Attractors with Vanishing Central Charge

TL;DR

This paper derives explicit formulas for non-BPS attractors with vanishing central charge in specific supergravity models, revealing their relation to BPS attractors and interpreting these findings within extended supergravity frameworks.

Contribution

It provides explicit, general expressions for non-BPS attractors with zero central charge in $st^{2}$ and $stu$ models without simplifying assumptions, and relates these to BPS solutions and higher supergravity invariants.

Findings

Explicit formulas for non-BPS attractors with vanishing central charge.

Relation between non-BPS and BPS attractors via moduli conjugation.

Connection between Cayley's hyperdeterminant and supergravity invariants.

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.