Charge Orbits of Symmetric Special Geometries and Attractors

Abstract

We study the critical points of the black hole scalar potential V BH in N = 2, d = 4 supergravity coupled to n V vector multiplets, in an asymptotically flat extremal black hole background described by a 2 (n V + 1)-dimensional dyonic charge vector and (complex) scalar fields which are coordinates of a special Kähler manifold.For the case of homogeneous symmetric spaces, we find three general classes of regular attractor solutions with non-vanishing Bekenstein-Hawking entropy. They correspond to three (inequivalent) classes of orbits of the charge vector, which is in a 2 (n V + 1)-dimensional representation R V of the U-duality group. Such orbits are non-degenerate, namely they have non-vanishing quartic invariant (for rank-3 spaces). Other than the 1 2 -BPS one, there are two other distinct non-BPS classes of charge orbits, one of which has vanishing central charge.The three species of solutions to the N = 2 extremal black hole attractor equations give rise to different mass spectra of the scalar fluctuations, whose pattern can be inferred by using invariance properties of the critical points of V BH and some group theoretical considerations on homogeneous symmetric special Kähler geometry.

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

“…In Section 3 we give 4d attractors in terms of 5d ones, and we compute the corresponding BH entropy. For symmetric spaces, the interpolation yields a clear relation between the N = 2, d = 4 and d = 5 1 2 -BPS and non-BPS BH charge orbits studied in the literature [26,40,43,48,49], which is developed in Section 4. Finally, some further comments and outlooks are given in Section 5.…”

“…For theories with a symmetric scalar manifold, these methods have led to a general classification of BPS and non-BPS attractors, as well as studies of their classical stability and entropy [26,40,43,48,49]. For N = 2, d = 4 theories, some of these results have also been extended to more general scalar manifolds based on cubic holomorphic prepotentials.…”

Enriques and octonionic magic supergravity models

“…In an analogous way the hypermultiplet moduli space, that includes deformations of the Kähler structure of CY FHSV ≈ E × T 2 , is obtained by c-map [4] to be SO (12,4) SO(12) × SO (4) ( 1.3) and is a quaternionic subspace of the exceptional quaternionic manifold obtained by c-map from the Octonionic magic model [4] SO ( and quite remarkably 16 is the rank of the gauge group in Type I and Heterotic models in D = 10. Both models correspond to self-mirror CY threefolds with h 11 = h 21 = 11 and h 11 = h 21 = 27, respectively, and admit an uplift to D = 6 with N = (1, 0) supersymmetry.…”

“…For the Octonionic theory the classification of attractors is as follows, the charge orbits are [12] BP S E 7(−25)…”

Iwasawa nilpotency degree of non compact symmetric cosets in \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$ \mathcal{N} $\end{document}‐extended supergravity

“…Moreover, non compact homogeneous spaces and their non compact duality group G appear in the description of the black hole orbits [9,10,11,12] as well as duality invariant Bekenstein-Hawking entropy formula [13,14,15,16] according to the attractor mechanism [17,18,19]. The study of the attractor mechanism in string theory has been pioneered in [20].…”