We present new results towards the construction of the most general black hole solutions in four-dimensional Fayet-Iliopoulos gauged supergravities. In these theories black holes can be asymptotically AdS and have arbitrary mass, angular momentum, electric and magnetic charges and NUT charge. Furthermore, a wide range of horizon topologies is allowed (compact and noncompact) and the complex scalar fields have a nontrivial radial and angular profile. We construct a large class of solutions in the simplest single scalar model with prepotential F = −iX 0 X 1 and discuss their thermodynamics. Moreover, various approaches and calculational tools for facing this problem with more general prepotentials are presented.
We consider extremal black hole attractors (both BPS and non-BPS) for N = 3 and N = 5 supergravity in d = 4 space-time dimensions.Attractors for matter-coupled N = 3 theory are similar to attractors in N = 2 supergravity minimally coupled to Abelian vector multiplets.On the other hand, N = 5 attractors are similar to attractors in N = 4 pure supergravity, and in such theories only 1 N -BPS non-degenerate solutions exist. All the above mentioned theories have a simple interpretation in the first order (fake supergravity) formalism. Furthermore, such theories do not have a d = 5 uplift.Finally we comment on the "duality" relations among the attractor solutions of N ≥ 2 supergravities sharing the same full bosonic sector. 8 Peculiarity of Pure N = 4 and N = 5 Supergravity 35 9 N 2 Supergravities with the same Bosonic Sector and "Dualities" 36 10 Conclusion 38 A Appendix I A Counterexample : N = 4 Matter-Coupled Supergravity 40 • N = 5 supergravity [74], with U -invariant quartic in BH charges.It is worth pointing out that N = 2 supergravity minimally coupled to one Abelian vector multiplet, corresponding to the (U (1)) 6 → (U (1)) 2 gauge truncation of N = 4 pure supergravity, is nothing but the so-called Maxwell-Einstein-axion-dilaton system, studied in [77,78] and recently discussed in [71] and in [63]. As stated above, the formula (1.3) indeed holds true (actually, with suitable changes, also in the non-extremal case).Furthermore, it is interesting to notice that all the above mentioned theories are all the N 2, d = 4 supergravities based on symmetric scalar manifolds which do not admit an uplift 1 to d = 5 space-time dimensions [81].The present paper is organized as follows.In Sect. 2 we briefly intoduce the fundamentals of the first order (fake supergravity) formalism for the non-degenerate attractor flows (both BPS and non-BPS) of extremal BHs in d = 4 space-time dimensions.Sect. 3 is thus devoted to a detailed study of N = 2, d = 4 supergravity minimally coupled to Abelian vector multiplets. In Subsect. 3.1 the related Attractor Equations are explicitly solved, for both the classes of non-degenerate critical points of V BH : the 1 2 -BPS one (Subsubsect. 3.1.1) and the non-BPS Z = 0 one, this latter with related moduli space (Subsubsect. 3.1.2). By exploiting the first order (fake supergravity) formalism, in Subsects. 3.2 and 3.3 the ADM (Arnowitt-Deser-Misner) mass M ADM [82], covariant scalar charges Σ i and (square) effective horizon radius R 2 H are explicitly computed respectively for 1 2 -BPS and non-BPS Z = 0 attractor flows, proving that the second line of Eq. (1.3) holds true. This latter result, already proved in [63], generalizes the findings of [77,78], also holding in the non-extremal case.Sect. 4 deals with N = 3, d = 4 supergravity coupled to matter (Abelian vector) multiplets. In Subsect. 4.1 the related Attractor Equations are explicitly solved, for both the classes of non-degenerate critical points of V BH : the 1 3 -BPS one (Subsubsect. 4.1.1) and the non-BPS Z AB = 0 one (Subsubsect. 4.1.2...
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