Black Hole Attractors in Extended Supergravity

Abstract

We review some aspects of the attractor mechanism for extremal black holes of (not necessarily supersymmetric) theories coupling Einstein gravity to scalars and Maxwell vector fields. Thence, we consider N = 2 and N = 8, d = 4 supergravities, reporting some recent advances on the moduli spaces associated to BPS and non-BPS attractor solutions supported by charge orbits with non-compact stabilizers.

d=4black hole attractors inN=2supergravity with Fayet-Iliopoulos terms

“…Consistently, in the ungauged limitα SO(8) → 0 the well-known expression of the BH entropy is recovered [80,10,35]: 47) whereas S BH,SO (8) vanishes in the limitα SO(8) → ∞.…”

Topological solutions in ungauged supergravity

“…From the very same solution ðU; ϕ i Þ, i ¼ 1; …; n s of the system of differential equations (2.18), (2.19), and (2.20) we can build three different four-dimensional solutions s 1 ∈ C 1 , s 2 ∈ C 2 =Z 2 and s 3 ∈ C 3 of the original theory. Since the class C 1 corresponds to spherically symmetric, static, asymptotically flat black holes, the flow of the corresponding scalars may exhibit attractors, or fixed points, at τ → −∞ [1,11,[13][14][15]19,[33][34][35][36][37][38][39]. This is, in particular, ensured for supersymmetric black holes.…”

“…It is a class of solutions which can be easily embedded in string theory, for example, by means of type-II fluxless Calabi-Yau compactifications, and therefore they correspond to states in the full-fledged string theory, after being appropriately corrected. In addition, they are a nontrivial example which exhibits the attractor mechanism, different from all the previous solutions where the attractor mechanism was proven to hold [1,[11][12][13][14][15]. The attractor mechanism was of outermost importance in supergravity and string theory in order to check the macroscopic computation, at strong coupling, of the entropy of a black hole versus the microscopic calculation, at weak coupling, where the black hole becomes a configuration of D-branes and other objects [16,17].…”

supersymmetric black attractors in six and seven dimensions

“…Following [53], this effective potential is given by the Weinhold potential, 52) which can be also put in the form…”

“…Here, we shall follow the approach used in the works [51,52,53,54]. The effective scalar potential V ef f of the 6D black object is expressed as a quadratic form of the central charges (7.1).…”