4d/5d correspondence for the black hole potential and its critical points

Abstract

We express the d = 4, N = 2 black hole effective potential for cubic holomorphic F functions and generic dyonic charges in terms of d = 5 real special geometry data. The 4d critical points are computed from the 5d ones, and their relation is elucidated. For symmetric spaces, we identify the BPS and non-BPS classes of attractors and the respective entropies. These always derive from simple interpolating formulae between four and five dimensions, depending on the volume modulus and on the 4d magnetic (or electric) charges.

5D black holes and non-linear sigma models

“…Using for example the identities found in [34], one may check that (2.13) is consistent with the general form of the scalar potential induced by Fayet-Iliopoulos terms in four dimensions [35,36],…”

“…Plugging into (2.37), and using identities in [34], we find that the scalar potential (2.37) does indeed reproduce (2.19),…”

More onN=8attractors

“…where the following shorthand notation has been introduced: Notice that this formula becomes identical to the corresponding one of [22] concerning (purely cubic) N ¼ 2 geometries [33,34], where a IJ ¼ 4e 4 g ij and V e 6 . The potential (4.10), because of the definitions (4.8), can be seen to be a polynomial of a degree up to the sixth in the axion fields, whose general solutions are hard to determine.…”

“…In order to extract the stu model, we notice that its d ¼ 5 uplift is the ðSOð1; 1ÞÞ 2 model with cubic hypersurface [33,34] (see e.g. the treatment given in [22])…”

Black Hole Attractors in Extended Supergravity

“…The moduli spaces associated to the N = 2, d = 4 non-BPS solutions with Z = 0 and Z = 0 and to the N = 2, d = 5 non-BPS solutions have been recently determined in [42] (see also [45]); they are respectively given by Tables 2, 3 and 4 of [42].…”