Attractors with Vanishing Central Charge

Abstract

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 formul\ae 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 Cayley's hyperdeterminant (the invariant of the stu model) to the invariants of the st^{2} and of the so-called t^{3} model.