d=4attractors, effective horizon radius, and fake supergravity

Abstract

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...

Degeneration of groups of type E 7 and minimal coupling in supergravity

“…As discussed in section 8 of [41], the quartic invariant of the R = 20 of SU(1, 5), after skew-diagonalization in the scalar-dressed R-symmetry U(5)-basis and use of the HuaBloch-Messiah-Zumino Theorem [50][51][52], is a perfect square. On this respect the couples (SU(1, 5), R = 20) and (SL(2, R) × SO(6), R = (2, 6)) (this latter pertaining to N = 4 "pure" supergravity) stand on a particular footing among simple and respectively semisimple groups "of type E 7 " [16].…”

“…In turn, the "pure" N = 5 theory admits two maximal "degenerative" truncations, which precisely match the kinematical decomposition of the N = 5 gravity multiplet into matter N = 2 multiplets. 34) and it admits two possible interpretations, due to the fact that N = 3 supergravity coupled to 1 vector multiplet and N = 2 supergravity minimally coupled to 3 vector multiplets share the very same bosonic sector (namely, they are "twin" theories; see the discussion in section 9 of [41]). In the N = 3 interpretation, one gets a theory with 1 vector multiplets, and the SU(2) commuting factor can be interpreted as the part of the R-symmetry truncated away in the supersymmetry reduction N = 5 → N = 3.…”

“…(3.31). Thence, one can proceed by truncating to the (R ⊕ Γ 1,5 )-based theory still enjoys a "twin" interpretation [41,70], either N = 2 or N = 4 supergravity; in the former case, the second SU(2) commuting factor also be interpreted as the global hyper R-symmetry, whereas in the latter case it is the R-symmetry truncated away in the supersymmetry reduction N = 6 → N = 4. In both cases, the truncation condition is The resulting theory still exhibits a quartic U -invariant I 4 , but it can be further truncated to a theory with U -duality group U(1, 3) with quadratic invariant.…”

“…The J H 3 -based theory can either be interpreted as N = 2 or as its "twin" N = 6 [39,41,69,70]; in the former case, the SU(2) commuting factor is the global hyper R-symmetry, whereas in the latter case it is the R-symmetry truncated away in the supersymmetry reduction N = 8 → N = 6 (a further truncation N = 6 → N = 3 is considered in subsection 4.1). In both cases, the truncation condition on the twoform Abelian is given in eq.…”

More onN=8attractors

“…Notice that in the basis of bare charges I 4;n¼0 , as given by Eq. (5.25), is (weakly) positive due to the Schwarz inequality, and not because it is a nontrivial perfect square of an expression of the bare magnetic and electric charges [37].…”