On the Moduli Space of non-BPS Attractors for N=2 Symmetric Manifolds

TL;DR

This work classifies the flat directions of non-BPS attractors in $ N=2$, $d=4$ homogeneous symmetric supergravities by analyzing the Hessian of the black-hole potential $V_{BH}$ at non-BPS critical points. It distinguishes between $Z eq 0$ and $Z=0$ cases, identifying moduli spaces governed by real special geometry (linked to the $d=5$ parent) and by Kahler homogeneous symmetric manifolds, respectively, and it provides explicit moduli-space structures for $ N=8$ and $ N=2$ theories across $d=4$ and $d=5$ dimensions. The results connect four- and five-dimensional theories through charge-orbit stabilizers, and they reveal that non-BPS attractors possess stable but with flat directions that are tied to non-compact stabilizers, with implications for classical entropy and potential quantum lifting. The paper also highlights the consistency with known degeneracy patterns and extends the discussion to non-symmetric or non-supersymmetric settings in the broader Maxwell-Einstein framework.

Abstract

We study the ``flat'' directions of non-BPS extremal black hole attractors for N=2, d=4 supergravities whose vector multiplets' scalar manifold is endowed with homogeneous symmetric special Kahler geometry. The non-BPS attractors with non-vanishing central charge have a moduli space described by real special geometry (and thus related to the d=5 parent theory), whereas the moduli spaces of non-BPS attractors with vanishing central charge are certain Kahler homogeneous symmetric manifolds. The moduli spaces of the non-BPS attractors of the corresponding N=2, d=5 theories are also indicated, and shown to be rank-1 homogeneous symmetric manifolds.