1. The Attractor Mechanism in Extremal Black Holes Alessio MARRANI Museo Storico della Fisica e Centro Studi e Ricerche “Enrico Fermi”, Rome, Italy & INFN – LNF, Frascati, Italy Prima Conferenza di Progetto del Museo Storico della Fisica e Centro Studi e Ricerche “Enrico Fermi”, Salone delle Conferenze, Ministero dell’Interno, Viminale, Rome, 30 November 2007

2. A. Marrani, SIF 20062 Collaborators: Dr. S. Bellucci  Dr. S. Bellucci (INFN -- LNF, Italy) Dr. A. Ceresole  Dr. A. Ceresole (Turin Univ. & INFN – Turin, Italy) Prof. S. Ferrara  Prof. S. Ferrara (CERN, Switzerland & UCLA, USA & INFN -- LNF, Italy) Prof. M. Gunaydin  Prof. M. Gunaydin (Penn State Univ., PA, USA) Dr. E. Orazi  Dr. E. Orazi (Turin Politecnico & INFN – Turin, Italy) Dr. A. Shcherbakov  Dr. A. Shcherbakov (JINR, Dubna, Russia & INFN – LNF, Italy) Dr. A. Yeranyan  Dr. A. Yeranyan (Yerevan State Univ., Armenia & INFN – LNF, Italy) Papers on Attractors: On some properties of the attractor equations, PLB 635, 172 (2006), hep-th/0602161 On some properties of the attractor equations, PLB 635, 172 (2006), hep-th/0602161 Charge orbits of symmetric special geometries and attractors, Charge orbits of symmetric special geometries and attractors, IJMP A21, 5043 (2006), hep-th/0606209 IJMP A21, 5043 (2006), hep-th/0606209 Mirror Fermat Calabi-Yau Threefolds and Landau-Ginzburg Black Hole Attractors, Mirror Fermat Calabi-Yau Threefolds and Landau-Ginzburg Black Hole Attractors, Riv. Nuovo Cim. 029, 1 (2006), hep-th/0608091 Riv. Nuovo Cim. 029, 1 (2006), hep-th/0608091 Attractor Horizon Geometries of Extremal Black Holes, Attractor Horizon Geometries of Extremal Black Holes, XVII SIGRAV Conf. 2006, hep-th/0702019 XVII SIGRAV Conf. 2006, hep-th/0702019 N=8 non-BPS Attractors, Fixed Scalars and Magic Supergravities, N=8 non-BPS Attractors, Fixed Scalars and Magic Supergravities, NPB 2007, in press, arXiv:0705.3866 NPB 2007, in press, arXiv:0705.3866 On the Moduli Space of non-BPS Attractors for N=2 Symmetric Manifolds, On the Moduli Space of non-BPS Attractors for N=2 Symmetric Manifolds, PLB 652, 111 (2007), arXiv:0706.1667 PLB 652, 111 (2007), arXiv:0706.1667 4d/5d Correspondence for the Black Hole Potential and its Critical Points, 4d/5d Correspondence for the Black Hole Potential and its Critical Points, CQG 2007, in press, arXiv:0707.0964 CQG 2007, in press, arXiv:0707.0964 Attractors with Vanishing Central Charge, PLB 2007 (in press), arXiv:0707.2730 Attractors with Vanishing Central Charge, PLB 2007 (in press), arXiv:0707.2730 Black Hole Attractors in Extended Supergravity, PASCOS07, arXiv:0708.1268 Black Hole Attractors in Extended Supergravity, PASCOS07, arXiv:0708.1268 Supersymmetric mechanics. Vol. 2: The attractor mechanism and space time singularities, Book : Supersymmetric mechanics. Vol. 2: The attractor mechanism and space time singularities, LNP 701, Springer – Verlag, 2006 LNP 701, Springer – Verlag, 2006

3. A. Marrani, SIF 20063 N=2, d=4 Supergravity coupled to n V Abelian vector multiplets: Maxwell-Einstein Supergravity Theory (MESGT) Field content : Supergravity Multiplet  Supergravity Multiplet Vielbein SU(2) doublet of gravitinos Graviphoton U(1) gauge boson Doublet of gauginos Complex scalar fields (MODULI) n V Abelian Vector Multiplets  n V Abelian Vector Multiplets Overall Gauge Symmetry No Hypermultiplets will be considered : they decouple from the system Only scalars from vector multiplets are relevant for the ATTRACTOR MECHANISM

4. A. Marrani, SIF 20064  Bekenstein – Hawking Entropy – Area Formula Macroscopic Approach to Black Hole Thermodynamics (Macroscopic Approach to Black Hole Thermodynamics) What is the Attractor Mechanism?  We consider an Extremal (T=0), dyonic, asymptotically flat, Extremal (T=0), dyonic, asymptotically flat, spherically symmetric, static Black Hole (BH) spherically symmetric, static Black Hole (BH) A priori the BH entropy will depend on the following variables : BH Electric charges BH Magnetic charges Values of the moduli fields at the Event Horizon of the black hole: they will in general depend on the initial data of their deterministic, classical evolution dynamics, i.e. on the asymptotical values Notice : Notice : are UNCONSTRAINED; they can take any possible complex value

