to the 1/2-BPS ones by complex conjugation of some moduli. By uplifting to $mathcal{N}=8$, d=4 supergravity, we give an interpretation of such a relation as an exchange of two of the four eigenvalues of the $mathcal{N}=8$ central charge matrix $Z_{AB}$. We also consider non-BPS attractors with non-vanishing $mathcal{Z}$; for peculiar charge configurations, we derive solutions violating the Ansatz usually formulated in literature. Finally, by group-theoretical considerations we relate Cayleys hyperdeterminant (the invariant of the stu model) to the invariants of the st^{2} and of the so-called t^{3} model.
