we can say that the attractor equations (plus the positivity of the mass, which is not guaranteed) seem to be sufficient conditions to have regular, static, spherically symmetric black holes.

Finally, observe that Eq. (\[eq:linearBPSmass\]) plus the identification, which will be established later on, between the NUT charge and the linear expression of the charges

$$\label{eq:linearBPSNUT}
N=  \langle\, \mathcal{I}_{\infty} \mid q \, \rangle\, ,$$

lead to a complex BPS relation

$$\label{eq:complexlinearBPS}
M+iN= \langle\, (\mathcal{V}/X)_{\infty} \mid q \, \rangle\, .$$

We will argue that supersymmetry requires $N$ to vanish, whence the above relation reads

$$M= \pm\sqrt{2} |\mathcal{Z}_{\infty}|\, ,$$

which is the standard BPS relation between mass and central charge. Of course, only the positive sign will be admissible.

Relations between the $N=2,d=4$ KSIs, attractors and sources {#sec-n2d4}
============================================================

The equations of motion[^12] for supersymmetric configurations of supergravity theories satisfy certain relations known as *Killing spinor identities* (*KSI*s), which can also be derived from the integrability conditions of the Killing spinor equations [@Kallosh:1993wx; @Bellorin:2005hy]. We have unbroken supersymmetry wherever the Killing spinors exist, and these exist, locally, wherever the KSIs are satisfied. Thus, if we are to have unbroken supersymmetry everywhere we must demand the KSIs to be satisfied everywhere. In this 