# Supersymmetric Mechanics – Vol. 2: The Attractor Mechanism by Stefano Bellucci, Sergio Ferrara, Alessio Marrani

By Stefano Bellucci, Sergio Ferrara, Alessio Marrani

This is the second one quantity in a sequence of books at the common topic of Supersymmetric Mechanics; the sequence is predicated on lectures and discussions held in 2005 and 2006 on the INFN-Laboratori Nazionali di Frascati. the 1st quantity seems as Lect. Notes Physics, Vol. 698 "Supersymmetric Mechanics , Vol .1: Supersymmetry, Noncommutativity and Matrix versions" (2006) ISBN: 3-540-33313-4.

The current large lecture offers a pedagogical creation, on the non-expert point, to the attractor mechanism in space-time singularities. In one of these framework, supersymmetry looks relating to dynamical platforms with fastened issues, describing the equilibrium nation and the steadiness good points of the thermodynamics of black holes. After a qualitative review, specific examples understanding the attractor mechanism are taken care of at a few size; they contain appropriate circumstances of asymptotically flat, maximal and non-maximal, prolonged supergravities in four and five dimensions. a few fresh advances alongside quite a few instructions of analysis at the attractor mechanism also are given.

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73) It is interesting to notice that, despite the symmetry of ∂i NΛΣ and X Λ X Σ in the symplectic indices, the dependence of NΛΣ on the X’s is such as to make the product (∂i NΛΣ ) X Λ X Σ vanish. 74) (∂i NΛΣ ) X Λ X Σ = 0 . 24); the S-dependent factor e−fS (z) corresponds to a (an holomorphic) K¨ ahler transformation. We may naturally divide S(z) in (nV + 1)-d subblocks A(z) B(z) . 77) AT C − C T A = B T D − DT B = 0. 80) X Λ FΛΣΞ = 0 , 7 Attention should be paid to carefully distinguish between: 2 F 1) the quantities F , FΛ , FΛΣ ≡ ∂X∂Λ ∂X Σ ≡ F and 2) the quantities F −Λ , F +Λ , F Λ and ∗ F Λ , which are related to the Abelian vector ﬁeld strengths in the N = 2, d = 4 nV -fold MESGT; they will be introduced in Subsect.

74) (∂i NΛΣ ) X Λ X Σ = 0 . 24); the S-dependent factor e−fS (z) corresponds to a (an holomorphic) K¨ ahler transformation. We may naturally divide S(z) in (nV + 1)-d subblocks A(z) B(z) . 77) AT C − C T A = B T D − DT B = 0. 80) X Λ FΛΣΞ = 0 , 7 Attention should be paid to carefully distinguish between: 2 F 1) the quantities F , FΛ , FΛΣ ≡ ∂X∂Λ ∂X Σ ≡ F and 2) the quantities F −Λ , F +Λ , F Λ and ∗ F Λ , which are related to the Abelian vector ﬁeld strengths in the N = 2, d = 4 nV -fold MESGT; they will be introduced in Subsect.

113) we ﬁnally get ∂i ∂ j ln (det (ImF)) = −Cilp C jlp Gll Gpp . 62) exists and it is invertible. 2 Electric–Magnetic Duality, Central Charge, and Attractor Mechanism In this subsection we will brieﬂy report how, in N = 2, d = 4 SUGRA coupled with nV Abelian vector multiplets (and nH hypermultiplets), the phenomenon of the doubling of preserved supersymmetries (and therefore of the restoration of maximal SUSY) occurs near the EH of the 12 -BPS stable soliton metric solution, whose simplest example is represented by the previously considered extremal RN BH.