5. A. Marrani, SIF 20065 Can the moduli be stabilized at the Event Horizon of the BH? Can the moduli be stabilized at the Event Horizon of the BH? Can they be made INDEPENDENT on the UNCONSTRAINED asymptotical values ? Regardless of the initial conditions, the Horizon values depend ONLY on the charges, but nevertheless the evolution remains DETERMINISTIC! ATTRACTOR MECHANISM : In approaching the Event Horizon, the moduli completely lose memory of the initial data, and take values dependent ONLY on the electric/magnetic charges of the BH: S. Ferrara, R. Kallosh, Phys.Rev. D54 (1996),1514, Phys.Rev. D54 (1996),1525, Phys.Rev. D54 (1996),1525, S. Ferrara, G. Gibbons, R. Kallosh, Nucl.Phys. B500 (1997),75 Conserved charges, from gauge-inv.

6. A. Marrani, SIF 20066 Thus, which is the criterion to determine the purely charge-dependent configs. of the moduli? How can the ATTRACTOR MECHANISM be implemented? of the moduli? How can the ATTRACTOR MECHANISM be implemented? Critical implementation (Ferrara, Gibbons, Kallosh, Nucl. Phys. B500 (1997), 75) : actually are non-degenerate critical points of an “effective black hole potential” Regular contravariant metric of the Special Kaehler moduli space : Covariant derivative of Z: Z is the N=2 Kaehler-covariantly holomorphic “central charge function” Horizon moduli configs. characterized as critical pts. of V BH CLASSICAL BLACK HOLE ENTROPY

7. A. Marrani, SIF 20067 General classification of BH Attractors in N=2, d=4 MESGT: 1.½-BPS Attractors:they preserve the maximum number of SUSYs (4 out of 8), and 1.½-BPS Attractors: they preserve the maximum number of SUSYs (4 out of 8), and they do saturate the Bogomol’ny – Prasad – Sommerfeld (BPS) bound: they do saturate the Bogomol’ny – Prasad – Sommerfeld (BPS) bound: Characterizing conditions: Known since the mid 90’s, starting from the cited seminal paper by Ferrara, Gibbons and Kallosh. 2. non-BPS Attractors with non-vanishing central charge 2. non-BPS Attractors with non-vanishing central charge: they do not preserve any NOT saturate SUSY of the asymptotical Minkowskian bkgd.,and do NOT saturate the BPS bound: Characterizing conditions: Recently discovered (Goldstein et al., hep-th/0507096, Tripathy and Trivedi, hep-th/0511117, and many others…), they correspond to BH backgrounds breaking all SUSYs, but in the framework of a SUSY theory : important phenomenological implications!

8. A. Marrani, SIF 20068 Characterizing conditions: 3. non-BPS Attractors with vanishing central charge 3. non-BPS Attractors with vanishing central charge: they do not preserve any SUSY of the asymptotical Minkowskian background,and do NOT saturate the BPS bound: It can be traced back to the regularity of the SKG of the moduli space Until June 2006, the unique explicit example of such a kind of extremal BH Attractors was given by Giryavets in hep-th/0511215. Such a kind of Attractors turns out to be really interesting, since it gives rise to a BH background breaking all butwithout central extension SUSYs in the framework of N=2, d=4 MESGT, but without central extension of the N=2 SUSY algebra pertaining the asymptotical Minkowskian background.  Homogeneous Symmetric Special Kaeheler Geometries;  Special Kaeheler Moduli Spaces arising from compactifications of d=10 Superstrings on Fermat CY3; Since then, the non-BPS, Z=0 Attractors have been studied by S.Bellucci, S.Ferrara, A.M., E. Orazi and A. Shcherbakov in a number of frameworks:  Peculiar Homogeneous Symmetric Models, the so-called st2 (nV =2) and stu (nV =3) Models.

9. A. Marrani, SIF 20069 Outlook and further developments: Study of BH Attractors in particular classes of Special Kaeheler Geometries (SKGs) of the scalar manifold of Maxwell-Einstein supergravities, such as:  SUSY extension of the moduli space: What is the supersymmetrized analogue of the Attractor Mechanism? Analysis of the (SK?) geometries of the moduli spaces arising from compactifications on (CY) Supermanifolds SKGs with “deformed” periods, arising from dimensional compactifications only LOCALLY on complex 3-folds which are CY3s only LOCALLY (for recent studies on SKG related to LOCAL CY 3 s, see e.g. Bilal and Metzger, hep-th/0503173) Going beyond the Static case : Rotating Rotating and/or Asymptotically non-flat (AdS) and/or Asymptotically non-flat (AdS) and/or with non-vanishing Cosmological Constant and/or with non-vanishing Cosmological Constant Extremal BHs Extremal BHs  for recent advances on compactifications on supermanifolds, see e.g. Grassi and Marescotti, hep-th/0607243 see e.g. Grassi and Marescotti, hep-th/0607243  Need for extension of Symplectic Geometry on Supermanifolds (recently studied, see Lavrov and Radchenko, arXiV : 0708.3778)

10. A. Marrani, SIF 200610 Thank You!