Attractors with vanishing central charge

We consider the attractor equations of particular N=2, d=4 supergravity models whose vector multiplets' scalar manifold is endowed with homogeneous symmetric cubic special Kähler geometry, namely of the so-called st2 and stu models. In this framework, we derive explicit expressions for the critical moduli corresponding to non-BPS attractors with vanishing N=2 central charge. Such formulæ hold for a generic black hole charge configuration, and they are obtained without formulating any ad hoc simplifying assumption. We find that such attractors are related to the 12-BPS ones by complex conjugation of some moduli. By uplifting to N=8, d=4 supergravity, we give an interpretation of such a relation as an exchange of two of the four eigenvalues of the N=8 central charge matrix ZAB. We also consider non-BPS attractors with non-vanishing Z; for peculiar charge configurations, we derive solutions violating the ansatz usually formulated in literature. Finally, by group-theoretical considerations we relate Cayley's hyperdeterminant (the invariant of the stu model) to the invariants of the st2 and of the so-called t3 model.